Subgroups — #31797481

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578428	C₂	1	2	yes	maximal, complemented
2	#22578428	C₂	1	2	yes	maximal, complemented
3	#22578426	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578425	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578422	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578421	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578409	C₂	1	2	yes	maximal, complemented
8	#22578375	C₂	1	1	yes	maximal, characteristic
9	C₂×#17984145	C₂	1	1	yes	maximal, characteristic
10	#22578367	C₂	1	2	yes	maximal, complemented
11	#117×D₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14555519	C₂²	1	2	yes	—
13	C₂²×#11764047	C₂²	1	2	yes	—
14	#17984155	C₂²	1	2	yes	complemented
15	#17984155	C₂²	1	2	yes	complemented
16	C₄⋊#11764047	C₂²	1	2	yes	complemented
17	C₄⋊#11764047	C₂²	1	2	yes	complemented
18	#17984145	C₂²	1	1	yes	characteristic
19	#17984145	C₂²	1	1	yes	characteristic
20	#17984145	C₂²	1	2	yes	—
21	#17984145	C₂²	1	1	yes	characteristic
22	#17984145	C₂²	1	1	yes	characteristic
23	#17984145	C₂²	1	2	yes	—
24	#17984143	C₂²	1	2	yes	complemented
25	#17984143	C₂²	1	2	yes	complemented
26	C₄×#11764047	C₂²	1	2	yes	complemented
27	C₄×#11764047	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
29	#17984047	C₄	1	1	yes	characteristic
30	#17984046	C₂²	1	1	yes	characteristic
31	#17984046	C₂²	1	1	yes	characteristic
32	#17984045	C₄	1	1	yes	characteristic
33	#17984045	C₄	1	1	yes	characteristic
34	#17984044	C₂²	1	1	yes	characteristic
35	#17984044	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
38	C₂²⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
39	C₂²⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
40	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
41	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
42	#117×C₁₃₇	C₄	1	1	yes	characteristic
43	#14555519	—	2	4	no	—
44	C₂×#11764047	—	2	4	no	—
45	C₂×#11764047	C₂³	1	2	yes	—
46	C₂×#11764047	C₂³	1	2	yes	—
47	C₂×#11764047	C₂³	1	2	yes	—
48	C₂×#11764047	C₂³	1	2	yes	—
49	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₃₇	—	2	4	no	—
52	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
62	#14555489	C₂×C₄	1	1	yes	characteristic
63	#14555489	C₂×C₄	1	1	yes	characteristic
64	#14555489	C₂×C₄	1	1	yes	characteristic
65	#14555489	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
68	#14555487	C₂×C₄	1	1	yes	characteristic
69	#14555487	C₂×C₄	1	1	yes	characteristic
70	#14555487	C₂×C₄	1	1	yes	characteristic
71	#14555487	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
85	#11764047	—	2	2	no	—
86	#11764047	—	2	2	no	—
87	#11764047	—	2	2	no	—
88	#11764047	—	2	2	no	—
89	C₂²×D₁₃₇	—	2	2	no	—
90	C₂²×D₁₃₇	—	2	2	no	—
91	C₂²×D₁₃₇	—	4	1	no	—
92	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₃₇	#41	1	4	yes	—
100	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
101	C₄×D₁₃₇	—	2	2	no	—
