Subgroups — C₁₃₇×#26482 #46875868

Order 35072.

846 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2454×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2065×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2025×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1970×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1970×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1942×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1942×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1864×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
15	#499×C₁₃₇	—	2	4	no	—
16	C₄×D₈×C₁₃₇	—	2	4	no	—
17	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	#399×C₁₃₇	—	2	2	no	—
29	#396×C₁₃₇	—	2	2	no	—
30	#390×C₁₃₇	C₂²	1	2	yes	—
31	#390×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	C₂²	1	2	yes	—
33	#389×C₁₃₇	C₂²	1	2	yes	—
34	#385×C₁₃₇	—	2	2	no	—
35	#380×C₁₃₇	—	2	2	no	—
36	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
37	#340×C₁₃₇	C₂²	1	2	yes	—
38	#340×C₁₃₇	C₂²	1	2	yes	—
39	#336×C₁₃₇	C₂²	1	1	yes	characteristic
40	#336×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₁₃₇	—	2	2	no	—
43	C₂×D₈×C₁₃₇	—	2	2	no	—
44	C₂×D₈×C₁₃₇	—	2	2	no	—
45	C₂×D₈×C₁₃₇	Q₈	1	4	yes	—
46	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
50	#125×C₁₃₇	—	2	4	no	—
51	#125×C₁₃₇	—	4	2	no	—
52	C₄×D₄×C₁₃₇	—	2	4	no	—
53	C₄×D₄×C₁₃₇	—	4	2	no	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
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×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
71	#107×C₁₃₇	—	2	2	no	—
72	#107×C₁₃₇	—	2	2	no	—
73	#107×C₁₃₇	D₄	1	4	yes	—
74	#106×C₁₃₇	—	2	4	no	—
75	C₄⋊C₈×C₁₃₇	—	2	2	no	—
76	#102×C₁₃₇	—	2	4	no	—
77	#102×C₁₃₇	—	2	4	no	—
78	#102×C₁₃₇	—	4	2	no	—
79	C₄×C₈×C₁₃₇	—	2	2	no	—
80	C₄×C₈×C₁₃₇	—	2	2	no	—
81	C₄×C₈×C₁₃₇	—	2	2	no	—
82	#95×C₁₃₇	—	2	2	no	—
83	#95×C₁₃₇	—	2	2	no	—
84	#95×C₁₃₇	D₄	1	1	yes	characteristic
85	#95×C₁₃₇	D₄	1	1	yes	characteristic
86	C₂⁴×C₁₃₇	—	2	2	no	—
87	C₂×D₄×C₁₃₇	—	2	2	no	—
88	C₂×D₄×C₁₃₇	—	2	2	no	—
89	C₂×D₄×C₁₃₇	—	2	2	no	—
90	C₂×D₄×C₁₃₇	—	2	2	no	—
91	C₂×D₄×C₁₃₇	—	2	2	no	—
92	C₂×D₄×C₁₃₇	C₂×Q₈	1	4	yes	—
93	C₂²×C₄×C₁₃₇	—	2	2	no	—
94	C₂²×C₄×C₁₃₇	—	2	2	no	—
95	C₂²×C₄×C₁₃₇	—	2	2	no	—
96	C₂²×C₄×C₁₃₇	—	4	4	no	—
97	C₂²×C₄×C₁₃₇	—	4	4	no	—
98	C₂²×C₄×C₁₃₇	—	8	2	no	—
99	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
106	D₈×C₁₃₇	—	2	8	no	—
107	C₂×C₈×C₁₃₇	—	2	1	no	—
108	C₂×C₈×C₁₃₇	—	2	1	no	—
109	C₂×C₈×C₁₃₇	—	2	1	no	—
110	C₂×C₈×C₁₃₇	—	2	1	no	—
111	C₂×C₈×C₁₃₇	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₁₃₇	#41	1	2	yes	—
118	C₂×C₈×C₁₃₇	#41	1	2	yes	—
119	C₄⋊C₄×C₁₃₇	—	2	2	no	—
120	C₄⋊C₄×C₁₃₇	—	2	2	no	—
121	C₄⋊C₄×C₁₃₇	—	2	2	no	—
122	C₄⋊C₄×C₁₃₇	—	2	2	no	—
123	C₄⋊C₄×C₁₃₇	—	2	2	no	—
124	C₄⋊C₄×C₁₃₇	—	2	2	no	—
125	C₄⋊C₄×C₁₃₇	—	2	2	no	—
126	C₄⋊C₄×C₁₃₇	—	4	4	no	—
127	C₄⋊C₄×C₁₃₇	—	4	4	no	—
128	C₄⋊C₄×C₁₃₇	—	8	2	no	—
129	C₄⋊C₄×C₁₃₇	—	8	2	no	—
130	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
