Subgroups — C₁₃₇×#18815 #46868201

Order 35072.

1228 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3498×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3235×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3208×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#3116×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1641×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1639×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1631×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1629×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#1555×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1496×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#1488×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₁₃₇	—	2	1	no	—
13	#550×C₁₃₇	—	2	1	no	—
14	#550×C₁₃₇	C₂²	1	2	yes	—
15	D₄²×C₁₃₇	—	2	2	no	—
16	D₄²×C₁₃₇	—	2	2	no	—
17	D₄²×C₁₃₇	—	2	1	no	—
18	D₄²×C₁₃₇	C₂²	1	2	yes	—
19	#535×C₁₃₇	—	2	2	no	—
20	#535×C₁₃₇	—	2	1	no	—
21	#533×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
22	#532×C₁₃₇	—	2	1	no	—
23	#532×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#127×C₁₃₇	—	2	1	no	—
25	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
27	#501×C₁₃₇	—	2	1	no	—
28	#492×C₁₃₇	C₂²	1	2	yes	complemented
29	#490×C₁₃₇	—	2	1	no	—
30	#486×C₁₃₇	C₂²	1	2	yes	complemented
31	#485×C₁₃₇	—	2	1	no	—
32	#482×C₁₃₇	—	2	2	no	—
33	#477×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
34	#476×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
35	#476×C₁₃₇	C₂²	1	2	yes	complemented
36	#474×C₁₃₇	C₂²	1	2	yes	complemented
37	#462×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
38	#461×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
39	#460×C₁₃₇	—	2	2	no	—
40	#452×C₁₃₇	—	2	1	no	—
41	#450×C₁₃₇	—	2	2	no	—
42	#441×C₁₃₇	—	2	1	no	—
43	#440×C₁₃₇	—	2	1	no	—
44	#432×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
45	#432×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
46	#431×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
47	#331×C₁₃₇	C₂²	1	2	yes	complemented
48	#331×C₁₃₇	C₂²	1	2	yes	complemented
49	Q₈⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
50	D₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
51	D₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
52	D₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
53	H(5,𝔽₂)×C₁₃₇	—	4	4	no	—
54	H(5,𝔽₂)×C₁₃₇	—	4	4	no	—
55	C₂×#41×C₁₃₇	—	2	2	no	—
56	C₂×#41×C₁₃₇	—	2	2	no	—
57	C₂×#41×C₁₃₇	—	2	1	no	—
58	C₂×#41×C₁₃₇	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₁₃₇	—	2	2	no	—
60	C₂²×D₄×C₁₃₇	—	2	4	no	—
61	C₂²×D₄×C₁₃₇	—	2	2	no	—
62	C₂²×D₄×C₁₃₇	—	2	2	no	—
63	C₂²×D₄×C₁₃₇	—	2	2	no	—
64	C₂²×D₄×C₁₃₇	—	2	1	no	—
65	C₂²×D₄×C₁₃₇	—	2	1	no	—
66	C₂²×D₄×C₁₃₇	D₄	1	2	yes	—
