Subgroups — #38752251

Order 25216.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263351	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20263346	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161648	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161301	C₂	1	2	yes	maximal, complemented
5	#27161294	C₂	1	2	yes	maximal, complemented
6	#27161279	C₂	1	2	yes	maximal, complemented
7	#27161274	C₂	1	2	yes	maximal, complemented
8	#27161106	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161078	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20263246	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic
13	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
14	#20263351	—	2	4	no	—
15	#20263346	—	2	4	no	—
16	C₂×#15691348	C₂²	1	1	yes	characteristic, complemented
17	C₂×#15691346	C₂²	1	1	yes	characteristic, complemented
18	C₂×#15691345	C₂²	1	2	yes	—
19	C₂×#15691345	C₂²	1	1	yes	characteristic, complemented
20	C₂×#15691343	C₂²	1	2	yes	—
21	C₂×#15691343	C₂²	1	1	yes	characteristic, complemented
22	C₂×#15691342	C₂²	1	1	yes	characteristic
23	C₂²×#13096802	C₂²	1	1	yes	characteristic
24	#20263323	C₂²	1	2	yes	—
25	#20263314	C₂²	1	2	yes	—
26	#20263301	—	2	1	no	—
27	#20263301	—	2	1	no	—
28	#20263296	—	2	2	no	—
29	C₄⋊#13096802	C₂²	1	2	yes	complemented
30	C₄⋊#13096802	C₂²	1	2	yes	complemented
31	#20263249	C₂²	1	2	yes	complemented
32	#20263246	C₂²	1	2	yes	—
33	#20263246	C₂²	1	2	yes	—
34	#20263246	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₉₇	C₂²	1	2	yes	complemented
37	C₄×D₄×C₁₉₇	—	2	2	no	—
38	C₄×D₄×C₁₉₇	C₂²	1	2	yes	complemented
39	#117×C₁₉₇	—	2	1	no	—
40	#117×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
43	C₂³×D₁₉₇	—	2	1	no	—
44	#15691358	—	2	2	no	—
45	#15691358	—	2	2	no	—
46	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
47	C₂×D₇₈₈	—	2	2	no	—
48	C₂×D₇₈₈	—	2	1	no	—
49	C₂×D₇₈₈	—	2	2	no	—
50	C₂×D₇₈₈	—	2	2	no	—
51	C₂×D₇₈₈	—	2	2	no	—
52	C₂×D₇₈₈	C₂³	1	2	yes	—
53	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
55	C₂×C₄×D₁₉₇	—	2	1	no	—
56	#15691348	—	2	2	no	—
57	#15691346	—	2	2	no	—
58	#15691345	—	2	4	no	—
59	#15691345	—	2	2	no	—
60	#15691343	—	2	4	no	—
61	#15691343	—	2	2	no	—
62	#15691342	—	2	2	no	—
63	C₂×#13096802	—	2	2	no	—
64	C₂×#13096802	C₂³	1	1	yes	characteristic
65	C₂×#13096802	C₂³	1	1	yes	characteristic
66	C₂×#13096802	C₂³	1	2	yes	—
67	#15691335	—	2	2	no	—
68	#15691335	—	2	2	no	—
69	#41×C₁₉₇	—	2	2	no	—
70	#41×C₁₉₇	D₄	1	4	yes	complemented
71	C₂×Q₈×C₁₉₇	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₉₇	—	2	1	no	—
73	C₂×D₄×C₁₉₇	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₉₇	—	2	1	no	—
75	C₂²×C₄×C₁₉₇	—	2	1	no	—
76	C₂²×C₄×C₁₉₇	—	2	2	no	—
77	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉₇	—	2	1	no	—
79	C₄⋊C₄×C₁₉₇	—	2	2	no	—
80	C₄⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
84	C₂²⋊C₄×C₁₉₇	D₄	1	2	yes	—
85	C₄²×C₁₉₇	—	2	2	no	—
86	C₄²×C₁₉₇	C₂³	1	2	yes	—
87	C₂²×D₁₉₇	—	2	2	no	—
88	C₂²×D₁₉₇	—	2	1	no	—
89	C₂²×D₁₉₇	—	4	2	no	—
90	C₂²×D₁₉₇	—	8	1	no	—
91	C₂²×D₁₉₇	—	8	1	no	—
92	Dic₃₉₄	—	2	1	no	—
93	Dic₃₉₄	—	2	1	no	—
94	D₇₈₈	—	2	2	no	—
95	D₇₈₈	—	2	1	no	—
96	D₇₈₈	—	2	2	no	—
97	D₇₈₈	—	2	1	no	—
98	D₇₈₈	—	4	2	no	—
