Subgroups — #31797549

Order 17536.

492 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	#22578471	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578470	C₂	1	1	yes	maximal, characteristic
3	#22578442	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#10587739	C₂	1	4	yes	maximal, complemented
5	#22578413	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578396	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578395	C₂	1	1	yes	maximal, characteristic
8	#22578366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
10	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#10587739	—	2	2	no	—
12	Q₈×#10587739	C₂²	1	4	yes	complemented
13	D₄×#10587739	—	2	2	no	—
14	D₄×#10587739	C₂²	1	4	yes	—
15	#17984167	—	2	2	no	—
16	#17984167	C₂²	1	1	yes	characteristic
17	#17984167	C₂²	1	1	yes	characteristic
18	#17984167	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#10587739	—	2	2	no	—
20	#17984157	C₂²	1	2	yes	complemented
21	#17984156	C₂²	1	2	yes	complemented
22	#17984151	C₂²	1	4	yes	complemented
23	#17984150	C₂²	1	4	yes	—
24	#17984142	C₂²	1	2	yes	complemented
25	C₈×#10587739	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#14555499	C₄	1	1	yes	characteristic, complemented
29	C₂×#14555498	C₄	1	1	yes	characteristic, complemented
30	C₂×#14555497	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₁₃₇	C₄	1	2	yes	complemented
32	SD₁₆×D₁₃₇	C₄	1	2	yes	complemented
33	SD₁₆×D₁₃₇	C₂²	1	2	yes	—
34	SD₁₆×D₁₃₇	C₂²	1	2	yes	—
35	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
37	C₂²×#10587739	—	2	4	no	—
38	#14555520	—	2	1	no	—
39	#14555519	—	2	1	no	—
40	#14555518	C₂³	1	2	yes	—
41	C₄⋊#10587739	—	2	4	no	—
42	C₄⋊#10587739	D₄	1	2	yes	—
43	C₄⋊#10587739	C₂³	1	2	yes	—
44	C₄⋊#10587739	C₂³	1	2	yes	—
45	C₂²⋊#10587739	—	2	4	no	—
46	C₂²⋊#10587739	—	2	1	no	—
47	C₂²⋊#10587739	—	2	1	no	—
48	C₂²⋊#10587739	D₄	1	2	yes	—
49	C₄×#10587739	—	2	4	no	—
50	C₄×#10587739	—	2	1	no	—
51	C₄×#10587739	—	2	1	no	—
52	C₄×#10587739	C₂³	1	2	yes	—
53	C₂³×D₁₃₇	—	2	1	no	—
54	C₂×Dic₂₇₄	—	2	1	no	—
55	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₁₃₇	—	2	1	no	—
57	Q₈×D₁₃₇	—	2	1	no	—
58	Q₈×D₁₃₇	C₂×C₄	1	2	yes	complemented
59	Q₈×D₁₃₇	C₂³	1	2	yes	—
60	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
62	D₄×D₁₃₇	—	2	1	no	—
63	D₄×D₁₃₇	—	2	1	no	—
64	D₄×D₁₃₇	C₂×C₄	1	2	yes	—
65	D₄×D₁₃₇	C₂³	1	2	yes	—
66	C₂×C₄×D₁₃₇	—	2	1	no	—
67	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
68	#14555499	C₂×C₄	1	2	yes	complemented
69	#14555499	C₂×C₄	1	2	yes	complemented
70	#14555498	C₂×C₄	1	2	yes	complemented
71	#14555498	C₂×C₄	1	2	yes	complemented
72	#14555497	C₂×C₄	1	2	yes	complemented
73	#14555497	C₂×C₄	1	2	yes	complemented
74	C₂×#11764040	C₂×C₄	1	1	yes	characteristic, complemented
75	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
76	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
77	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
78	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
82	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
83	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂×#10587739	—	2	2	no	—
85	C₂×#10587739	—	2	2	no	—
86	C₂×#10587739	—	8	4	no	—
87	C₂×#10587739	C₂×D₄	1	4	yes	—
88	#11764047	—	2	2	no	—
89	C₂²×D₁₃₇	—	2	1	no	—
90	C₂²×D₁₃₇	—	2	2	no	—
91	C₂²×D₁₃₇	—	2	2	no	—
92	C₂²×D₁₃₇	—	8	2	no	—
93	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
94	Dic₂₇₄	—	2	1	no	—
95	Dic₂₇₄	—	2	2	no	—
96	Dic₂₇₄	—	2	2	no	—
97	Dic₂₇₄	C₂²×C₄	1	2	yes	—
98	D₅₄₈	—	2	1	no	—
