Subgroups — #38751583

Order 25216.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263346	C₂	1	1	yes	maximal, characteristic
2	#27161562	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161560	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161182	C₂	1	2	yes	maximal, complemented
5	#27161176	C₂	1	2	yes	maximal, complemented
6	#27161171	C₂	1	2	yes	maximal, complemented
7	#27161159	C₂	1	2	yes	maximal, complemented
8	C₂×#20263259	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161081	C₂	1	1	yes	maximal, characteristic, complemented
10	#27161037	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic
13	#20263346	C₄	1	8	yes	complemented
14	C₂×#15691348	C₂²	1	1	yes	characteristic
15	C₂×#15691346	C₂²	1	1	yes	characteristic
16	C₂×#15691345	C₂²	1	1	yes	characteristic
17	C₂×#15691343	C₂²	1	1	yes	characteristic
18	C₂²×#13096802	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₉₇	C₂²	1	2	yes	—
20	#20263291	C₂²	1	2	yes	—
21	D₄×Dic₁₉₇	—	2	2	no	—
22	D₄×Dic₁₉₇	C₂²	1	2	yes	—
23	#20263288	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₉₇	—	2	2	no	—
25	#20263285	—	2	1	no	—
26	#20263285	C₂²	1	1	yes	characteristic
27	#20263284	—	2	1	no	—
28	C₂×C₄⋊Dic₁₉₇	—	2	1	no	—
29	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₉₇	—	2	1	no	—
31	#20263259	C₂²	1	2	yes	—
32	#20263259	C₂²	1	2	yes	—
33	#20263251	C₂²	1	2	yes	—
34	#20263248	C₂²	1	2	yes	—
35	#20263236	C₂²	1	2	yes	—
36	C₈×Dic₁₉₇	C₂²	1	2	yes	—
37	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
40	#103×C₁₉₇	C₂²	1	2	yes	—
41	#102×C₁₉₇	C₂²	1	2	yes	—
42	#15691358	—	2	2	no	—
43	#15691358	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
45	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
47	C₂²×Dic₁₉₇	—	2	2	no	—
48	C₂²×Dic₁₉₇	—	2	1	no	—
49	C₂×C₄×D₁₉₇	—	2	2	no	—
50	C₂×C₄×D₁₉₇	—	2	1	no	—
51	#15691348	C₂×C₄	1	4	yes	complemented
52	#15691346	C₂×C₄	1	4	yes	complemented
53	#15691345	C₂×C₄	1	4	yes	complemented
54	#15691343	C₂×C₄	1	4	yes	complemented
55	C₂×#13096802	C₂×C₄	1	4	yes	complemented
56	C₂×#13096802	C₂³	1	1	yes	characteristic
57	C₂×#13096802	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₉₇	—	2	2	no	—
59	C₄⋊Dic₁₉₇	—	2	1	no	—
60	C₄⋊Dic₁₉₇	D₄	1	1	yes	characteristic
61	C₄⋊Dic₁₉₇	D₄	1	1	yes	characteristic
62	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
63	#15691337	—	2	2	no	—
64	C₂²⋊Dic₁₉₇	—	2	2	no	—
65	C₂²⋊Dic₁₉₇	D₄	1	2	yes	—
66	#15691335	—	2	2	no	—
67	C₄×Dic₁₉₇	—	2	2	no	—
68	C₄×Dic₁₉₇	—	2	2	no	—
69	C₄×Dic₁₉₇	C₂³	1	2	yes	—
70	#41×C₁₉₇	—	2	2	no	—
71	#41×C₁₉₇	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₁₉₇	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₁₉₇	—	2	1	no	—
74	C₂×D₄×C₁₉₇	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₉₇	—	2	1	no	—
76	C₂²×C₄×C₁₉₇	—	2	1	no	—
77	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
78	M₁₆×C₁₉₇	—	2	2	no	—
79	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₉₇	—	2	1	no	—
81	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
82	C₂²×D₁₉₇	—	2	1	no	—
83	Dic₃₉₄	—	2	1	no	—
84	Dic₃₉₄	C₂²×C₄	1	2	yes	—
85	D₇₈₈	—	2	1	no	—
