Subgroups — C₁₃₇×#22658 #46872044

Order 35072.

982 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2354×C₁₃₇	C₂	1	2	yes	maximal
4	#1948×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1854×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1808×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₃₇	C₂²	1	4	yes	—
12	C₈×D₄×C₁₃₇	C₄	1	8	yes	complemented
13	#427×C₁₃₇	—	2	8	no	—
14	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#390×C₁₃₇	—	2	2	no	—
23	#388×C₁₃₇	—	2	4	no	—
24	#384×C₁₃₇	C₂²	1	2	yes	—
25	#380×C₁₃₇	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
28	#340×C₁₃₇	C₂²	1	2	yes	—
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₇	—	2	4	no	—
33	C₂²×C₈×C₁₃₇	—	2	4	no	—
34	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₁₃₇	Q₈	1	4	yes	—
40	#117×C₁₃₇	—	2	8	no	—
41	#117×C₁₃₇	—	4	4	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	—	2	2	no	—
52	C₂×C₄²×C₁₃₇	—	2	4	no	—
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	#107×C₁₃₇	—	2	4	no	—
55	#107×C₁₃₇	—	2	2	no	—
56	#107×C₁₃₇	—	4	2	no	—
57	#107×C₁₃₇	D₄	1	4	yes	complemented
58	#106×C₁₃₇	—	2	8	no	—
59	C₄⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
60	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
62	#95×C₁₃₇	—	2	4	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₁₃₇	—	2	2	no	—
66	C₂×D₄×C₁₃₇	—	2	2	no	—
67	C₂×D₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	—	2	4	no	—
75	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	1	no	—
84	C₂×C₈×C₁₃₇	—	2	1	no	—
85	C₂×C₈×C₁₃₇	—	2	16	no	—
86	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃₇	#41	1	2	yes	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	2	2	no	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	4	4	no	—
97	C₄⋊C₄×C₁₃₇	—	8	2	no	—
98	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
100	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
101	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
102	C₂²⋊C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
103	C₄²×C₁₃₇	—	4	8	no	—
104	C₄²×C₁₃₇	—	4	4	no	—
105	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
107	C₂³×C₁₃₇	—	2	2	no	—
108	C₂³×C₁₃₇	—	2	2	no	—
109	C₂³×C₁₃₇	—	2	2	no	—
110	C₂³×C₁₃₇	—	2	2	no	—
111	C₂³×C₁₃₇	—	2	2	no	—
112	C₂³×C₁₃₇	—	2	2	no	—
113	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₁₃₇	—	2	8	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	1	no	—
119	C₂×C₄×C₁₃₇	—	2	1	no	—
120	C₂×C₄×C₁₃₇	—	2	4	no	—
121	C₂×C₄×C₁₃₇	—	2	4	no	—
122	C₂×C₄×C₁₃₇	—	2	4	no	—
123	C₂×C₄×C₁₃₇	—	2	4	no	—
124	C₂×C₄×C₁₃₇	—	2	4	no	—
125	C₂×C₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	—	4	2	no	—
127	C₂×C₄×C₁₃₇	—	4	4	no	—
128	C₂×C₄×C₁₃₇	—	4	2	no	—
129	C₂×C₄×C₁₃₇	—	4	2	no	—
130	C₂×C₄×C₁₃₇	—	4	4	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	4	no	—
133	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₃₇	#135	1	2	yes	—
137	C₂×C₄×C₁₃₇	#135	1	2	yes	—
138	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₁₃₇	—	2	4	no	—
145	C₈×C₁₃₇	—	2	2	no	—
146	C₈×C₁₃₇	C₄×D₄	1	4	yes	—
147	C₂²×C₁₃₇	—	2	4	no	—
148	C₂²×C₁₃₇	—	2	4	no	—
149	C₂²×C₁₃₇	—	2	4	no	—
150	C₂²×C₁₃₇	—	4	2	no	—
151	C₂²×C₁₃₇	—	4	2	no	—
152	C₂²×C₁₃₇	—	4	2	no	—
153	C₂²×C₁₃₇	—	4	2	no	—
154	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
156	C₂²×C₁₃₇	#547	1	1	yes	characteristic
157	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
158	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
159	C₂²×C₁₃₇	#576	1	1	yes	characteristic
160	C₂²×C₁₃₇	#578	1	1	yes	characteristic
161	C₄×C₁₃₇	—	2	4	no	—
162	C₄×C₁₃₇	—	2	2	no	—
163	C₄×C₁₃₇	—	4	4	no	—
164	C₄×C₁₃₇	—	8	2	no	—
165	C₄×C₁₃₇	—	8	4	no	—
166	C₄×C₁₃₇	#427	1	4	yes	complemented
167	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₁₃₇	#520	1	1	yes	characteristic
169	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
170	C₄×C₁₃₇	#554	1	1	yes	characteristic
171	C₂×C₁₃₇	—	4	4	no	—
172	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₁₃₇	C₂×#427	1	1	yes	characteristic
174	C₂×C₁₃₇	#2913	1	1	yes	characteristic
175	C₂×C₁₃₇	#3283	1	1	yes	characteristic
176	C₂×C₁₃₇	#3287	1	1	yes	characteristic
177	C₂×C₁₃₇	#3341	1	1	yes	characteristic
178	C₂×C₁₃₇	#3342	1	1	yes	characteristic
179	#22658	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₁₃₇	#22658	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×C₈×D₄	C₂×C₁₃₇	1	1	yes	characteristic
182	C₂×#427	C₂×C₁₃₇	1	2	yes	complemented
183	#2354	C₂×C₁₃₇	1	2	yes	—
184	#1948	C₂×C₁₃₇	1	2	yes	complemented
185	#1854	C₂×C₁₃₇	1	4	yes	complemented
186	#1845	C₂×C₁₃₇	1	1	yes	characteristic, complemented
187	#1808	C₂×C₁₃₇	1	2	yes	complemented
188	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
189	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
190	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
191	C₂×#117	C₂²×C₁₃₇	1	4	yes	—
192	C₈×D₄	C₄×C₁₃₇	1	8	yes	complemented
193	#427	—	2	8	no	—