67	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
68	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
69	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
70	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
73	#127×C₁₃₇	—	2	2	no	—
74	#127×C₁₃₇	—	2	2	no	—
75	#127×C₁₃₇	—	2	2	no	—
76	#127×C₁₃₇	—	2	1	no	—
77	#127×C₁₃₇	—	2	1	no	—
78	#127×C₁₃₇	D₄	1	2	yes	—
79	#127×C₁₃₇	D₄	1	1	yes	characteristic
80	#127×C₁₃₇	C₂³	1	2	yes	—
81	#125×C₁₃₇	—	2	1	no	—
82	#124×C₁₃₇	—	2	2	no	—
83	#124×C₁₃₇	—	2	1	no	—
84	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
85	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
86	#121×C₁₃₇	—	2	2	no	—
87	#121×C₁₃₇	—	2	4	no	—
88	#121×C₁₃₇	—	2	4	no	—
89	#121×C₁₃₇	—	2	1	no	—
90	#121×C₁₃₇	—	2	2	no	—
91	#121×C₁₃₇	—	2	2	no	—
92	#121×C₁₃₇	—	2	2	no	—
93	#121×C₁₃₇	D₄	1	2	yes	—
94	#120×C₁₃₇	—	2	4	no	—
95	#120×C₁₃₇	—	2	4	no	—
96	#120×C₁₃₇	—	2	4	no	—
97	C₄×Q₈×C₁₃₇	—	2	1	no	—
98	C₄×Q₈×C₁₃₇	—	2	1	no	—
99	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
100	C₄×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₁₃₇	—	2	2	no	—
102	C₄×D₄×C₁₃₇	—	2	1	no	—
103	C₄×D₄×C₁₃₇	—	2	1	no	—
104	C₄×D₄×C₁₃₇	—	2	2	no	—
105	C₄×D₄×C₁₃₇	—	2	1	no	—
106	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
107	C₄×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
108	C₄×D₄×C₁₃₇	C₂³	1	2	yes	—
109	#117×C₁₃₇	—	2	1	no	—
110	#117×C₁₃₇	—	2	1	no	—
111	#117×C₁₃₇	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₁₃₇	—	2	1	no	—
113	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
114	#107×C₁₃₇	—	2	1	no	—
115	#107×C₁₃₇	—	2	2	no	—
116	#106×C₁₃₇	—	2	1	no	—
117	#106×C₁₃₇	—	2	2	no	—
118	C₄⋊C₈×C₁₃₇	—	2	1	no	—
119	C₄⋊C₈×C₁₃₇	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₁₃₇	C₂³	1	1	yes	characteristic
121	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
122	#103×C₁₃₇	—	2	1	no	—
123	#103×C₁₃₇	—	2	1	no	—
124	#103×C₁₃₇	—	2	1	no	—
125	#102×C₁₃₇	—	2	1	no	—
126	#102×C₁₃₇	—	2	4	no	—
127	#102×C₁₃₇	—	2	2	no	—
128	#102×C₁₃₇	—	2	2	no	—
129	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
130	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
131	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
132	#97×C₁₃₇	—	2	1	no	—
133	#97×C₁₃₇	—	2	1	no	—
134	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
135	C₂⁴×C₁₃₇	—	2	4	no	—
136	C₂⁴×C₁₃₇	—	2	2	no	—
137	#41×C₁₃₇	—	2	2	no	—
138	#41×C₁₃₇	—	2	2	no	—
139	#41×C₁₃₇	—	4	8	no	—
140	#41×C₁₃₇	—	4	2	no	—
141	#41×C₁₃₇	—	4	2	no	—
142	#41×C₁₃₇	—	4	4	no	—
143	#41×C₁₃₇	—	4	4	no	—
144	C₂×Q₈×C₁₃₇	—	2	1	no	—
145	C₂×Q₈×C₁₃₇	—	2	1	no	—
146	C₂×Q₈×C₁₃₇	—	2	1	no	—
147	C₂×Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
148	C₂×D₄×C₁₃₇	—	2	1	no	—
149	C₂×D₄×C₁₃₇	—	2	4	no	—
150	C₂×D₄×C₁₃₇	—	2	4	no	—
151	C₂×D₄×C₁₃₇	—	2	2	no	—
152	C₂×D₄×C₁₃₇	—	2	4	no	—