99	D₇₈₈	—	8	2	no	—
100	D₇₈₈	—	8	2	no	—
101	C₂²⋊D₁₉₇	—	4	2	no	—
102	C₂×Dic₁₉₇	—	2	1	no	—
103	C₄×D₁₉₇	—	4	2	no	—
104	#13096802	—	2	1	no	—
105	#13096802	—	2	1	no	—
106	#13096802	—	2	2	no	—
107	C₂³×C₁₉₇	—	2	1	no	—
108	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₉₇	—	2	1	no	—
110	Q₈×C₁₉₇	C₂×D₄	1	2	yes	—
111	D₄×C₁₉₇	—	2	4	no	—
112	D₄×C₁₉₇	—	2	1	no	—
113	D₄×C₁₉₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₁₉₇	—	2	1	no	—
115	C₂×C₄×C₁₉₇	—	2	4	no	—
116	C₂×C₄×C₁₉₇	—	2	2	no	—
117	C₂×C₄×C₁₉₇	—	8	2	no	—
118	C₂×C₄×C₁₉₇	—	8	1	no	—
119	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₁₉₇	#41	1	2	yes	—
124	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
125	C₂²×C₁₉₇	—	2	1	no	—
126	C₂²×C₁₉₇	—	2	1	no	—
127	C₂²×C₁₉₇	—	2	2	no	—
128	C₂²×C₁₉₇	—	4	1	no	—
129	C₂²×C₁₉₇	—	4	1	no	—
130	C₂²×C₁₉₇	#121	1	2	yes	—
131	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₉₇	—	4	1	no	—
133	C₂×D₁₉₇	—	4	2	no	—
134	C₂×D₁₉₇	—	4	1	no	—
135	C₂×D₁₉₇	—	4	2	no	—
136	C₂×D₁₉₇	—	4	1	no	—
137	C₂×D₁₉₇	—	4	2	no	—
138	C₂×D₁₉₇	—	8	4	no	—
139	C₄×C₁₉₇	—	2	2	no	—
140	C₄×C₁₉₇	—	2	2	no	—
141	C₄×C₁₉₇	—	2	2	no	—
142	C₄×C₁₉₇	—	4	2	no	—
143	C₄×C₁₉₇	#121	1	2	yes	—
144	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
146	Dic₁₉₇	—	4	1	no	—
147	C₂×C₁₉₇	—	2	2	no	—
148	C₂×C₁₉₇	—	4	2	no	—
149	C₂×C₁₉₇	C₂×#121	1	1	yes	characteristic
150	C₂×C₁₉₇	#576	1	1	yes	characteristic
151	C₂×C₁₉₇	#577	1	1	yes	characteristic
152	D₁₉₇	—	8	1	no	—
153	D₁₉₇	—	8	2	no	—
154	C₁₉₇	#3045	1	1	yes	characteristic, complemented, Sylow
155	#3045	—	197	1	no	maximal, Sylow
156	C₂×AGL(1,ℤ/8)	—	197	1	no	—
157	C₂×#135	—	197	1	no	—
158	#535	—	197	1	no	—
159	#520	D₁₉₇	1	1	yes	characteristic, complemented
160	#464	—	197	2	no	—
161	#463	—	197	2	no	—
162	#460	—	197	2	no	—
163	#459	—	197	2	no	—
164	#424	—	197	1	no	—
165	#417	—	197	1	no	—
166	C₂×#102	—	197	1	no	—
167	C₂×#41	—	197	1	no	—
168	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
169	C₂²×D₄	—	197	1	no	—
170	AGL(1,ℤ/8)	—	394	4	no	—
171	#135	—	394	4	no	—
172	C₂×Q₁₆	—	197	1	no	—
173	C₂×SD₁₆	—	197	1	no	—
174	C₂×SD₁₆	—	197	2	no	—
175	C₂×SD₁₆	—	197	1	no	—
176	C₂×D₈	—	197	1	no	—
177	C₂×D₈	—	197	2	no	—
178	C₂×M₁₆	—	197	1	no	—
179	C₂²×C₈	—	197	1	no	—
180	#127	—	197	2	no	—
181	#124	—	197	2	no	—
182	#121	—	394	1	no	—
183	#121	—	394	1	no	—
184	#120	—	394	2	no	—
185	C₄×Q₈	C₂×D₁₉₇	1	2	yes	complemented
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	C₂×D₁₉₇	1	2	yes	complemented
188	#117	—	2	1	no	—
189	#117	C₂×D₁₉₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
192	C₄⋊C₈	—	197	2	no	—
193	C₄⋊C₈	—	197	2	no	—
194	#103	—	197	2	no	—
195	#102	—	197	2	no	—
196	#102	—	197	2	no	—
197	#102	—	197	2	no	—
198	C₂⁴	—	394	1	no	—
199	#41	—	2	2	no	—
200	#41	—	394	2	no	—
201	#41	—	394	2	no	—
202	#41	D₇₈₈	1	4	yes	complemented
203	C₂×Q₈	—	197	1	no	—
204	C₂×Q₈	C₂²×D₁₉₇	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	394	1	no	—
207	C₂×D₄	—	394	1	no	—
208	C₂×D₄	—	394	2	no	—
209	C₂×D₄	—	394	2	no	—
210	C₂×D₄	—	394	2	no	—
211	C₂×D₄	—	394	2	no	—
212	C₂×D₄	—	197	2	no	—
213	C₂×D₄	—	197	1	no	—
214	C₂×D₄	C₂²×D₁₉₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	394	1	no	—
