Subgroups — C₁₃₇×#19266 #46868652

Order 35072.

786 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3079×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3038×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1666×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1665×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1566×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1549×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1545×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₁₃₇	—	2	1	no	—
11	C₂²×SD₁₆×C₁₃₇	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
15	#492×C₁₃₇	—	2	1	no	—
16	#492×C₁₃₇	—	2	1	no	—
17	#492×C₁₃₇	C₂²	1	4	yes	—
18	#477×C₁₃₇	—	2	1	no	—
19	#476×C₁₃₇	—	2	1	no	—
20	#475×C₁₃₇	C₂²	1	2	yes	complemented
21	#474×C₁₃₇	C₂²	1	2	yes	complemented
22	#460×C₁₃₇	—	2	1	no	—
23	#460×C₁₃₇	C₂²	1	2	yes	—
24	#459×C₁₃₇	—	2	1	no	—
25	#459×C₁₃₇	C₂²	1	2	yes	—
26	C₈×Q₈×C₁₃₇	C₂²	1	2	yes	complemented
27	C₈×D₄×C₁₃₇	—	2	2	no	—
28	C₈×D₄×C₁₃₇	C₂²	1	2	yes	complemented
29	#431×C₁₃₇	—	2	1	no	—
30	#431×C₁₃₇	C₂²	1	1	yes	characteristic
31	#428×C₁₃₇	—	2	1	no	—
32	#423×C₁₃₇	C₂²	1	1	yes	characteristic
33	#423×C₁₃₇	C₂²	1	1	yes	characteristic
34	#418×C₁₃₇	—	2	1	no	—
35	#417×C₁₃₇	—	2	1	no	—
36	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
37	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
38	#334×C₁₃₇	C₂²	1	4	yes	complemented
39	#332×C₁₃₇	C₂²	1	4	yes	complemented
40	#331×C₁₃₇	C₂²	1	4	yes	—
41	C₂×#41×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₁₃₇	—	2	1	no	—
43	C₂²×D₄×C₁₃₇	—	2	1	no	—
44	C₂²×D₄×C₁₃₇	—	2	1	no	—
45	C₂²×D₄×C₁₃₇	—	2	1	no	—
46	AGL(1,ℤ/8)×C₁₃₇	—	4	4	no	—
47	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
48	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
49	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
50	C₂×SD₁₆×C₁₃₇	—	4	8	no	—
51	C₂×D₈×C₁₃₇	—	2	2	no	—
52	C₂×M₁₆×C₁₃₇	—	2	1	no	—
53	C₂×M₁₆×C₁₃₇	—	2	1	no	—
54	C₂²×C₈×C₁₃₇	—	2	2	no	—
55	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
56	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
57	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
58	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
59	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
60	#127×C₁₃₇	—	2	1	no	—
61	#127×C₁₃₇	—	2	1	no	—
62	#127×C₁₃₇	—	2	1	no	—
63	#127×C₁₃₇	C₂³	1	2	yes	—
64	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₁₃₇	—	2	1	no	—
67	C₄×D₄×C₁₃₇	—	2	1	no	—
68	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
69	C₄×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
70	#117×C₁₃₇	—	2	1	no	—
71	#117×C₁₃₇	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄²×C₁₃₇	—	2	1	no	—
75	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
76	#107×C₁₃₇	—	2	2	no	—
77	#106×C₁₃₇	—	2	2	no	—
78	C₄⋊C₈×C₁₃₇	—	2	2	no	—
79	C₄⋊C₈×C₁₃₇	—	2	1	no	—
80	C₄⋊C₈×C₁₃₇	—	2	1	no	—
81	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