131	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
132	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
133	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
134	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
135	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
136	C₄²×C₁₃₇	—	2	1	no	—
137	C₄²×C₁₃₇	—	2	1	no	—
138	C₄²×C₁₃₇	—	2	1	no	—
139	C₄²×C₁₃₇	—	2	1	no	—
140	C₄²×C₁₃₇	—	2	1	no	—
141	C₄²×C₁₃₇	—	2	1	no	—
142	C₂³×C₁₃₇	—	2	4	no	—
143	C₂³×C₁₃₇	—	4	2	no	—
144	C₂³×C₁₃₇	—	4	2	no	—
145	C₂³×C₁₃₇	—	4	2	no	—
146	C₂³×C₁₃₇	—	4	2	no	—
147	C₂³×C₁₃₇	—	4	2	no	—
148	C₂³×C₁₃₇	—	4	2	no	—
149	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
150	D₄×C₁₃₇	—	2	4	no	—
151	D₄×C₁₃₇	—	2	4	no	—
152	D₄×C₁₃₇	—	4	2	no	—
153	D₄×C₁₃₇	—	4	2	no	—
154	C₂×C₄×C₁₃₇	—	2	1	no	—
155	C₂×C₄×C₁₃₇	—	2	1	no	—
156	C₂×C₄×C₁₃₇	—	2	1	no	—
157	C₂×C₄×C₁₃₇	—	2	1	no	—
158	C₂×C₄×C₁₃₇	—	2	1	no	—
159	C₂×C₄×C₁₃₇	—	2	1	no	—
160	C₂×C₄×C₁₃₇	—	2	1	no	—
161	C₂×C₄×C₁₃₇	—	2	1	no	—
162	C₂×C₄×C₁₃₇	—	2	4	no	—
163	C₂×C₄×C₁₃₇	—	2	1	no	—
164	C₂×C₄×C₁₃₇	—	2	1	no	—
165	C₂×C₄×C₁₃₇	—	2	1	no	—
166	C₂×C₄×C₁₃₇	—	2	1	no	—
167	C₂×C₄×C₁₃₇	—	2	1	no	—
168	C₂×C₄×C₁₃₇	—	2	1	no	—
169	C₂×C₄×C₁₃₇	—	2	1	no	—
170	C₂×C₄×C₁₃₇	—	2	1	no	—
171	C₂×C₄×C₁₃₇	—	4	2	no	—
172	C₂×C₄×C₁₃₇	—	4	2	no	—
173	C₂×C₄×C₁₃₇	—	4	2	no	—
174	C₂×C₄×C₁₃₇	—	4	2	no	—
175	C₂×C₄×C₁₃₇	—	4	2	no	—
176	C₂×C₄×C₁₃₇	—	4	2	no	—
177	C₂×C₄×C₁₃₇	—	4	2	no	—
178	C₂×C₄×C₁₃₇	—	4	2	no	—
179	C₂×C₄×C₁₃₇	—	8	8	no	—
180	C₂×C₄×C₁₃₇	#135	1	2	yes	—
181	C₂×C₄×C₁₃₇	#135	1	2	yes	—
182	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
188	C₈×C₁₃₇	—	2	2	no	—
189	C₈×C₁₃₇	—	4	1	no	—
190	C₈×C₁₃₇	—	4	1	no	—
191	C₈×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
192	C₂²×C₁₃₇	—	4	4	no	—
193	C₂²×C₁₃₇	—	4	4	no	—
194	C₂²×C₁₃₇	—	4	4	no	—
195	C₂²×C₁₃₇	—	8	2	no	—
196	C₂²×C₁₃₇	—	8	2	no	—
197	C₂²×C₁₃₇	—	8	2	no	—
198	C₂²×C₁₃₇	—	8	2	no	—
199	C₂²×C₁₃₇	#532	1	1	yes	characteristic
200	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
201	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
202	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
203	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
204	C₂²×C₁₃₇	#576	1	1	yes	characteristic
205	C₂²×C₁₃₇	#576	1	1	yes	characteristic
206	C₄×C₁₃₇	—	2	2	no	—
207	C₄×C₁₃₇	—	2	2	no	—
208	C₄×C₁₃₇	—	4	1	no	—
209	C₄×C₁₃₇	—	4	1	no	—
210	C₄×C₁₃₇	—	4	4	no	—
211	C₄×C₁₃₇	—	4	1	no	—
212	C₄×C₁₃₇	—	4	1	no	—
213	C₄×C₁₃₇	—	8	2	no	—
214	C₄×C₁₃₇	—	8	2	no	—
215	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
216	C₄×C₁₃₇	#542	1	1	yes	characteristic
217	C₄×C₁₃₇	#548	1	1	yes	characteristic
218	C₄×C₁₃₇	#554	1	1	yes	characteristic
219	C₂×C₁₃₇	—	8	4	no	—
220	C₂×C₁₃₇	#2452	1	1	yes	characteristic
221	C₂×C₁₃₇	#3168	1	1	yes	characteristic
222	C₂×C₁₃₇	#3176	1	1	yes	characteristic
223	C₂×C₁₃₇	#3280	1	1	yes	characteristic
224	C₂×C₁₃₇	#3287	1	1	yes	characteristic
