Subgroups — #38756101

Order 25216.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161698	C₂	1	4	yes	maximal, complemented
2	#27161690	C₂	1	2	yes	maximal, complemented
3	#27161683	C₂	1	2	yes	maximal, complemented
4	#27161674	C₂	1	4	yes	maximal, complemented
5	#27161666	C₂	1	4	yes	maximal, complemented
6	#27161636	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161609	C₂	1	8	yes	maximal, complemented
8	#27161586	C₂	1	2	yes	maximal, complemented
9	#41×Dic₁₉₇	C₂	1	2	yes	maximal, complemented
10	C₂×#20263283	C₂	1	1	yes	maximal, characteristic, complemented
11	#532×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
13	#20263326	C₂²	1	2	yes	complemented
14	#20263321	C₂²	1	8	yes	—
15	#20263319	C₂²	1	8	yes	complemented
16	#20263318	C₂²	1	2	yes	complemented
17	#20263316	C₂²	1	8	yes	complemented
18	#20263312	C₂²	1	8	yes	—
19	#20263310	C₂²	1	8	yes	—
20	#20263307	C₂²	1	4	yes	—
21	#20263307	C₂²	1	8	yes	complemented
22	#20263305	C₂²	1	4	yes	—
23	#20263303	C₂²	1	8	yes	complemented
24	#20263302	C₂²	1	4	yes	—
25	#20263298	C₂²	1	8	yes	—
26	#20263297	C₂²	1	4	yes	—
27	#20263294	C₂²	1	4	yes	—
28	Q₈×Dic₁₉₇	C₂²	1	4	yes	complemented
29	C₄×Dic₃₉₄	C₂²	1	4	yes	complemented
30	D₄×Dic₁₉₇	C₂²	1	8	yes	—
31	D₄×Dic₁₉₇	C₂²	1	4	yes	complemented
32	C₄×C₂²⋊D₁₉₇	C₂²	1	8	yes	—
33	#20263285	C₂²	1	2	yes	—
34	#20263285	C₂²	1	4	yes	complemented
35	#20263283	C₂²	1	4	yes	—
36	#20263283	C₂²	1	2	yes	—
37	#20263283	C₂²	1	2	yes	—
38	C₂×#15691337	C₂²	1	2	yes	—
39	C₂×#15691335	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
41	C₂×C₄×Dic₁₉₇	C₂²	1	4	yes	complemented
42	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
43	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
44	#127×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
45	#124×C₁₉₇	C₂²	1	2	yes	complemented
46	#121×C₁₉₇	C₂²	1	4	yes	—
47	C₄×D₄×C₁₉₇	C₂²	1	4	yes	complemented
48	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
49	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
50	C₂×Dic₃₉₄	C₂³	1	4	yes	—
51	C₂×C₂²⋊D₁₉₇	C₂³	1	8	yes	—
52	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
54	C₂²×Dic₁₉₇	C₂³	1	8	yes	—
55	C₂×C₄×D₁₉₇	—	2	4	no	—
56	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
57	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
58	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
59	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
60	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
61	#15691337	C₂³	1	4	yes	—
62	#15691337	C₂³	1	4	yes	—
63	#15691337	C₂³	1	8	yes	—
64	#15691337	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₉₇	C₂³	1	4	yes	—
66	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
67	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
68	#15691335	C₂³	1	4	yes	—
69	#15691335	C₂³	1	4	yes	—
70	#15691335	C₂³	1	8	yes	—
71	C₄×Dic₁₉₇	C₂³	1	2	yes	—