82	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
83	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
84	#103×C₁₃₇	—	2	2	no	—
85	#102×C₁₃₇	—	2	2	no	—
86	#102×C₁₃₇	—	2	2	no	—
87	#102×C₁₃₇	—	2	2	no	—
88	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
89	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
90	C₄×C₈×C₁₃₇	—	2	2	no	—
91	C₄×C₈×C₁₃₇	—	2	1	no	—
92	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
93	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
94	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
95	C₂⁴×C₁₃₇	—	4	1	no	—
96	#41×C₁₃₇	—	2	2	no	—
97	#41×C₁₃₇	SD₁₆	1	4	yes	complemented
98	C₂×Q₈×C₁₃₇	—	2	2	no	—
99	C₂×Q₈×C₁₃₇	—	4	4	no	—
100	C₂×Q₈×C₁₃₇	—	4	1	no	—
101	C₂×Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₁₃₇	—	2	1	no	—
103	C₂×D₄×C₁₃₇	—	2	2	no	—
104	C₂×D₄×C₁₃₇	—	2	2	no	—
105	C₂×D₄×C₁₃₇	—	2	2	no	—
106	C₂×D₄×C₁₃₇	—	4	4	no	—
107	C₂×D₄×C₁₃₇	—	4	1	no	—
108	C₂×D₄×C₁₃₇	—	4	1	no	—
109	C₂×D₄×C₁₃₇	—	4	1	no	—
110	C₂×D₄×C₁₃₇	—	4	2	no	—
111	C₂×D₄×C₁₃₇	—	4	2	no	—
112	C₂×D₄×C₁₃₇	—	8	2	no	—
113	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₁₃₇	—	2	1	no	—
115	C₂²×C₄×C₁₃₇	—	2	1	no	—
116	C₂²×C₄×C₁₃₇	—	2	1	no	—
117	C₂²×C₄×C₁₃₇	—	4	1	no	—
118	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
121	SD₁₆×C₁₃₇	—	4	8	no	—
122	SD₁₆×C₁₃₇	—	4	4	no	—
123	SD₁₆×C₁₃₇	—	4	8	no	—
124	D₈×C₁₃₇	—	4	4	no	—
125	M₁₆×C₁₃₇	—	4	2	no	—
126	M₁₆×C₁₃₇	—	4	2	no	—
127	C₂×C₈×C₁₃₇	—	2	4	no	—
128	C₂×C₈×C₁₃₇	—	2	2	no	—
129	C₂×C₈×C₁₃₇	—	2	2	no	—
130	C₂×C₈×C₁₃₇	—	2	2	no	—
131	C₂×C₈×C₁₃₇	—	4	4	no	—
132	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₁₃₇	—	2	1	no	—
135	C₄⋊C₄×C₁₃₇	—	2	1	no	—
136	C₄⋊C₄×C₁₃₇	—	2	2	no	—
137	C₄⋊C₄×C₁₃₇	—	2	2	no	—
138	C₄⋊C₄×C₁₃₇	—	4	1	no	—
139	C₄⋊C₄×C₁₃₇	—	4	1	no	—
140	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
142	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
143	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
144	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
145	C₄²×C₁₃₇	—	2	1	no	—
146	C₄²×C₁₃₇	—	2	1	no	—
147	C₄²×C₁₃₇	—	2	1	no	—
148	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₁₃₇	#41	1	1	yes	characteristic
151	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
152	C₂³×C₁₃₇	—	2	1	no	—
153	C₂³×C₁₃₇	—	4	2	no	—
154	C₂³×C₁₃₇	—	8	2	no	—
155	C₂³×C₁₃₇	—	8	2	no	—
156	C₂³×C₁₃₇	—	8	2	no	—
157	C₂³×C₁₃₇	#121	1	1	yes	characteristic
158	Q₈×C₁₃₇	—	2	1	no	—
159	Q₈×C₁₃₇	—	4	4	no	—
160	Q₈×C₁₃₇	—	8	2	no	—
161	Q₈×C₁₃₇	C₂×SD₁₆	1	2	yes	complemented
162	D₄×C₁₃₇	—	2	4	no	—
163	D₄×C₁₃₇	—	2	1	no	—
164	D₄×C₁₃₇	—	4	2	no	—
165	D₄×C₁₃₇	—	4	2	no	—
166	D₄×C₁₃₇	—	4	4	no	—
167	D₄×C₁₃₇	—	8	2	no	—
168	D₄×C₁₃₇	—	8	2	no	—
169	D₄×C₁₃₇	—	8	2	no	—
170	D₄×C₁₃₇	—	8	2	no	—
171	D₄×C₁₃₇	C₂×SD₁₆	1	2	yes	complemented
172	C₂×C₄×C₁₃₇	—	2	1	no	—
173	C₂×C₄×C₁₃₇	—	2	2	no	—
174	C₂×C₄×C₁₃₇	—	2	4	no	—
175	C₂×C₄×C₁₃₇	—	2	2	no	—
176	C₂×C₄×C₁₃₇	—	2	1	no	—
177	C₂×C₄×C₁₃₇	—	2	1	no	—
178	C₂×C₄×C₁₃₇	—	4	2	no	—