225	C₂×C₁₃₇	#3380	1	1	yes	characteristic
226	C₂×C₁₃₇	#3380	1	1	yes	characteristic
227	#26482	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₁₃₇	#26482	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
230	C₂×C₄×D₈	C₂×C₁₃₇	1	1	yes	characteristic
231	#2454	C₂×C₁₃₇	1	2	yes	complemented
232	#2065	C₂×C₁₃₇	1	2	yes	complemented
233	#2025	C₂×C₁₃₇	1	2	yes	complemented
234	#1970	C₂×C₁₃₇	1	1	yes	characteristic
235	#1970	C₂×C₁₃₇	1	1	yes	characteristic
236	#1942	C₂×C₁₃₇	1	1	yes	characteristic, complemented
237	#1942	C₂×C₁₃₇	1	1	yes	characteristic, complemented
238	#1864	C₂×C₁₃₇	1	2	yes	complemented
239	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
240	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
241	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
242	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
243	#499	—	2	4	no	—
244	C₄×D₈	—	2	4	no	—
245	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
246	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
247	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
248	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
249	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
250	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
251	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
252	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
253	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
256	#399	—	2	2	no	—
257	#396	—	2	2	no	—
258	#390	C₂²×C₁₃₇	1	2	yes	—
259	#390	C₂²×C₁₃₇	1	2	yes	—
260	#389	C₂²×C₁₃₇	1	2	yes	—
261	#389	C₂²×C₁₃₇	1	2	yes	—
262	#385	—	2	2	no	—
263	#380	—	2	2	no	—
264	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
265	#340	C₂²×C₁₃₇	1	2	yes	—
266	#340	C₂²×C₁₃₇	1	2	yes	—
267	#336	C₂²×C₁₃₇	1	1	yes	characteristic
268	#336	C₂²×C₁₃₇	1	1	yes	characteristic
269	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
270	C₂³×C₄	—	2	2	no	—
271	C₂×D₈	—	2	2	no	—
272	C₂×D₈	—	2	2	no	—
273	C₂×D₈	Q₈×C₁₃₇	1	4	yes	—
274	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
278	#125	—	2	4	no	—
279	#125	—	4	2	no	—
280	C₄×D₄	—	2	4	no	—
281	C₄×D₄	—	4	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
299	#107	—	2	2	no	—
300	#107	—	2	2	no	—
301	#107	D₄×C₁₃₇	1	4	yes	—
302	#106	—	2	4	no	—
303	C₄⋊C₈	—	2	2	no	—
304	#102	—	2	4	no	—
305	#102	—	2	4	no	—
306	#102	—	4	2	no	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	—	2	2	no	—
309	C₄×C₈	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	D₄×C₁₃₇	1	1	yes	characteristic
313	#95	D₄×C₁₃₇	1	1	yes	characteristic
314	C₂⁴	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	C₂×Q₈×C₁₃₇	1	4	yes	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	4	4	no	—
325	C₂²×C₄	—	4	4	no	—
326	C₂²×C₄	—	8	2	no	—
327	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
333	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
334	D₈	—	2	8	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