194	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
195	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
196	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
197	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
198	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
199	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
200	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
201	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
202	#390	—	2	2	no	—
203	#388	—	2	4	no	—
204	#384	C₂²×C₁₃₇	1	2	yes	—
205	#380	C₂²×C₁₃₇	1	4	yes	—
206	C₄×C₄⋊C₄	—	2	2	no	—
207	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
208	#340	C₂²×C₁₃₇	1	4	yes	—
209	#340	C₂²×C₁₃₇	1	2	yes	—
210	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
211	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
212	C₂²×C₈	—	2	4	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
215	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
216	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
218	C₄×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
219	C₄×D₄	Q₈×C₁₃₇	1	4	yes	—
220	#117	—	2	8	no	—
221	#117	—	4	4	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
228	C₂×C₂²⋊C₄	—	2	4	no	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
231	C₂×C₄²	—	2	4	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
234	#107	—	2	4	no	—
235	#107	—	2	2	no	—
236	#107	—	4	2	no	—
237	#107	D₄×C₁₃₇	1	4	yes	complemented
238	#106	—	2	8	no	—
239	C₄⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
240	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
241	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
242	#95	—	2	4	no	—
243	#95	—	2	2	no	—
244	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
256	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
260	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	16	no	—
266	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
267	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
268	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
270	C₂×C₈	#41×C₁₃₇	1	2	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	8	2	no	—
278	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
279	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
280	C₂²⋊C₄	—	4	8	no	—
281	C₂²⋊C₄	—	8	4	no	—
282	C₂²⋊C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
283	C₄²	—	4	8	no	—
284	C₄²	—	4	4	no	—
285	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
286	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
294	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
295	D₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
315	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
316	C₂×C₄	#135×C₁₃₇	1	2	yes	—
317	C₂×C₄	#135×C₁₃₇	1	2	yes	—
318	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
321	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
322	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
323	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	2	no	—
326	C₈	C₄×D₄×C₁₃₇	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
335	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
336	C₂²	#547×C₁₃₇	1	1	yes	characteristic
337	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
338	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
339	C₂²	#576×C₁₃₇	1	1	yes	characteristic
340	C₂²	#578×C₁₃₇	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	4	no	—
346	C₄	#427×C₁₃₇	1	4	yes	complemented
347	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
348	C₄	#520×C₁₃₇	1	1	yes	characteristic
349	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
350	C₄	#554×C₁₃₇	1	1	yes	characteristic
351	C₂	—	4	4	no	—
352	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
353	C₂	C₂×#427×C₁₃₇	1	1	yes	characteristic
354	C₂	#2913×C₁₃₇	1	1	yes	characteristic
355	C₂	#3283×C₁₃₇	1	1	yes	characteristic
356	C₂	#3287×C₁₃₇	1	1	yes	characteristic
357	C₂	#3341×C₁₃₇	1	1	yes	characteristic
358	C₂	#3342×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₁₃₇	C₂	semidirect
4. #1948×C₁₃₇	C₂	semidirect
5. #1854×C₁₃₇	C₂	semidirect
6. #1845×C₁₃₇	C₂	semidirect
7. #1808×C₁₃₇	C₂	semidirect
8. C₄×#107×C₁₃₇	C₂	semidirect
12. C₈×D₄×C₁₃₇	C₄	semidirect
57. #107×C₁₃₇	D₄	semidirect
166. C₄×C₁₃₇	#427	semidirect
179. #22658	C₁₃₇	direct (N × H)
180. C₁₃₇	#22658	direct (N × H)
182. C₂×#427	C₂×C₁₃₇	semidirect
184. #1948	C₂×C₁₃₇	semidirect
185. #1854	C₂×C₁₃₇	semidirect
186. #1845	C₂×C₁₃₇	semidirect
187. #1808	C₂×C₁₃₇	semidirect
188. C₄×#107	C₂×C₁₃₇	semidirect
192. C₈×D₄	C₄×C₁₃₇	semidirect
237. #107	D₄×C₁₃₇	semidirect
346. C₄	#427×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₃₇	2
C₂×#427×C₁₃₇	2
#2354×C₁₃₇	2
#1948×C₁₃₇	2
#1854×C₁₃₇	2
#1845×C₁₃₇	2
#1808×C₁₃₇	2
C₄×#107×C₁₃₇	2
#22658	137

Subgroups — C137×#22658 #46872044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22658 #46872044