179	C₂×C₄×C₁₃₇	—	4	2	no	—
180	C₂×C₄×C₁₃₇	—	4	2	no	—
181	C₂×C₄×C₁₃₇	—	8	2	no	—
182	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
183	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
184	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
185	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
186	C₂×C₄×C₁₃₇	#135	1	2	yes	—
187	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
190	C₈×C₁₃₇	—	2	2	no	—
191	C₈×C₁₃₇	—	4	2	no	—
192	C₈×C₁₃₇	—	4	2	no	—
193	C₈×C₁₃₇	—	4	2	no	—
194	C₈×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	complemented
195	C₂²×C₁₃₇	—	2	1	no	—
196	C₂²×C₁₃₇	—	2	1	no	—
197	C₂²×C₁₃₇	—	2	2	no	—
198	C₂²×C₁₃₇	—	4	1	no	—
199	C₂²×C₁₃₇	—	4	1	no	—
200	C₂²×C₁₃₇	—	8	2	no	—
201	C₂²×C₁₃₇	—	8	2	no	—
202	C₂²×C₁₃₇	—	8	2	no	—
203	C₂²×C₁₃₇	—	16	4	no	—
204	C₂²×C₁₃₇	#465	1	2	yes	—
205	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
206	C₄×C₁₃₇	—	2	1	no	—
207	C₄×C₁₃₇	—	2	2	no	—
208	C₄×C₁₃₇	—	2	1	no	—
209	C₄×C₁₃₇	—	4	1	no	—
210	C₄×C₁₃₇	—	4	1	no	—
211	C₄×C₁₃₇	—	8	2	no	—
212	C₄×C₁₃₇	#465	1	2	yes	—
213	C₄×C₁₃₇	#471	1	2	yes	—
214	C₄×C₁₃₇	C₂²×SD₁₆	1	1	yes	characteristic
215	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
216	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
217	C₂×C₁₃₇	—	2	2	no	—
218	C₂×C₁₃₇	—	4	2	no	—
219	C₂×C₁₃₇	—	16	2	no	—
220	C₂×C₁₃₇	#3038	1	1	yes	characteristic
221	C₂×C₁₃₇	C₂×#465	1	1	yes	characteristic
222	C₂×C₁₃₇	C₂×#471	1	1	yes	characteristic
223	#19266	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₁₃₇	#19266	1	1	yes	characteristic, direct product factor, Sylow
225	C₂×#492	C₂×C₁₃₇	1	1	yes	characteristic, complemented
226	#3079	C₂×C₁₃₇	1	1	yes	characteristic, complemented
227	#3038	C₂×C₁₃₇	1	1	yes	characteristic
228	C₈×#41	C₂×C₁₃₇	1	1	yes	characteristic, complemented
229	#1666	C₂×C₁₃₇	1	4	yes	complemented
230	#1665	C₂×C₁₃₇	1	4	yes	complemented
231	#1566	C₂×C₁₃₇	1	1	yes	characteristic, complemented
232	#1549	C₂×C₁₃₇	1	1	yes	characteristic, complemented
233	#1545	C₂×C₁₃₇	1	1	yes	characteristic, complemented
234	C₂×AGL(1,ℤ/8)	—	2	1	no	—
235	C₂²×SD₁₆	—	2	2	no	—
236	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
237	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
238	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
239	#492	—	2	1	no	—
240	#492	—	2	1	no	—
241	#492	C₂²×C₁₃₇	1	4	yes	—
242	#477	—	2	1	no	—
243	#476	—	2	1	no	—
244	#475	C₂²×C₁₃₇	1	2	yes	complemented
245	#474	C₂²×C₁₃₇	1	2	yes	complemented
246	#460	—	2	1	no	—
247	#460	C₂²×C₁₃₇	1	2	yes	—
248	#459	—	2	1	no	—
249	#459	C₂²×C₁₃₇	1	2	yes	—
250	C₈×Q₈	C₂²×C₁₃₇	1	2	yes	complemented
251	C₈×D₄	—	2	2	no	—
252	C₈×D₄	C₂²×C₁₃₇	1	2	yes	complemented
253	#431	—	2	1	no	—
254	#431	C₂²×C₁₃₇	1	1	yes	characteristic
255	#428	—	2	1	no	—
256	#423	C₂²×C₁₃₇	1	1	yes	characteristic
257	#423	C₂²×C₁₃₇	1	1	yes	characteristic
258	#418	—	2	1	no	—
259	#417	—	2	1	no	—
260	C₂×C₄×C₈	—	2	1	no	—
261	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