102	C₄×D₁₃₇	—	2	2	no	—
103	C₄×D₁₃₇	—	2	2	no	—
104	C₄×D₁₃₇	—	2	2	no	—
105	C₄×D₁₃₇	—	2	2	no	—
106	C₄×D₁₃₇	—	2	2	no	—
107	C₄×D₁₃₇	—	4	1	no	—
108	C₄×D₁₃₇	—	4	1	no	—
109	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₁₃₇	—	2	1	no	—
118	C₂²×C₁₃₇	—	2	1	no	—
119	C₂²×C₁₃₇	#117	1	2	yes	—
120	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₁₃₇	—	2	1	no	—
122	C₂×D₁₃₇	—	2	1	no	—
123	C₂×D₁₃₇	—	2	1	no	—
124	C₂×D₁₃₇	—	2	1	no	—
125	C₂×D₁₃₇	—	2	1	no	—
126	C₂×D₁₃₇	—	4	4	no	—
127	C₂×D₁₃₇	#131	1	2	yes	—
128	C₄×C₁₃₇	—	2	1	no	—
129	C₄×C₁₃₇	—	2	1	no	—
130	C₄×C₁₃₇	—	2	1	no	—
131	C₄×C₁₃₇	—	2	1	no	—
132	C₄×C₁₃₇	—	2	1	no	—
133	C₄×C₁₃₇	—	2	1	no	—
134	Dic₁₃₇	—	2	1	no	—
135	Dic₁₃₇	—	2	1	no	—
136	Dic₁₃₇	—	2	1	no	—
137	Dic₁₃₇	—	2	1	no	—
138	Dic₁₃₇	#117	1	2	yes	—
139	Dic₁₃₇	C₂×#41	1	2	yes	—
140	C₂×C₁₃₇	—	2	2	no	—
141	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
142	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
143	C₂×C₁₃₇	#568	1	1	yes	characteristic
144	D₁₃₇	—	2	2	no	—
145	D₁₃₇	—	4	1	no	—
146	C₁₃₇	#2924	1	1	yes	characteristic, complemented, Sylow
147	#2924	—	137	1	no	maximal, Sylow
148	C₂×#117	—	137	1	no	—
149	#433	—	137	2	no	—
150	#433	—	137	2	no	—
151	#432	—	137	1	no	—
152	#432	—	137	1	no	—
153	#431	—	137	1	no	—
154	#431	—	137	1	no	—
155	#424	—	137	2	no	—
156	#407	—	137	1	no	—
157	C₂×C₂²⋊C₈	—	137	1	no	—
158	#405	—	137	2	no	—
159	C₂³×C₄	—	137	1	no	—
160	C₂×M₁₆	—	137	2	no	—
161	C₂²×C₈	—	137	2	no	—
162	#117	—	137	1	no	—
163	#117	—	137	1	no	—
164	#117	—	137	1	no	—
165	#117	—	137	1	no	—
166	#117	—	137	1	no	—
167	#117	—	137	1	no	—
168	#117	—	137	1	no	—
169	#117	#10587739	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	137	1	no	—
171	C₂×C₄⋊C₄	—	137	1	no	—
172	C₂×C₂²⋊C₄	—	137	1	no	—
173	C₂×C₂²⋊C₄	—	137	1	no	—
174	C₂×C₄²	—	137	1	no	—
175	C₂×C₄²	—	137	1	no	—
176	C₄⋊C₈	—	137	2	no	—
177	C₄⋊C₈	—	137	2	no	—
178	C₄⋊C₈	—	137	2	no	—
179	C₄⋊C₈	—	137	2	no	—
180	C₂²⋊C₈	—	137	1	no	—
181	C₂²⋊C₈	—	137	1	no	—
182	C₂²⋊C₈	—	137	2	no	—
183	C₂²⋊C₈	—	137	1	no	—
184	C₂²⋊C₈	—	137	1	no	—
185	C₂²⋊C₈	—	137	2	no	—
186	#97	—	137	2	no	—
187	#97	—	137	2	no	—
188	C₄×C₈	—	137	2	no	—
189	C₄×C₈	—	137	2	no	—
190	C₂⁴	—	137	1	no	—
191	C₂²×C₄	—	274	4	no	—
192	C₂²×C₄	—	137	1	no	—
193	C₂²×C₄	—	137	1	no	—
194	C₂²×C₄	—	137	1	no	—
195	C₂²×C₄	—	137	1	no	—
196	C₂²×C₄	—	137	1	no	—
197	C₂²×C₄	—	137	1	no	—
198	C₂²×C₄	—	137	1	no	—
199	C₂²×C₄	—	137	1	no	—
200	C₂²×C₄	—	137	1	no	—
201	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
202	M₁₆	—	274	4	no	—
203	C₂×C₈	—	274	4	no	—
204	C₂×C₈	—	137	2	no	—
205	C₂×C₈	—	137	2	no	—
206	C₂×C₈	—	137	2	no	—
207	C₂×C₈	—	137	2	no	—
208	C₄⋊C₄	—	137	1	no	—
209	C₄⋊C₄	—	137	1	no	—
210	C₄⋊C₄	—	137	1	no	—
211	C₄⋊C₄	—	137	1	no	—
212	C₄⋊C₄	—	137	1	no	—
213	C₄⋊C₄	—	137	1	no	—
214	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
215	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
216	C₂²⋊C₄	—	137	1	no	—