344	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
345	C₂×C₈	#41×C₁₃₇	1	2	yes	—
346	C₂×C₈	#41×C₁₃₇	1	2	yes	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	4	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	8	2	no	—
357	C₄⋊C₄	—	8	2	no	—
358	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
359	C₂²⋊C₄	—	4	4	no	—
360	C₂²⋊C₄	—	4	4	no	—
361	C₂²⋊C₄	—	8	2	no	—
362	C₂²⋊C₄	—	8	2	no	—
363	C₂²⋊C₄	—	8	2	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	1	no	—
367	C₄²	—	2	1	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₂³	—	2	4	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
378	D₄	—	2	4	no	—
379	D₄	—	2	4	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	8	no	—
408	C₂×C₄	#135×C₁₃₇	1	2	yes	—
409	C₂×C₄	#135×C₁₃₇	1	2	yes	—
410	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
411	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
414	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
415	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	4	1	no	—
418	C₈	—	4	1	no	—
419	C₈	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	2	no	—
427	C₂²	#532×C₁₃₇	1	1	yes	characteristic
428	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
429	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
430	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
431	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
432	C₂²	#576×C₁₃₇	1	1	yes	characteristic
433	C₂²	#576×C₁₃₇	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	4	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	8	2	no	—
442	C₄	—	8	2	no	—
443	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
444	C₄	#542×C₁₃₇	1	1	yes	characteristic
445	C₄	#548×C₁₃₇	1	1	yes	characteristic
446	C₄	#554×C₁₃₇	1	1	yes	characteristic
447	C₂	—	8	4	no	—
448	C₂	#2452×C₁₃₇	1	1	yes	characteristic
449	C₂	#3168×C₁₃₇	1	1	yes	characteristic
450	C₂	#3176×C₁₃₇	1	1	yes	characteristic
451	C₂	#3280×C₁₃₇	1	1	yes	characteristic
452	C₂	#3287×C₁₃₇	1	1	yes	characteristic
453	C₂	#3380×C₁₃₇	1	1	yes	characteristic
454	C₂	#3380×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₇	C₂	semidirect
3. #2454×C₁₃₇	C₂	semidirect
4. #2065×C₁₃₇	C₂	semidirect
5. #2025×C₁₃₇	C₂	semidirect
8. #1942×C₁₃₇	C₂	semidirect
9. #1942×C₁₃₇	C₂	semidirect
10. #1864×C₁₃₇	C₂	semidirect
11. C₈×C₄⋊C₄×C₁₃₇	C₂	semidirect
227. #26482	C₁₃₇	direct (N × H)
228. C₁₃₇	#26482	direct (N × H)
229. C₂×#499	C₂×C₁₃₇	semidirect
231. #2454	C₂×C₁₃₇	semidirect
232. #2065	C₂×C₁₃₇	semidirect
233. #2025	C₂×C₁₃₇	semidirect
236. #1942	C₂×C₁₃₇	semidirect
237. #1942	C₂×C₁₃₇	semidirect
238. #1864	C₂×C₁₃₇	semidirect
239. C₈×C₄⋊C₄	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
C₂×C₄×D₈×C₁₃₇	2
#2454×C₁₃₇	2
#2065×C₁₃₇	2
#2025×C₁₃₇	2
#1970×C₁₃₇	2
#1942×C₁₃₇	2
#1864×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#26482	137

Subgroups — C137×#26482 #46875868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#26482 #46875868