219	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
220	Q₁₆	—	394	2	no	—
221	SD₁₆	—	394	2	no	—
222	SD₁₆	—	394	4	no	—
223	SD₁₆	—	394	2	no	—
224	D₈	—	394	2	no	—
225	D₈	—	394	4	no	—
226	M₁₆	—	394	2	no	—
227	C₂×C₈	—	394	2	no	—
228	C₂×C₈	—	197	1	no	—
229	C₂×C₈	—	197	1	no	—
230	C₂×C₈	—	197	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
234	C₄⋊C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
235	C₄⋊C₄	C₂²×D₁₉₇	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	394	2	no	—
238	C₂²⋊C₄	—	394	2	no	—
239	C₂²⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	C₂²×D₁₉₇	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	—	788	2	no	—
244	C₂³	—	1576	1	no	—
245	C₂³	—	1576	1	no	—
246	C₂³	—	394	2	no	—
247	C₂³	—	394	1	no	—
248	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	394	1	no	—
251	Q₈	—	394	1	no	—
252	Q₈	C₂×D₇₈₈	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	788	2	no	—
256	D₄	—	788	2	no	—
257	D₄	—	1576	2	no	—
258	D₄	—	1576	2	no	—
259	D₄	—	394	2	no	—
260	D₄	—	394	1	no	—
261	D₄	—	394	2	no	—
262	D₄	—	394	1	no	—
263	D₄	C₂×D₇₈₈	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	788	2	no	—
270	C₂×C₄	—	394	1	no	—
271	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
274	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
275	C₂×C₄	#15691358	1	2	yes	—
276	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
277	C₈	—	394	1	no	—
278	C₈	—	394	1	no	—
279	C₈	—	394	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	788	1	no	—
286	C₂²	—	788	2	no	—
287	C₂²	—	788	1	no	—
288	C₂²	—	788	2	no	—
289	C₂²	—	788	1	no	—
290	C₂²	—	788	2	no	—
291	C₂²	—	1576	4	no	—
292	C₂²	#20263300	1	2	yes	—
293	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	788	1	no	—
299	C₄	#20263300	1	2	yes	—
300	C₄	C₂²×D₇₈₈	1	1	yes	characteristic
301	C₄	C₂×#15691358	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	1576	2	no	—
305	C₂	—	1576	1	no	—
306	C₂	C₂×#20263300	1	1	yes	characteristic
307	C₂	#27161882	1	1	yes	characteristic
308	C₂	#27161887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#520×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	#520	D₁₉₇
hypocenter	C₁₉₇	#3045
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#20263351	C₂	semidirect
2. C₂×#20263346	C₂	semidirect
3. #27161648	C₂	semidirect
4. #27161301	C₂	semidirect
5. #27161294	C₂	semidirect
6. #27161279	C₂	semidirect
7. #27161274	C₂	semidirect
8. #27161106	C₂	semidirect
9. #27161078	C₂	semidirect
10. C₂×#20263246	C₂	semidirect
11. #520×C₁₉₇	C₂	semidirect
16. C₂×#15691348	C₂²	semidirect
17. C₂×#15691346	C₂²	semidirect
19. C₂×#15691345	C₂²	semidirect
21. C₂×#15691343	C₂²	semidirect
29. C₄⋊#13096802	C₂²	semidirect
30. C₄⋊#13096802	C₂²	semidirect
31. #20263249	C₂²	semidirect
34. #20263246	C₂²	semidirect
36. C₄×Q₈×C₁₉₇	C₂²	semidirect
38. C₄×D₄×C₁₉₇	C₂²	semidirect
70. #41×C₁₉₇	D₄	semidirect
154. C₁₉₇	#3045	semidirect
159. #520	D₁₉₇	semidirect
185. C₄×Q₈	C₂×D₁₉₇	semidirect
187. C₄×D₄	C₂×D₁₉₇	semidirect
202. #41	D₇₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263351	2
C₂×#20263346	2
#27161648	2
#27161301	2
#27161294	2
#27161279	2
#27161274	2
#27161106	2
#27161078	2
C₂×#20263246	2
#520×C₁₉₇	2
#3045	197