153	C₂×D₄×C₁₃₇	—	2	4	no	—
154	C₂×D₄×C₁₃₇	—	2	2	no	—
155	C₂×D₄×C₁₃₇	—	2	1	no	—
156	C₂×D₄×C₁₃₇	—	2	2	no	—
157	C₂×D₄×C₁₃₇	—	2	1	no	—
158	C₂×D₄×C₁₃₇	—	2	2	no	—
159	C₂×D₄×C₁₃₇	—	2	2	no	—
160	C₂×D₄×C₁₃₇	—	2	2	no	—
161	C₂×D₄×C₁₃₇	—	4	8	no	—
162	C₂×D₄×C₁₃₇	—	4	2	no	—
163	C₂×D₄×C₁₃₇	—	4	2	no	—
164	C₂×D₄×C₁₃₇	—	4	16	no	—
165	C₂×D₄×C₁₃₇	—	4	4	no	—
166	C₂×D₄×C₁₃₇	—	4	4	no	—
167	C₂×D₄×C₁₃₇	—	8	2	no	—
168	C₂×D₄×C₁₃₇	—	8	4	no	—
169	C₂×D₄×C₁₃₇	—	8	2	no	—
170	C₂×D₄×C₁₃₇	—	8	4	no	—
171	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
172	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
173	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₁₃₇	—	2	1	no	—
175	C₂²×C₄×C₁₃₇	—	2	4	no	—
176	C₂²×C₄×C₁₃₇	—	2	1	no	—
177	C₂²×C₄×C₁₃₇	—	2	1	no	—
178	C₂²×C₄×C₁₃₇	—	2	2	no	—
179	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
182	Q₁₆×C₁₃₇	—	4	2	no	—
183	SD₁₆×C₁₃₇	—	4	2	no	—
184	SD₁₆×C₁₃₇	—	8	4	no	—
185	C₂×C₈×C₁₃₇	—	2	2	no	—
186	C₂×C₈×C₁₃₇	—	2	2	no	—
187	C₂×C₈×C₁₃₇	—	2	1	no	—
188	C₂×C₈×C₁₃₇	—	2	1	no	—
189	C₄⋊C₄×C₁₃₇	—	2	1	no	—
190	C₄⋊C₄×C₁₃₇	—	2	1	no	—
191	C₄⋊C₄×C₁₃₇	—	2	1	no	—
192	C₄⋊C₄×C₁₃₇	—	2	1	no	—
193	C₄⋊C₄×C₁₃₇	—	2	1	no	—
194	C₄⋊C₄×C₁₃₇	—	2	1	no	—
195	C₄⋊C₄×C₁₃₇	—	2	1	no	—
196	C₄⋊C₄×C₁₃₇	—	4	1	no	—
197	C₄⋊C₄×C₁₃₇	—	4	1	no	—
198	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
201	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
202	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
203	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
204	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
205	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
206	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
207	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
208	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
209	C₄²×C₁₃₇	—	2	1	no	—
210	C₄²×C₁₃₇	—	2	1	no	—
211	C₄²×C₁₃₇	—	2	1	no	—
212	C₄²×C₁₃₇	—	4	1	no	—
213	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
217	C₂³×C₁₃₇	—	2	1	no	—
218	C₂³×C₁₃₇	—	2	2	no	—
219	C₂³×C₁₃₇	—	2	4	no	—
220	C₂³×C₁₃₇	—	4	4	no	—
221	C₂³×C₁₃₇	—	4	8	no	—
222	C₂³×C₁₃₇	—	8	2	no	—
223	C₂³×C₁₃₇	—	8	2	no	—
224	C₂³×C₁₃₇	—	8	4	no	—
225	C₂³×C₁₃₇	—	8	4	no	—
226	C₂³×C₁₃₇	#120	1	1	yes	characteristic
227	Q₈×C₁₃₇	—	2	1	no	—
228	Q₈×C₁₃₇	—	4	2	no	—
229	Q₈×C₁₃₇	—	4	2	no	—
230	Q₈×C₁₃₇	—	4	1	no	—
231	Q₈×C₁₃₇	—	4	1	no	—
232	Q₈×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
233	D₄×C₁₃₇	—	2	4	no	—
234	D₄×C₁₃₇	—	2	1	no	—
235	D₄×C₁₃₇	—	2	4	no	—
236	D₄×C₁₃₇	—	4	4	no	—
237	D₄×C₁₃₇	—	4	4	no	—
238	D₄×C₁₃₇	—	4	2	no	—
239	D₄×C₁₃₇	—	4	8	no	—
240	D₄×C₁₃₇	—	4	4	no	—
241	D₄×C₁₃₇	—	4	2	no	—
242	D₄×C₁₃₇	—	4	4	no	—
243	D₄×C₁₃₇	—	4	8	no	—
244	D₄×C₁₃₇	—	4	2	no	—
245	D₄×C₁₃₇	—	4	2	no	—
246	D₄×C₁₃₇	—	4	4	no	—
247	D₄×C₁₃₇	—	8	2	no	—
248	D₄×C₁₃₇	—	8	4	no	—