262	#334	C₂²×C₁₃₇	1	4	yes	complemented
263	#332	C₂²×C₁₃₇	1	4	yes	complemented
264	#331	C₂²×C₁₃₇	1	4	yes	—
265	C₂×#41	D₄×C₁₃₇	1	1	yes	characteristic
266	C₂²×Q₈	—	2	1	no	—
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	AGL(1,ℤ/8)	—	4	4	no	—
271	C₂×SD₁₆	—	2	2	no	—
272	C₂×SD₁₆	—	2	4	no	—
273	C₂×SD₁₆	—	2	4	no	—
274	C₂×SD₁₆	—	4	8	no	—
275	C₂×D₈	—	2	2	no	—
276	C₂×M₁₆	—	2	1	no	—
277	C₂×M₁₆	—	2	1	no	—
278	C₂²×C₈	—	2	2	no	—
279	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
284	#127	—	2	1	no	—
285	#127	—	2	1	no	—
286	#127	—	2	1	no	—
287	#127	C₂³×C₁₃₇	1	2	yes	—
288	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
289	C₄×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
293	C₄×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
294	#117	—	2	1	no	—
295	#117	D₄×C₁₃₇	1	1	yes	characteristic
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄²	—	2	1	no	—
299	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
300	#107	—	2	2	no	—
301	#106	—	2	2	no	—
302	C₄⋊C₈	—	2	2	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
306	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
307	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
308	#103	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	2	2	no	—
312	C₂²⋊C₈	—	2	1	no	—
313	C₂²⋊C₈	—	2	2	no	—
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	—	2	1	no	—
316	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
317	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
318	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
319	C₂⁴	—	4	1	no	—
320	#41	—	2	2	no	—
321	#41	SD₁₆×C₁₃₇	1	4	yes	complemented
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	4	4	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	—	4	1	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	4	2	no	—
336	C₂×D₄	—	8	2	no	—
337	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
344	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
345	SD₁₆	—	4	8	no	—
346	SD₁₆	—	4	4	no	—
347	SD₁₆	—	4	8	no	—
348	D₈	—	4	4	no	—
349	M₁₆	—	4	2	no	—
350	M₁₆	—	4	2	no	—
351	C₂×C₈	—	2	4	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	4	4	no	—
356	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
357	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
365	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
366	C₂²⋊C₄	—	2	1	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	1	no	—
372	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
373	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
374	C₄²	#41×C₁₃₇	1	1	yes	characteristic
375	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
376	C₂³	—	2	1	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	8	2	no	—
379	C₂³	—	8	2	no	—
380	C₂³	—	8	2	no	—
381	C₂³	#121×C₁₃₇	1	1	yes	characteristic
382	Q₈	—	2	1	no	—
383	Q₈	—	4	4	no	—
384	Q₈	—	8	2	no	—
385	Q₈	C₂×SD₁₆×C₁₃₇	1	2	yes	complemented
386	D₄	—	2	4	no	—
