Subgroups — C₁₃₇×#26318 #46875704

Order 35072.

802 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2548×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2094×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2071×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1984×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1982×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1957×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1948×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1926×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	#1926×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₁₃₇	—	2	1	no	—
14	C₂×C₄×Q₈×C₁₃₇	—	2	1	no	—
15	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
21	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
22	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	#394×C₁₃₇	—	2	2	no	—
26	#390×C₁₃₇	—	2	1	no	—
27	#390×C₁₃₇	C₂²	1	2	yes	—
28	#390×C₁₃₇	C₂²	1	2	yes	—
29	#389×C₁₃₇	—	2	1	no	—
30	#389×C₁₃₇	—	2	2	no	—
31	#384×C₁₃₇	C₂²	1	2	yes	—
32	#384×C₁₃₇	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
36	#340×C₁₃₇	C₂²	1	1	yes	characteristic
37	#340×C₁₃₇	C₂²	1	1	yes	characteristic
38	#340×C₁₃₇	C₂²	1	2	yes	—
39	#336×C₁₃₇	C₂²	1	2	yes	—
40	#336×C₁₃₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₁₃₇	—	2	1	no	—
43	C₂²×D₄×C₁₃₇	—	2	1	no	—
44	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	#127×C₁₃₇	—	2	2	no	—
49	#127×C₁₃₇	—	2	2	no	—
50	#124×C₁₃₇	—	4	4	no	—
51	C₄×Q₈×C₁₃₇	—	2	2	no	—
52	C₄×Q₈×C₁₃₇	—	2	2	no	—
53	C₄×Q₈×C₁₃₇	—	4	4	no	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄²×C₁₃₇	—	2	2	no	—
70	C₂×C₄²×C₁₃₇	—	2	1	no	—
71	C₂×C₄²×C₁₃₇	—	2	1	no	—
72	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
74	#106×C₁₃₇	—	2	2	no	—
75	#106×C₁₃₇	—	2	2	no	—
76	#106×C₁₃₇	—	4	2	no	—
77	C₄⋊C₈×C₁₃₇	—	2	2	no	—
78	#102×C₁₃₇	—	2	4	no	—
79	#102×C₁₃₇	—	2	4	no	—
80	#102×C₁₃₇	—	4	2	no	—
81	#102×C₁₃₇	—	4	2	no	—
82	C₄×C₈×C₁₃₇	—	2	2	no	—
83	#95×C₁₃₇	—	2	2	no	—
84	#95×C₁₃₇	—	2	1	no	—
85	#95×C₁₃₇	—	2	1	no	—
86	C₂⁴×C₁₃₇	—	2	2	no	—
87	C₂×Q₈×C₁₃₇	—	2	2	no	—
88	C₂×Q₈×C₁₃₇	—	2	2	no	—
89	C₂×Q₈×C₁₃₇	SD₁₆	1	4	yes	—
90	C₂×D₄×C₁₃₇	—	2	2	no	—
91	C₂×D₄×C₁₃₇	—	2	2	no	—
92	C₂×D₄×C₁₃₇	—	2	2	no	—
93	C₂×D₄×C₁₃₇	—	2	2	no	—
94	C₂×D₄×C₁₃₇	—	2	2	no	—
95	C₂×D₄×C₁₃₇	—	2	2	no	—
96	C₂×D₄×C₁₃₇	—	4	2	no	—
97	C₂×D₄×C₁₃₇	—	4	2	no	—
98	C₂×D₄×C₁₃₇	—	4	2	no	—
99	C₂×D₄×C₁₃₇	—	4	2	no	—
100	C₂×D₄×C₁₃₇	—	8	1	no	—
101	C₂×D₄×C₁₃₇	—	8	1	no	—
102	C₂²×C₄×C₁₃₇	—	2	1	no	—
103	C₂²×C₄×C₁₃₇	—	2	1	no	—
104	C₂²×C₄×C₁₃₇	—	2	1	no	—
105	C₂²×C₄×C₁₃₇	—	2	2	no	—
106	C₂²×C₄×C₁₃₇	—	2	1	no	—
107	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₁₃₇	—	2	1	no	—
113	C₂×C₈×C₁₃₇	—	2	1	no	—
114	C₂×C₈×C₁₃₇	—	2	1	no	—
115	C₂×C₈×C₁₃₇	—	2	2	no	—
116	C₂×C₈×C₁₃₇	—	2	2	no	—
117	C₂×C₈×C₁₃₇	—	2	1	no	—
118	C₂×C₈×C₁₃₇	—	2	2	no	—
119	C₂×C₈×C₁₃₇	—	2	1	no	—
120	C₂×C₈×C₁₃₇	#41	1	2	yes	—
121	C₄⋊C₄×C₁₃₇	—	2	4	no	—
122	C₄⋊C₄×C₁₃₇	—	2	2	no	—
123	C₄⋊C₄×C₁₃₇	—	2	1	no	—
124	C₄⋊C₄×C₁₃₇	—	2	2	no	—
125	C₄⋊C₄×C₁₃₇	—	2	1	no	—