249	D₄×C₁₃₇	—	8	4	no	—
250	D₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
251	C₂×C₄×C₁₃₇	—	2	4	no	—
252	C₂×C₄×C₁₃₇	—	2	1	no	—
253	C₂×C₄×C₁₃₇	—	2	1	no	—
254	C₂×C₄×C₁₃₇	—	2	1	no	—
255	C₂×C₄×C₁₃₇	—	2	2	no	—
256	C₂×C₄×C₁₃₇	—	2	1	no	—
257	C₂×C₄×C₁₃₇	—	2	1	no	—
258	C₂×C₄×C₁₃₇	—	2	2	no	—
259	C₂×C₄×C₁₃₇	—	4	1	no	—
260	C₂×C₄×C₁₃₇	—	4	8	no	—
261	C₂×C₄×C₁₃₇	—	4	2	no	—
262	C₂×C₄×C₁₃₇	—	4	2	no	—
263	C₂×C₄×C₁₃₇	—	4	2	no	—
264	C₂×C₄×C₁₃₇	—	4	2	no	—
265	C₂×C₄×C₁₃₇	—	4	1	no	—
266	C₂×C₄×C₁₃₇	—	4	1	no	—
267	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
268	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
269	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
270	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
273	C₈×C₁₃₇	—	4	2	no	—
274	C₈×C₁₃₇	—	4	1	no	—
275	C₈×C₁₃₇	—	4	1	no	—
276	C₈×C₁₃₇	—	4	2	no	—
277	C₂²×C₁₃₇	—	2	1	no	—
278	C₂²×C₁₃₇	—	2	2	no	—
279	C₂²×C₁₃₇	—	2	1	no	—
280	C₂²×C₁₃₇	—	2	1	no	—
281	C₂²×C₁₃₇	—	4	1	no	—
282	C₂²×C₁₃₇	—	4	1	no	—
283	C₂²×C₁₃₇	—	4	4	no	—
284	C₂²×C₁₃₇	—	4	2	no	—
285	C₂²×C₁₃₇	—	4	2	no	—
286	C₂²×C₁₃₇	—	4	4	no	—
287	C₂²×C₁₃₇	—	4	4	no	—
288	C₂²×C₁₃₇	—	4	2	no	—
289	C₂²×C₁₃₇	—	16	4	no	—
290	C₂²×C₁₃₇	—	16	8	no	—
291	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
292	C₄×C₁₃₇	—	2	1	no	—
293	C₄×C₁₃₇	—	2	2	no	—
294	C₄×C₁₃₇	—	2	1	no	—
295	C₄×C₁₃₇	—	2	1	no	—
296	C₄×C₁₃₇	—	2	1	no	—
297	C₄×C₁₃₇	—	4	2	no	—
298	C₄×C₁₃₇	—	4	1	no	—
299	C₄×C₁₃₇	—	4	1	no	—
300	C₄×C₁₃₇	—	8	1	no	—
301	C₄×C₁₃₇	—	8	1	no	—
302	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₄×C₁₃₇	#577	1	2	yes	—
305	C₂×C₁₃₇	—	4	1	no	—
306	C₂×C₁₃₇	—	4	2	no	—
307	C₂×C₁₃₇	—	8	2	no	—
308	C₂×C₁₃₇	—	8	4	no	—
309	C₂×C₁₃₇	#3003	1	1	yes	characteristic
310	C₂×C₁₃₇	#3009	1	1	yes	characteristic
311	C₂×C₁₃₇	#3019	1	1	yes	characteristic
312	#18815	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
313	C₁₃₇	#18815	1	1	yes	characteristic, direct product factor, Sylow
314	#3498	C₂×C₁₃₇	1	1	yes	characteristic
315	#3235	C₂×C₁₃₇	1	1	yes	characteristic, complemented
316	#3208	C₂×C₁₃₇	1	1	yes	characteristic, complemented
317	#3116	C₂×C₁₃₇	1	1	yes	characteristic, complemented
318	#1641	C₂×C₁₃₇	1	2	yes	complemented
319	#1639	C₂×C₁₃₇	1	2	yes	complemented
320	#1631	C₂×C₁₃₇	1	2	yes	complemented
321	#1629	C₂×C₁₃₇	1	2	yes	complemented
322	#1555	C₂×C₁₃₇	1	1	yes	characteristic, complemented
323	#1496	C₂×C₁₃₇	1	1	yes	characteristic, complemented
324	#1488	C₂×C₁₃₇	1	1	yes	characteristic, complemented
325	C₂×H(5,𝔽₂)	—	2	1	no	—
326	#550	—	2	1	no	—
327	#550	C₂²×C₁₃₇	1	2	yes	—
328	D₄²	—	2	2	no	—
329	D₄²	—	2	2	no	—
330	D₄²	—	2	1	no	—
331	D₄²	C₂²×C₁₃₇	1	2	yes	—
332	#535	—	2	2	no	—
333	#535	—	2	1	no	—
334	#533	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
335	#532	—	2	1	no	—
336	#532	C₂²×C₁₃₇	1	1	yes	characteristic