99	D₅₄₈	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₁₃₇	—	2	2	no	—
101	C₂²⋊D₁₃₇	—	2	2	no	—
102	C₂×Dic₁₃₇	—	2	1	no	—
103	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
104	C₄×D₁₃₇	—	2	2	no	—
105	C₄×D₁₃₇	—	2	2	no	—
106	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
108	C₄×D₁₃₇	#41	1	1	yes	characteristic
109	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
110	#11764040	C₂²×C₄	1	1	yes	characteristic
111	#11764040	C₂²×C₄	1	1	yes	characteristic
112	C₂³×C₁₃₇	—	2	1	no	—
113	Q₈×C₁₃₇	—	2	1	no	—
114	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
115	D₄×C₁₃₇	—	2	1	no	—
116	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₃₇	—	2	1	no	—
118	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
120	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	—	8	1	no	—
123	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂×D₁₃₇	—	2	2	no	—
125	C₂×D₁₃₇	—	2	2	no	—
126	C₂×D₁₃₇	—	4	4	no	—
127	C₂×D₁₃₇	—	8	2	no	—
128	C₂×D₁₃₇	—	8	1	no	—
129	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
130	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
131	#10587739	—	2	4	no	—
132	#10587739	—	4	2	no	—
133	#10587739	—	4	2	no	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
137	Dic₁₃₇	—	2	2	no	—
138	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
139	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₁₃₇	—	2	1	no	—
141	C₂×C₁₃₇	—	4	2	no	—
142	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
143	D₁₃₇	—	2	1	no	—
144	D₁₃₇	—	4	2	no	—
145	D₁₃₇	#577	1	2	yes	—
146	C₁₃₇	#2949	1	1	yes	characteristic, complemented, Sylow
147	#2949	—	137	1	no	maximal, Sylow
148	C₂²×SD₁₆	—	137	1	no	—
149	#519	—	137	1	no	—
150	#518	—	137	1	no	—
151	#440	—	137	4	no	—
152	C₄×SD₁₆	—	137	4	no	—
153	#427	—	137	1	no	—
154	#419	—	137	1	no	—
155	#418	—	137	1	no	—
156	#405	—	137	1	no	—
157	C₂²×Q₈	—	137	1	no	—
158	C₂²×D₄	—	137	1	no	—
159	C₂×SD₁₆	—	137	1	no	—
160	C₂×SD₁₆	—	137	1	no	—
161	C₂×SD₁₆	—	137	2	no	—
162	C₂×SD₁₆	—	137	2	no	—
163	C₂×SD₁₆	—	137	2	no	—
164	C₂×SD₁₆	—	137	2	no	—
165	C₂×SD₁₆	—	137	1	no	—
166	C₂×SD₁₆	#10587739	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	137	1	no	—
168	C₂²×C₈	—	137	1	no	—
169	C₄×Q₈	—	274	2	no	—
170	C₄×Q₈	—	137	4	no	—
171	C₄×D₄	—	274	2	no	—
172	C₄×D₄	—	137	4	no	—
173	#117	—	274	2	no	—
174	#117	—	137	1	no	—
175	#117	—	137	1	no	—
176	#117	—	137	1	no	—
177	C₂×C₂²⋊C₄	—	274	2	no	—
178	#107	—	137	2	no	—
179	#106	—	137	2	no	—
180	#103	—	137	4	no	—
181	#102	—	137	4	no	—
182	#97	—	137	2	no	—
183	C₄×C₈	—	137	2	no	—
184	C₂⁴	—	274	1	no	—
185	C₂×Q₈	—	274	1	no	—
186	C₂×Q₈	—	274	1	no	—
187	C₂×Q₈	—	274	1	no	—
188	C₂×Q₈	—	137	2	no	—
189	C₂×Q₈	—	137	2	no	—
190	C₂×Q₈	—	137	1	no	—
191	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic, complemented
192	C₂×D₄	—	274	1	no	—
193	C₂×D₄	—	274	1	no	—
194	C₂×D₄	—	274	1	no	—
195	C₂×D₄	—	137	2	no	—
196	C₂×D₄	—	137	2	no	—
197	C₂×D₄	—	137	1	no	—
198	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
199	C₂²×C₄	—	274	1	no	—
200	C₂²×C₄	—	274	4	no	—
201	C₂²×C₄	—	137	1	no	—
202	SD₁₆	—	137	2	no	—
203	SD₁₆	—	137	2	no	—
204	SD₁₆	—	137	2	no	—
205	SD₁₆	—	137	2	no	—
206	SD₁₆	—	137	2	no	—
207	SD₁₆	—	137	2	no	—
208	SD₁₆	C₂×#10587739	1	2	yes	complemented
209	SD₁₆	C₂×#10587739	1	2	yes	complemented
210	M₁₆	—	274	1	no	—
211	M₁₆	—	274	1	no	—
212	C₂×C₈	—	137	2	no	—
213	C₂×C₈	—	137	1	no	—
214	C₂×C₈	—	137	1	no	—
215	C₂×C₈	—	137	1	no	—
216	C₂×C₈	—	137	1	no	—
217	C₂×C₈	—	137	1	no	—