217	C₂²⋊C₄	—	137	1	no	—
218	C₂²⋊C₄	—	137	1	no	—
219	C₂²⋊C₄	—	137	1	no	—
220	C₂²⋊C₄	—	137	1	no	—
221	C₂²⋊C₄	—	137	1	no	—
222	C₂²⋊C₄	C₂×#10587739	1	1	yes	characteristic
223	C₂²⋊C₄	C₂×#10587739	1	1	yes	characteristic
224	C₄²	—	137	1	no	—
225	C₄²	—	137	1	no	—
226	C₄²	—	137	1	no	—
227	C₄²	—	137	1	no	—
228	C₄²	—	137	1	no	—
229	C₄²	—	137	1	no	—
230	C₄²	C₂×#10587739	1	1	yes	characteristic
231	C₄²	C₂×#10587739	1	1	yes	characteristic
232	C₂³	—	274	2	no	—
233	C₂³	—	274	2	no	—
234	C₂³	—	548	1	no	—
235	C₂³	—	137	1	no	—
236	C₂³	—	137	1	no	—
237	C₂³	C₂²×#10587739	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	274	2	no	—
240	C₂×C₄	—	274	2	no	—
241	C₂×C₄	—	274	2	no	—
242	C₂×C₄	—	274	2	no	—
243	C₂×C₄	—	274	2	no	—
244	C₂×C₄	—	274	2	no	—
245	C₂×C₄	—	548	1	no	—
246	C₂×C₄	—	548	1	no	—
247	C₂×C₄	—	137	1	no	—
248	C₂×C₄	—	137	1	no	—
249	C₂×C₄	—	137	1	no	—
250	C₂×C₄	—	137	1	no	—
251	C₂×C₄	—	137	4	no	—
252	C₂×C₄	—	137	1	no	—
253	C₂×C₄	—	137	1	no	—
254	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
255	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
256	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
257	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
258	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
259	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
260	C₈	—	274	2	no	—
261	C₈	—	274	2	no	—
262	C₈	—	274	2	no	—
263	C₈	—	274	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	274	1	no	—
267	C₂²	—	274	1	no	—
268	C₂²	—	274	1	no	—
269	C₂²	—	274	1	no	—
270	C₂²	—	274	1	no	—
271	C₂²	—	548	4	no	—
272	C₂²	—	137	2	no	—
273	C₂²	#17984167	1	2	yes	—
274	C₂²	C₂³×#10587739	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	274	1	no	—
282	C₄	—	274	1	no	—
283	C₄	—	274	1	no	—
284	C₄	—	274	1	no	—
285	C₄	—	137	2	no	—
286	C₄	—	137	2	no	—
287	C₂	—	2	2	no	—
288	C₂	—	274	2	no	—
289	C₂	—	548	1	no	—
290	C₂	C₂×#17984167	1	1	yes	characteristic
291	C₂	D₁₃₇⋊#131	1	1	yes	characteristic
292	C₂	#22578482	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#117×C₁₃₇	C₄
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	#117	#10587739
hypocenter	C₁₃₇	#2924
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #22578428	C₂	semidirect
2. #22578428	C₂	semidirect
3. #22578426	C₂	semidirect
4. #22578425	C₂	semidirect
5. #22578422	C₂	semidirect
6. #22578421	C₂	semidirect
7. #22578409	C₂	semidirect
10. #22578367	C₂	semidirect
14. #17984155	C₂²	semidirect
15. #17984155	C₂²	semidirect
16. C₄⋊#11764047	C₂²	semidirect
17. C₄⋊#11764047	C₂²	semidirect
24. #17984143	C₂²	semidirect
25. #17984143	C₂²	semidirect
26. C₄×#11764047	C₂²	semidirect
27. C₄×#11764047	C₂²	semidirect
146. C₁₃₇	#2924	semidirect

Maximal subgroups

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

subgroup	index
#22578428	2
#22578426	2
#22578425	2
#22578422	2
#22578421	2
#22578409	2
#22578375	2
C₂×#17984145	2
#22578367	2
#117×D₁₃₇	2
#2924	137