86	D₇₈₈	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₉₇	—	2	4	no	—
88	C₂×Dic₁₉₇	—	2	1	no	—
89	C₂×Dic₁₉₇	—	2	2	no	—
90	C₂×Dic₁₉₇	—	8	2	no	—
91	C₂×Dic₁₉₇	—	8	1	no	—
92	C₂×Dic₁₉₇	C₂×D₄	1	2	yes	—
93	C₂×Dic₁₉₇	C₂×D₄	1	2	yes	—
94	C₄×D₁₉₇	—	2	4	no	—
95	C₄×D₁₉₇	—	8	2	no	—
96	#13096802	C₂²×C₄	1	2	yes	—
97	#13096802	C₂²×C₄	1	2	yes	—
98	C₂³×C₁₉₇	—	2	1	no	—
99	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₁₉₇	—	2	1	no	—
101	Q₈×C₁₉₇	C₂²×C₄	1	2	yes	—
102	D₄×C₁₉₇	—	2	4	no	—
103	D₄×C₁₉₇	—	2	1	no	—
104	D₄×C₁₉₇	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₁₉₇	—	2	1	no	—
106	C₂×C₄×C₁₉₇	—	2	4	no	—
107	C₂×C₄×C₁₉₇	—	2	2	no	—
108	C₂×C₄×C₁₉₇	—	8	1	no	—
109	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉₇	#41	1	2	yes	—
112	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
113	C₈×C₁₉₇	—	2	2	no	—
114	C₂²×C₁₉₇	—	2	1	no	—
115	C₂²×C₁₉₇	—	2	1	no	—
116	C₂²×C₁₉₇	—	2	2	no	—
117	C₂²×C₁₉₇	—	4	1	no	—
118	C₂²×C₁₉₇	—	4	1	no	—
119	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
120	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₉₇	—	2	2	no	—
122	C₂×D₁₉₇	—	4	1	no	—
123	C₂×D₁₉₇	—	4	1	no	—
124	C₄×C₁₉₇	—	2	2	no	—
125	C₄×C₁₉₇	—	4	2	no	—
126	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
127	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
129	Dic₁₉₇	—	2	2	no	—
130	Dic₁₉₇	—	2	2	no	—
131	Dic₁₉₇	—	2	2	no	—
132	Dic₁₉₇	—	4	2	no	—
133	C₂×C₁₉₇	—	2	2	no	—
134	C₂×C₁₉₇	—	4	2	no	—
135	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₉₇	#576	1	1	yes	characteristic
137	C₂×C₁₉₇	#578	1	1	yes	characteristic
138	D₁₉₇	—	4	2	no	—
139	C₁₉₇	#2951	1	1	yes	characteristic, complemented, Sylow
140	#2951	—	197	1	no	maximal, Sylow
141	C₂×#135	—	197	1	no	—
142	#520	—	197	1	no	—
143	#520	—	197	1	no	—
144	#440	—	197	2	no	—
145	#440	—	197	2	no	—
146	C₄×Q₁₆	—	197	2	no	—
147	C₄×D₈	—	197	2	no	—
148	C₂×#107	—	197	1	no	—
149	#417	—	197	1	no	—
150	#417	D₁₉₇	1	1	yes	characteristic, complemented
151	#405	—	197	1	no	—
152	C₂×#41	—	197	1	no	—
153	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
154	#135	—	197	8	no	—
155	C₂×Q₁₆	—	197	1	no	—
156	C₂×SD₁₆	—	197	1	no	—
157	C₂×SD₁₆	—	197	1	no	—
158	C₂×D₈	—	197	1	no	—
159	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
160	C₂²×C₈	—	197	1	no	—
161	C₄×Q₈	—	197	2	no	—
162	C₄×Q₈	—	197	2	no	—
163	C₄×D₄	—	394	2	no	—
164	C₄×D₄	—	394	2	no	—
165	C₄×D₄	—	197	2	no	—
166	C₄×D₄	—	197	2	no	—
167	#117	—	394	1	no	—
168	#117	—	394	1	no	—
169	#117	—	197	1	no	—
170	C₂×C₄⋊C₄	—	394	1	no	—
171	C₂×C₄⋊C₄	—	394	1	no	—
172	C₂×C₄⋊C₄	—	197	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
174	#107	—	197	2	no	—
175	#107	—	197	2	no	—
176	#103	—	197	2	no	—
177	#103	C₂×D₁₉₇	1	2	yes	—
178	#102	—	197	2	no	—
179	#102	C₂×D₁₉₇	1	2	yes	—
180	#97	—	197	2	no	—
181	C₄×C₈	—	197	2	no	—
182	#41	—	2	2	no	—
183	#41	—	394	2	no	—
184	#41	—	197	4	no	—
185	#41	C₄×D₁₉₇	1	4	yes	complemented
186	C₂×Q₈	—	197	1	no	—
187	C₂×Q₈	C₂²×D₁₉₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	394	1	no	—
190	C₂×D₄	—	197	1	no	—
191	C₂×D₄	C₂²×D₁₉₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	394	2	no	—