337	C₂×#127	—	2	1	no	—
338	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
339	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
340	#501	—	2	1	no	—
341	#492	C₂²×C₁₃₇	1	2	yes	complemented
342	#490	—	2	1	no	—
343	#486	C₂²×C₁₃₇	1	2	yes	complemented
344	#485	—	2	1	no	—
345	#482	—	2	2	no	—
346	#477	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
347	#476	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
348	#476	C₂²×C₁₃₇	1	2	yes	complemented
349	#474	C₂²×C₁₃₇	1	2	yes	complemented
350	#462	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
351	#461	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
352	#460	—	2	2	no	—
353	#452	—	2	1	no	—
354	#450	—	2	2	no	—
355	#441	—	2	1	no	—
356	#440	—	2	1	no	—
357	#432	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
358	#432	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
359	#431	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
360	#331	C₂²×C₁₃₇	1	2	yes	complemented
361	#331	C₂²×C₁₃₇	1	2	yes	complemented
362	Q₈⋊C₈	C₂²×C₁₃₇	1	2	yes	complemented
363	D₄⋊C₈	C₂²×C₁₃₇	1	2	yes	complemented
364	D₄⋊C₈	C₂²×C₁₃₇	1	2	yes	complemented
365	D₄⋊C₈	C₂²×C₁₃₇	1	2	yes	complemented
366	H(5,𝔽₂)	—	4	4	no	—
367	H(5,𝔽₂)	—	4	4	no	—
368	C₂×#41	—	2	2	no	—
369	C₂×#41	—	2	2	no	—
370	C₂×#41	—	2	1	no	—
371	C₂×#41	D₄×C₁₃₇	1	1	yes	characteristic
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	—	2	4	no	—
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	—	2	1	no	—
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	D₄×C₁₃₇	1	2	yes	—
380	C₂×Q₁₆	—	2	1	no	—
381	C₂×SD₁₆	—	2	1	no	—
382	C₂×SD₁₆	—	2	4	no	—
383	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
384	C₄⋊Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
385	C₄⋊Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
386	#127	—	2	2	no	—
387	#127	—	2	2	no	—
388	#127	—	2	2	no	—
389	#127	—	2	1	no	—
390	#127	—	2	1	no	—
391	#127	D₄×C₁₃₇	1	1	yes	characteristic
392	#127	D₄×C₁₃₇	1	2	yes	—
393	#127	C₂³×C₁₃₇	1	2	yes	—
394	#125	—	2	1	no	—
395	#124	—	2	2	no	—
396	#124	—	2	1	no	—
397	C₂²⋊Q₈	—	2	1	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	#121	—	2	2	no	—
400	#121	—	2	4	no	—
401	#121	—	2	4	no	—
402	#121	—	2	1	no	—
403	#121	—	2	2	no	—
404	#121	—	2	2	no	—
405	#121	—	2	2	no	—
406	#121	D₄×C₁₃₇	1	2	yes	—
407	#120	—	2	4	no	—
408	#120	—	2	4	no	—
409	#120	—	2	4	no	—
410	C₄×Q₈	—	2	1	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
413	C₄×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
420	C₄×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
421	C₄×D₄	C₂³×C₁₃₇	1	2	yes	—
422	#117	—	2	1	no	—
423	#117	—	2	1	no	—
424	#117	D₄×C₁₃₇	1	1	yes	characteristic
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
427	#107	—	2	1	no	—
428	#107	—	2	2	no	—