126	C₄⋊C₄×C₁₃₇	—	2	1	no	—
127	C₄⋊C₄×C₁₃₇	—	2	1	no	—
128	C₄⋊C₄×C₁₃₇	—	2	1	no	—
129	C₄⋊C₄×C₁₃₇	—	2	2	no	—
130	C₄⋊C₄×C₁₃₇	—	2	1	no	—
131	C₄⋊C₄×C₁₃₇	—	2	1	no	—
132	C₄⋊C₄×C₁₃₇	—	4	4	no	—
133	C₄⋊C₄×C₁₃₇	—	4	2	no	—
134	C₄⋊C₄×C₁₃₇	—	8	1	no	—
135	C₄⋊C₄×C₁₃₇	—	8	2	no	—
136	C₄⋊C₄×C₁₃₇	#41	1	2	yes	—
137	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
138	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
139	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
140	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
141	C₄²×C₁₃₇	—	2	2	no	—
142	C₄²×C₁₃₇	—	2	2	no	—
143	C₄²×C₁₃₇	—	4	4	no	—
144	C₄²×C₁₃₇	—	4	2	no	—
145	C₄²×C₁₃₇	—	4	2	no	—
146	C₄²×C₁₃₇	#41	1	2	yes	—
147	C₄²×C₁₃₇	#41	1	2	yes	—
148	C₂³×C₁₃₇	—	4	2	no	—
149	C₂³×C₁₃₇	—	4	2	no	—
150	C₂³×C₁₃₇	—	4	2	no	—
151	C₂³×C₁₃₇	—	4	2	no	—
152	C₂³×C₁₃₇	—	4	2	no	—
153	C₂³×C₁₃₇	—	4	2	no	—
154	C₂³×C₁₃₇	—	4	2	no	—
155	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₁₃₇	—	2	8	no	—
157	D₄×C₁₃₇	—	4	2	no	—
158	D₄×C₁₃₇	—	4	2	no	—
159	D₄×C₁₃₇	—	4	2	no	—
160	D₄×C₁₃₇	—	4	2	no	—
161	D₄×C₁₃₇	—	8	4	no	—
162	D₄×C₁₃₇	—	8	4	no	—
163	C₂×C₄×C₁₃₇	—	2	2	no	—
164	C₂×C₄×C₁₃₇	—	2	2	no	—
165	C₂×C₄×C₁₃₇	—	2	1	no	—
166	C₂×C₄×C₁₃₇	—	2	1	no	—
167	C₂×C₄×C₁₃₇	—	2	2	no	—
168	C₂×C₄×C₁₃₇	—	2	2	no	—
169	C₂×C₄×C₁₃₇	—	2	2	no	—
170	C₂×C₄×C₁₃₇	—	2	2	no	—
171	C₂×C₄×C₁₃₇	—	4	1	no	—
172	C₂×C₄×C₁₃₇	—	4	1	no	—
173	C₂×C₄×C₁₃₇	—	4	1	no	—
174	C₂×C₄×C₁₃₇	—	4	1	no	—
175	C₂×C₄×C₁₃₇	—	4	2	no	—
176	C₂×C₄×C₁₃₇	—	4	1	no	—
177	C₂×C₄×C₁₃₇	—	4	1	no	—
178	C₂×C₄×C₁₃₇	—	4	1	no	—
179	C₂×C₄×C₁₃₇	—	4	2	no	—
180	C₂×C₄×C₁₃₇	—	4	1	no	—
181	C₂×C₄×C₁₃₇	—	4	1	no	—
182	C₂×C₄×C₁₃₇	—	4	2	no	—
183	C₂×C₄×C₁₃₇	—	4	1	no	—
184	C₂×C₄×C₁₃₇	—	4	1	no	—
185	C₂×C₄×C₁₃₇	#123	1	2	yes	—
186	C₂×C₄×C₁₃₇	#123	1	2	yes	—
187	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	4	yes	—
188	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
189	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₁₃₇	—	2	2	no	—
197	C₈×C₁₃₇	—	4	1	no	—
198	C₈×C₁₃₇	—	4	2	no	—
199	C₂²×C₁₃₇	—	8	2	no	—
200	C₂²×C₁₃₇	—	8	2	no	—
201	C₂²×C₁₃₇	—	8	2	no	—
202	C₂²×C₁₃₇	—	8	2	no	—
203	C₂²×C₁₃₇	—	8	2	no	—
204	C₂²×C₁₃₇	—	8	2	no	—
205	C₂²×C₁₃₇	—	8	2	no	—
206	C₂²×C₁₃₇	C₂×#123	1	1	yes	characteristic
207	C₂²×C₁₃₇	#548	1	1	yes	characteristic
208	C₂²×C₁₃₇	#550	1	1	yes	characteristic
209	C₂²×C₁₃₇	C₂²×SD₁₆	1	1	yes	characteristic
210	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₂²×C₁₃₇	#576	1	1	yes	characteristic
212	C₂²×C₁₃₇	#577	1	1	yes	characteristic
213	C₄×C₁₃₇	—	2	4	no	—
214	C₄×C₁₃₇	—	2	2	no	—
215	C₄×C₁₃₇	—	4	2	no	—
216	C₄×C₁₃₇	—	4	2	no	—
217	C₄×C₁₃₇	—	8	1	no	—
218	C₄×C₁₃₇	—	8	1	no	—
219	C₄×C₁₃₇	—	8	2	no	—
220	C₄×C₁₃₇	—	8	1	no	—
221	C₄×C₁₃₇	#481	1	4	yes	—
222	C₄×C₁₃₇	#529	1	1	yes	characteristic
223	C₄×C₁₃₇	#542	1	1	yes	characteristic
224	C₄×C₁₃₇	#556	1	1	yes	characteristic
225	C₄×C₁₃₇	#559	1	1	yes	characteristic
226	C₂×C₁₃₇	—	16	2	no	—
227	C₂×C₁₃₇	#2448	1	1	yes	characteristic
228	C₂×C₁₃₇	C₂×#481	1	1	yes	characteristic
229	C₂×C₁₃₇	#3096	1	1	yes	characteristic
230	C₂×C₁₃₇	#3362	1	1	yes	characteristic