195	C₂²×C₄	—	394	1	no	—
196	C₂²×C₄	—	394	2	no	—
197	C₂²×C₄	—	394	1	no	—
198	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
199	Q₁₆	—	197	4	no	—
200	SD₁₆	—	197	4	no	—
201	SD₁₆	—	197	4	no	—
202	D₈	—	197	4	no	—
203	M₁₆	—	2	2	no	—
204	C₂×C₈	—	197	4	no	—
205	C₂×C₈	—	197	1	no	—
206	C₂×C₈	—	197	1	no	—
207	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	394	2	no	—
210	C₄⋊C₄	—	394	2	no	—
211	C₄⋊C₄	—	394	1	no	—
212	C₄⋊C₄	—	197	2	no	—
213	C₄⋊C₄	—	197	1	no	—
214	C₄⋊C₄	—	197	1	no	—
215	C₄⋊C₄	C₂²×D₁₉₇	1	2	yes	—
216	C₂²⋊C₄	—	394	2	no	—
217	C₂²⋊C₄	—	394	2	no	—
218	C₂²⋊C₄	—	197	2	no	—
219	C₄²	—	394	2	no	—
220	C₄²	—	394	2	no	—
221	C₄²	—	197	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	394	1	no	—
224	C₂³	D₄×D₁₉₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	394	1	no	—
227	Q₈	—	197	2	no	—
228	Q₈	C₂×C₄×D₁₉₇	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	394	4	no	—
232	D₄	—	394	1	no	—
233	D₄	—	197	2	no	—
234	D₄	C₂×C₄×D₁₉₇	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	1576	2	no	—
240	C₂×C₄	—	1576	1	no	—
241	C₂×C₄	—	1576	2	no	—
242	C₂×C₄	—	394	2	no	—
243	C₂×C₄	—	394	1	no	—
244	C₂×C₄	—	394	4	no	—
245	C₂×C₄	—	197	2	no	—
246	C₂×C₄	—	197	2	no	—
247	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
248	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
249	C₂×C₄	#15691357	1	2	yes	—
250	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	197	2	no	—
253	C₈	—	197	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	788	1	no	—
260	C₂²	—	788	1	no	—
261	C₂²	—	394	2	no	—
262	C₂²	#20263286	1	2	yes	—
263	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	788	2	no	—
267	C₄	—	394	2	no	—
268	C₄	—	394	2	no	—
269	C₄	—	394	2	no	—
270	C₄	#20263286	1	2	yes	—
271	C₄	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
272	C₄	C₂×#15691357	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	788	2	no	—
276	C₂	C₂×#20263286	1	1	yes	characteristic
277	C₂	#27161883	1	1	yes	characteristic
278	C₂	#27161895	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#417×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂×#20263346	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#417	D₁₉₇
hypocenter	C₁₉₇	#2951
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #27161562	C₂	semidirect
3. #27161560	C₂	semidirect
4. #27161182	C₂	semidirect
5. #27161176	C₂	semidirect
6. #27161171	C₂	semidirect
7. #27161159	C₂	semidirect
8. C₂×#20263259	C₂	semidirect
9. #27161081	C₂	semidirect
10. #27161037	C₂	semidirect
11. #417×C₁₉₇	C₂	semidirect
13. #20263346	C₄	semidirect
43. #15691358	C₂×C₄	semidirect
51. #15691348	C₂×C₄	semidirect
52. #15691346	C₂×C₄	semidirect
53. #15691345	C₂×C₄	semidirect
54. #15691343	C₂×C₄	semidirect
55. C₂×#13096802	C₂×C₄	semidirect
71. #41×C₁₉₇	C₂×C₄	semidirect
139. C₁₉₇	#2951	semidirect
150. #417	D₁₉₇	semidirect
185. #41	C₄×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263346	2
#27161562	2
#27161560	2
#27161182	2
#27161176	2
#27161171	2
#27161159	2
C₂×#20263259	2
#27161081	2
#27161037	2
#417×C₁₉₇	2
#2951	197