218	C₂×C₈	C₂×#10587739	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	274	4	no	—
220	C₄⋊C₄	—	137	2	no	—
221	C₄⋊C₄	—	137	2	no	—
222	C₄⋊C₄	—	137	2	no	—
223	C₂²⋊C₄	—	274	1	no	—
224	C₂²⋊C₄	—	274	1	no	—
225	C₂²⋊C₄	—	274	4	no	—
226	C₂²⋊C₄	—	137	2	no	—
227	C₄²	—	274	4	no	—
228	C₄²	—	274	1	no	—
229	C₄²	—	274	1	no	—
230	C₄²	—	137	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	274	2	no	—
233	C₂³	—	274	2	no	—
234	C₂³	—	274	1	no	—
235	C₂³	—	1096	2	no	—
236	C₂³	—	137	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	274	2	no	—
239	Q₈	—	274	2	no	—
240	Q₈	—	274	1	no	—
241	Q₈	—	137	2	no	—
242	Q₈	C₂²×#10587739	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	274	2	no	—
245	D₄	—	274	2	no	—
246	D₄	—	274	1	no	—
247	D₄	—	137	2	no	—
248	D₄	C₂²×#10587739	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	274	2	no	—
251	C₂×C₄	—	274	2	no	—
252	C₂×C₄	—	274	1	no	—
253	C₂×C₄	—	274	2	no	—
254	C₂×C₄	—	274	2	no	—
255	C₂×C₄	—	1096	4	no	—
256	C₂×C₄	—	137	4	no	—
257	C₂×C₄	—	137	1	no	—
258	C₂×C₄	—	137	1	no	—
259	C₂×C₄	—	137	1	no	—
260	C₂×C₄	—	137	1	no	—
261	C₂×C₄	—	137	1	no	—
262	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
263	C₈	—	274	2	no	—
264	C₈	—	137	1	no	—
265	C₈	—	137	1	no	—
266	C₈	C₂²×#10587739	1	1	yes	characteristic
267	C₈	C₂²×#10587739	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	274	2	no	—
271	C₂²	—	274	2	no	—
272	C₂²	—	548	4	no	—
273	C₂²	—	1096	1	no	—
274	C₂²	—	1096	2	no	—
275	C₂²	—	137	1	no	—
276	C₂²	—	137	1	no	—
277	C₂²	D₄×#10587739	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	274	4	no	—
280	C₄	—	274	2	no	—
281	C₄	—	548	2	no	—
282	C₄	—	548	2	no	—
283	C₄	—	137	1	no	—
284	C₄	—	137	1	no	—
285	C₄	D₄×#10587739	1	1	yes	characteristic
286	C₄	C₂³×#10587739	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	274	1	no	—
290	C₂	—	548	2	no	—
291	C₂	—	137	2	no	—
292	C₂	C₂×D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	C₂×SD₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	C₂×D₄×D₁₃₇	C₂²
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂×SD₁₆	#10587739
hypocenter	C₁₃₇	#2949
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #22578471	C₂	semidirect
3. #22578442	C₂	semidirect
4. SD₁₆×#10587739	C₂	semidirect
5. #22578413	C₂	semidirect
6. #22578396	C₂	semidirect
8. #22578366	C₂	semidirect
12. Q₈×#10587739	C₂²	semidirect
20. #17984157	C₂²	semidirect
21. #17984156	C₂²	semidirect
22. #17984151	C₂²	semidirect
24. #17984142	C₂²	semidirect
25. C₈×#10587739	C₂²	semidirect
28. C₂×#14555499	C₄	semidirect
29. C₂×#14555498	C₄	semidirect
30. C₂×#14555497	C₄	semidirect
31. SD₁₆×D₁₃₇	C₄	semidirect
32. SD₁₆×D₁₃₇	C₄	semidirect
36. C₂×SD₁₆×C₁₃₇	C₄	semidirect
55. C₂×Dic₂₇₄	C₂×C₄	semidirect
58. Q₈×D₁₃₇	C₂×C₄	semidirect
68. #14555499	C₂×C₄	semidirect
69. #14555499	C₂×C₄	semidirect
70. #14555498	C₂×C₄	semidirect
71. #14555498	C₂×C₄	semidirect
72. #14555497	C₂×C₄	semidirect
73. #14555497	C₂×C₄	semidirect
74. C₂×#11764040	C₂×C₄	semidirect
75. C₈×D₁₃₇	C₂×C₄	semidirect
76. C₈×D₁₃₇	C₂×C₄	semidirect
79. C₂×Q₈×C₁₃₇	C₂×C₄	semidirect
81. SD₁₆×C₁₃₇	C₂×C₄	semidirect
82. SD₁₆×C₁₃₇	C₂×C₄	semidirect
83. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
146. C₁₃₇	#2949	semidirect
166. C₂×SD₁₆	#10587739	semidirect
191. C₂×Q₈	C₂×#10587739	semidirect
208. SD₁₆	C₂×#10587739	semidirect
209. SD₁₆	C₂×#10587739	semidirect
218. C₂×C₈	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578471	2
#22578470	2
#22578442	2
SD₁₆×#10587739	2
#22578413	2
#22578396	2
#22578395	2
#22578366	2
C₂×SD₁₆×D₁₃₇	2
#2949	137