231	C₂×C₁₃₇	#3373	1	1	yes	characteristic
232	C₂×C₁₃₇	#3392	1	1	yes	characteristic
233	C₂×C₁₃₇	#3399	1	1	yes	characteristic
234	#26318	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₁₃₇	#26318	1	1	yes	characteristic, direct product factor, Sylow
236	#2548	C₂×C₁₃₇	1	1	yes	characteristic
237	Q₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
238	#2094	C₂×C₁₃₇	1	2	yes	complemented
239	#2071	C₂×C₁₃₇	1	2	yes	complemented
240	#1984	C₂×C₁₃₇	1	1	yes	characteristic, complemented
241	#1982	C₂×C₁₃₇	1	1	yes	characteristic
242	#1957	C₂×C₁₃₇	1	1	yes	characteristic
243	#1948	C₂×C₁₃₇	1	1	yes	characteristic, complemented
244	#1926	C₂×C₁₃₇	1	2	yes	complemented
245	#1926	C₂×C₁₃₇	1	2	yes	complemented
246	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
247	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
248	C₂×#124	—	2	1	no	—
249	C₂×C₄×Q₈	—	2	1	no	—
250	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
251	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
252	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
253	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
254	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
255	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
256	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
257	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
258	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
259	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
260	#394	—	2	2	no	—
261	#390	—	2	1	no	—
262	#390	C₂²×C₁₃₇	1	2	yes	—
263	#390	C₂²×C₁₃₇	1	2	yes	—
264	#389	—	2	1	no	—
265	#389	—	2	2	no	—
266	#384	C₂²×C₁₃₇	1	2	yes	—
267	#384	C₂²×C₁₃₇	1	1	yes	characteristic
268	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
269	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
270	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
271	#340	C₂²×C₁₃₇	1	1	yes	characteristic
272	#340	C₂²×C₁₃₇	1	2	yes	—
273	#340	C₂²×C₁₃₇	1	1	yes	characteristic
274	#336	C₂²×C₁₃₇	1	2	yes	—
275	#336	C₂²×C₁₃₇	1	2	yes	—
276	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
280	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
282	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
283	#127	—	2	2	no	—
284	#127	—	2	2	no	—
285	#124	—	4	4	no	—
286	C₄×Q₈	—	2	2	no	—
287	C₄×Q₈	—	2	2	no	—
288	C₄×Q₈	—	4	4	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
294	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
308	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
309	#106	—	2	2	no	—
310	#106	—	2	2	no	—
311	#106	—	4	2	no	—
312	C₄⋊C₈	—	2	2	no	—
313	#102	—	2	4	no	—
314	#102	—	2	4	no	—
315	#102	—	4	2	no	—
316	#102	—	4	2	no	—
317	C₄×C₈	—	2	2	no	—
318	#95	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	—	2	1	no	—
321	C₂⁴	—	2	2	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	SD₁₆×C₁₃₇	1	4	yes	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	4	2	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	8	1	no	—