429	#106	—	2	1	no	—
430	#106	—	2	2	no	—
431	C₄⋊C₈	—	2	1	no	—
432	C₄⋊C₈	C₂³×C₁₃₇	1	1	yes	characteristic
433	C₄⋊C₈	C₂³×C₁₃₇	1	1	yes	characteristic
434	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
435	#103	—	2	1	no	—
436	#103	—	2	1	no	—
437	#103	—	2	1	no	—
438	#102	—	2	1	no	—
439	#102	—	2	4	no	—
440	#102	—	2	2	no	—
441	#102	—	2	2	no	—
442	C₂²⋊C₈	—	2	1	no	—
443	C₂²⋊C₈	—	2	1	no	—
444	C₂²⋊C₈	—	2	1	no	—
445	#97	—	2	1	no	—
446	#97	—	2	1	no	—
447	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
448	C₂⁴	—	2	4	no	—
449	C₂⁴	—	2	2	no	—
450	#41	—	2	2	no	—
451	#41	—	2	2	no	—
452	#41	—	4	8	no	—
453	#41	—	4	2	no	—
454	#41	—	4	2	no	—
455	#41	—	4	4	no	—
456	#41	—	4	4	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	—	4	2	no	—
476	C₂×D₄	—	4	2	no	—
477	C₂×D₄	—	4	16	no	—
478	C₂×D₄	—	4	4	no	—
479	C₂×D₄	—	4	4	no	—
480	C₂×D₄	—	8	2	no	—
481	C₂×D₄	—	8	4	no	—
482	C₂×D₄	—	8	2	no	—
483	C₂×D₄	—	8	4	no	—
484	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
485	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
486	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
495	Q₁₆	—	4	2	no	—
496	SD₁₆	—	4	2	no	—
497	SD₁₆	—	8	4	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	4	1	no	—
510	C₄⋊C₄	—	4	1	no	—
511	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
512	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
513	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	1	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	1	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	4	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	4	1	no	—
526	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
527	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
528	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
529	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
530	C₂³	—	2	1	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	4	8	no	—
535	C₂³	—	8	2	no	—
536	C₂³	—	8	2	no	—
537	C₂³	—	8	4	no	—
538	C₂³	—	8	4	no	—
539	C₂³	#120×C₁₃₇	1	1	yes	characteristic
540	Q₈	—	2	1	no	—
541	Q₈	—	4	2	no	—
542	Q₈	—	4	2	no	—
543	Q₈	—	4	1	no	—
544	Q₈	—	4	1	no	—
545	Q₈	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
546	D₄	—	2	4	no	—
547	D₄	—	2	1	no	—
548	D₄	—	2	4	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	8	no	—
553	D₄	—	4	4	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	4	no	—
556	D₄	—	4	8	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	2	no	—
559	D₄	—	4	4	no	—
560	D₄	—	8	2	no	—
561	D₄	—	8	4	no	—
562	D₄	—	8	4	no	—
563	D₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	2	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
581	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
582	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
586	C₈	—	4	2	no	—