387	D₄	—	2	1	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	4	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	2	no	—
393	D₄	—	8	2	no	—
394	D₄	—	8	2	no	—
395	D₄	C₂×SD₁₆×C₁₃₇	1	2	yes	complemented
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	8	2	no	—
406	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
407	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
408	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
409	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
410	C₂×C₄	#135×C₁₃₇	1	2	yes	—
411	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
414	C₈	—	2	2	no	—
415	C₈	—	4	2	no	—
416	C₈	—	4	2	no	—
417	C₈	—	4	2	no	—
418	C₈	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	complemented
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	16	4	no	—
428	C₂²	#465×C₁₃₇	1	2	yes	—
429	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
430	C₄	—	2	1	no	—
431	C₄	—	2	2	no	—
432	C₄	—	2	1	no	—
433	C₄	—	4	1	no	—
434	C₄	—	4	1	no	—
435	C₄	—	8	2	no	—
436	C₄	#465×C₁₃₇	1	2	yes	—
437	C₄	#471×C₁₃₇	1	2	yes	—
438	C₄	C₂²×SD₁₆×C₁₃₇	1	1	yes	characteristic
439	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
440	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
441	C₂	—	2	2	no	—
442	C₂	—	4	2	no	—
443	C₂	—	16	2	no	—
444	C₂	#3038×C₁₃₇	1	1	yes	characteristic
445	C₂	C₂×#465×C₁₃₇	1	1	yes	characteristic
446	C₂	C₂×#471×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	#3038×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
1. C₂×#492×C₁₃₇	C₂	semidirect
2. #3079×C₁₃₇	C₂	semidirect
4. C₈×#41×C₁₃₇	C₂	semidirect
5. #1666×C₁₃₇	C₂	semidirect
6. #1665×C₁₃₇	C₂	semidirect
7. #1566×C₁₃₇	C₂	semidirect
8. #1549×C₁₃₇	C₂	semidirect
9. #1545×C₁₃₇	C₂	semidirect
20. #475×C₁₃₇	C₂²	semidirect
21. #474×C₁₃₇	C₂²	semidirect
26. C₈×Q₈×C₁₃₇	C₂²	semidirect
28. C₈×D₄×C₁₃₇	C₂²	semidirect
38. #334×C₁₃₇	C₂²	semidirect
39. #332×C₁₃₇	C₂²	semidirect
97. #41×C₁₃₇	SD₁₆	semidirect
161. Q₈×C₁₃₇	C₂×SD₁₆	semidirect
171. D₄×C₁₃₇	C₂×SD₁₆	semidirect
194. C₈×C₁₃₇	AGL(1,ℤ/8)	semidirect
223. #19266	C₁₃₇	direct (N × H)
224. C₁₃₇	#19266	direct (N × H)
225. C₂×#492	C₂×C₁₃₇	semidirect
226. #3079	C₂×C₁₃₇	semidirect
228. C₈×#41	C₂×C₁₃₇	semidirect
229. #1666	C₂×C₁₃₇	semidirect
230. #1665	C₂×C₁₃₇	semidirect
231. #1566	C₂×C₁₃₇	semidirect
232. #1549	C₂×C₁₃₇	semidirect
233. #1545	C₂×C₁₃₇	semidirect
244. #475	C₂²×C₁₃₇	semidirect
245. #474	C₂²×C₁₃₇	semidirect
250. C₈×Q₈	C₂²×C₁₃₇	semidirect
252. C₈×D₄	C₂²×C₁₃₇	semidirect
262. #334	C₂²×C₁₃₇	semidirect
263. #332	C₂²×C₁₃₇	semidirect
321. #41	SD₁₆×C₁₃₇	semidirect
385. Q₈	C₂×SD₁₆×C₁₃₇	semidirect
395. D₄	C₂×SD₁₆×C₁₃₇	semidirect
418. C₈	AGL(1,ℤ/8)×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₁₃₇	2
#3079×C₁₃₇	2
#3038×C₁₃₇	2
C₈×#41×C₁₃₇	2
#1666×C₁₃₇	2
#1665×C₁₃₇	2
#1566×C₁₃₇	2
#1549×C₁₃₇	2
#1545×C₁₃₇	2
#19266	137

Subgroups — C137×#19266 #46868652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19266 #46868652