336	C₂×D₄	—	8	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	#41×C₁₃₇	1	2	yes	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	8	1	no	—
370	C₄⋊C₄	—	8	2	no	—
371	C₄⋊C₄	#41×C₁₃₇	1	2	yes	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₂²⋊C₄	—	4	4	no	—
374	C₂²⋊C₄	—	8	2	no	—
375	C₂²⋊C₄	—	8	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	4	4	no	—
379	C₄²	—	4	2	no	—
380	C₄²	—	4	2	no	—
381	C₄²	#41×C₁₃₇	1	2	yes	—
382	C₄²	#41×C₁₃₇	1	2	yes	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
391	Q₈	—	2	8	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	8	4	no	—
397	D₄	—	8	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	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	#123×C₁₃₇	1	2	yes	—
421	C₂×C₄	#123×C₁₃₇	1	2	yes	—
422	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	4	yes	—
423	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
424	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
425	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
429	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
430	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	4	1	no	—
433	C₈	—	4	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	C₂×#123×C₁₃₇	1	1	yes	characteristic
442	C₂²	#548×C₁₃₇	1	1	yes	characteristic
443	C₂²	#550×C₁₃₇	1	1	yes	characteristic
444	C₂²	C₂²×SD₁₆×C₁₃₇	1	1	yes	characteristic
445	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
446	C₂²	#576×C₁₃₇	1	1	yes	characteristic
447	C₂²	#577×C₁₃₇	1	1	yes	characteristic
448	C₄	—	2	4	no	—
449	C₄	—	2	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	8	1	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	2	no	—
455	C₄	—	8	1	no	—
456	C₄	#481×C₁₃₇	1	4	yes	—
457	C₄	#529×C₁₃₇	1	1	yes	characteristic
458	C₄	#542×C₁₃₇	1	1	yes	characteristic
459	C₄	#556×C₁₃₇	1	1	yes	characteristic
460	C₄	#559×C₁₃₇	1	1	yes	characteristic
461	C₂	—	16	2	no	—
462	C₂	#2448×C₁₃₇	1	1	yes	characteristic
463	C₂	C₂×#481×C₁₃₇	1	1	yes	characteristic
464	C₂	#3096×C₁₃₇	1	1	yes	characteristic
465	C₂	#3362×C₁₃₇	1	1	yes	characteristic
466	C₂	#3373×C₁₃₇	1	1	yes	characteristic
467	C₂	#3392×C₁₃₇	1	1	yes	characteristic
468	C₂	#3399×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄⋊C₄×C₁₃₇	C₂	semidirect
3. #2094×C₁₃₇	C₂	semidirect
4. #2071×C₁₃₇	C₂	semidirect
5. #1984×C₁₃₇	C₂	semidirect
8. #1948×C₁₃₇	C₂	semidirect
9. #1926×C₁₃₇	C₂	semidirect
10. #1926×C₁₃₇	C₂	semidirect
11. C₄×#106×C₁₃₇	C₂	semidirect
234. #26318	C₁₃₇	direct (N × H)
235. C₁₃₇	#26318	direct (N × H)
237. Q₈×C₄⋊C₄	C₂×C₁₃₇	semidirect
238. #2094	C₂×C₁₃₇	semidirect
239. #2071	C₂×C₁₃₇	semidirect
240. #1984	C₂×C₁₃₇	semidirect
243. #1948	C₂×C₁₃₇	semidirect
244. #1926	C₂×C₁₃₇	semidirect
245. #1926	C₂×C₁₃₇	semidirect
246. C₄×#106	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2548×C₁₃₇	2
Q₈×C₄⋊C₄×C₁₃₇	2
#2094×C₁₃₇	2
#2071×C₁₃₇	2
#1984×C₁₃₇	2
#1982×C₁₃₇	2
#1957×C₁₃₇	2
#1948×C₁₃₇	2
#1926×C₁₃₇	2
C₄×#106×C₁₃₇	2
#26318	137

Subgroups — C137×#26318 #46875704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#26318 #46875704