587	C₈	—	4	1	no	—
588	C₈	—	4	1	no	—
589	C₈	—	4	2	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	16	4	no	—
603	C₂²	—	16	8	no	—
604	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
605	C₄	—	2	1	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	—	8	1	no	—
614	C₄	—	8	1	no	—
615	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
616	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
617	C₄	#577×C₁₃₇	1	2	yes	—
618	C₂	—	4	1	no	—
619	C₂	—	4	2	no	—
620	C₂	—	8	2	no	—
621	C₂	—	8	4	no	—
622	C₂	#3003×C₁₃₇	1	1	yes	characteristic
623	C₂	#3009×C₁₃₇	1	1	yes	characteristic
624	C₂	#3019×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	#3498×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #3235×C₁₃₇	C₂	semidirect
3. #3208×C₁₃₇	C₂	semidirect
4. #3116×C₁₃₇	C₂	semidirect
5. #1641×C₁₃₇	C₂	semidirect
6. #1639×C₁₃₇	C₂	semidirect
7. #1631×C₁₃₇	C₂	semidirect
8. #1629×C₁₃₇	C₂	semidirect
9. #1555×C₁₃₇	C₂	semidirect
10. #1496×C₁₃₇	C₂	semidirect
11. #1488×C₁₃₇	C₂	semidirect
21. #533×C₁₃₇	C₂²	semidirect
28. #492×C₁₃₇	C₂²	semidirect
30. #486×C₁₃₇	C₂²	semidirect
33. #477×C₁₃₇	C₂²	semidirect
34. #476×C₁₃₇	C₂²	semidirect
35. #476×C₁₃₇	C₂²	semidirect
36. #474×C₁₃₇	C₂²	semidirect
37. #462×C₁₃₇	C₂²	semidirect
38. #461×C₁₃₇	C₂²	semidirect
44. #432×C₁₃₇	C₂²	semidirect
45. #432×C₁₃₇	C₂²	semidirect
46. #431×C₁₃₇	C₂²	semidirect
47. #331×C₁₃₇	C₂²	semidirect
48. #331×C₁₃₇	C₂²	semidirect
49. Q₈⋊C₈×C₁₃₇	C₂²	semidirect
50. D₄⋊C₈×C₁₃₇	C₂²	semidirect
51. D₄⋊C₈×C₁₃₇	C₂²	semidirect
52. D₄⋊C₈×C₁₃₇	C₂²	semidirect
312. #18815	C₁₃₇	direct (N × H)
313. C₁₃₇	#18815	direct (N × H)
315. #3235	C₂×C₁₃₇	semidirect
316. #3208	C₂×C₁₃₇	semidirect
317. #3116	C₂×C₁₃₇	semidirect
318. #1641	C₂×C₁₃₇	semidirect
319. #1639	C₂×C₁₃₇	semidirect
320. #1631	C₂×C₁₃₇	semidirect
321. #1629	C₂×C₁₃₇	semidirect
322. #1555	C₂×C₁₃₇	semidirect
323. #1496	C₂×C₁₃₇	semidirect
324. #1488	C₂×C₁₃₇	semidirect
334. #533	C₂²×C₁₃₇	semidirect
341. #492	C₂²×C₁₃₇	semidirect
343. #486	C₂²×C₁₃₇	semidirect
346. #477	C₂²×C₁₃₇	semidirect
347. #476	C₂²×C₁₃₇	semidirect
348. #476	C₂²×C₁₃₇	semidirect
349. #474	C₂²×C₁₃₇	semidirect
350. #462	C₂²×C₁₃₇	semidirect
351. #461	C₂²×C₁₃₇	semidirect
357. #432	C₂²×C₁₃₇	semidirect
358. #432	C₂²×C₁₃₇	semidirect
359. #431	C₂²×C₁₃₇	semidirect
360. #331	C₂²×C₁₃₇	semidirect
361. #331	C₂²×C₁₃₇	semidirect
362. Q₈⋊C₈	C₂²×C₁₃₇	semidirect
363. D₄⋊C₈	C₂²×C₁₃₇	semidirect
364. D₄⋊C₈	C₂²×C₁₃₇	semidirect
365. D₄⋊C₈	C₂²×C₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#3498×C₁₃₇	2
#3235×C₁₃₇	2
#3208×C₁₃₇	2
#3116×C₁₃₇	2
#1641×C₁₃₇	2
#1639×C₁₃₇	2
#1631×C₁₃₇	2
#1629×C₁₃₇	2
#1555×C₁₃₇	2
#1496×C₁₃₇	2
#1488×C₁₃₇	2
#18815	137

Subgroups — C137×#18815 #46868201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#18815 #46868201