72	C₄×Dic₁₉₇	C₂³	1	2	yes	—
73	C₄×Dic₁₉₇	C₂³	1	4	yes	—
74	C₄×Dic₁₉₇	C₂³	1	8	yes	—
75	#41×C₁₉₇	—	2	8	no	—
76	C₂×Q₈×C₁₉₇	C₂³	1	2	yes	—
77	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
78	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
79	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
80	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
81	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
84	C₂²⋊C₄×C₁₉₇	C₂³	1	8	yes	—
85	C₄²×C₁₉₇	C₂³	1	2	yes	—
86	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
88	C₂²×D₁₉₇	—	2	2	no	—
89	C₂²×D₁₉₇	—	4	2	no	—
90	C₂²×D₁₉₇	C₂⁴	1	2	yes	—
91	Dic₃₉₄	—	4	4	no	—
92	C₂²⋊D₁₉₇	—	4	8	no	—
93	C₂×Dic₁₉₇	—	2	2	no	—
94	C₂×Dic₁₉₇	—	4	2	no	—
95	C₂×Dic₁₉₇	—	4	8	no	—
96	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
97	C₂×Dic₁₉₇	C₂⁴	1	8	yes	—
98	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
99	C₄×D₁₉₇	—	2	4	no	—
100	C₄×D₁₉₇	—	2	4	no	—
101	C₂³×C₁₉₇	C₂⁴	1	4	yes	—
102	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
103	Q₈×C₁₉₇	—	2	4	no	—
104	D₄×C₁₉₇	—	2	8	no	—
105	D₄×C₁₉₇	—	2	4	no	—
106	D₄×C₁₉₇	—	4	4	no	—
107	C₂×C₄×C₁₉₇	—	2	8	no	—
108	C₂×C₄×C₁₉₇	—	4	2	no	—
109	C₂×C₄×C₁₉₇	#41	1	4	yes	—
110	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
113	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
115	C₂²×C₁₉₇	—	2	2	no	—
116	C₂²×C₁₉₇	—	2	4	no	—
117	C₂²×C₁₉₇	—	2	8	no	—
118	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
119	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₉₇	—	2	2	no	—
121	C₂×D₁₉₇	—	2	4	no	—
122	C₂×D₁₉₇	—	4	4	no	—
123	C₄×C₁₉₇	—	2	4	no	—
124	C₄×C₁₉₇	—	2	2	no	—
125	C₄×C₁₉₇	—	2	2	no	—
126	C₄×C₁₉₇	C₂×#41	1	2	yes	—
127	C₄×C₁₉₇	C₂×#41	1	2	yes	—
128	Dic₁₉₇	—	2	2	no	—
129	Dic₁₉₇	—	2	4	no	—
130	Dic₁₉₇	—	2	8	no	—
131	C₂×C₁₉₇	—	2	2	no	—
132	C₂×C₁₉₇	—	4	4	no	—
133	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
134	C₂×C₁₉₇	#585	1	2	yes	—
135	D₁₉₇	—	4	2	no	—
136	C₁₉₇	#3455	1	1	yes	characteristic, complemented, Sylow
137	#3455	—	197	1	no	maximal, Sylow
138	#542	—	197	4	no	—
139	#541	—	197	2	no	—
140	#540	—	197	2	no	—
141	#539	—	197	4	no	—
142	#538	—	197	4	no	—
143	#533	—	197	1	no	—
144	#532	D₁₉₇	1	1	yes	characteristic, complemented
145	#529	—	197	8	no	—
146	#525	—	197	2	no	—
147	C₄×#41	—	197	2	no	—
148	C₂×#117	—	197	1	no	—
149	C₂×#41	C₂×D₁₉₇	1	2	yes	—
150	C₂³×C₄	—	197	1	no	—
151	C₄⋊Q₈	—	197	2	no	—
152	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
153	#127	C₂×D₁₉₇	1	1	yes	characteristic, complemented
154	#126	—	197	8	no	—
155	#126	—	197	8	no	—
156	#125	—	197	8	no	—
157	#125	—	197	2	no	—
158	#124	—	197	8	no	—
159	#124	C₂×D₁₉₇	1	2	yes	complemented
160	#123	—	197	4	no	—
161	#123	—	197	8	no	—
162	#123	—	197	8	no	—
163	C₂²⋊Q₈	—	197	4	no	—
164	C₂²⋊Q₈	—	197	4	no	—
165	C₂²⋊Q₈	—	197	8	no	—
166	#121	—	197	4	no	—
167	#121	—	197	8	no	—
168	#121	C₂×D₁₉₇	1	4	yes	—
169	#120	—	197	4	no	—
170	C₄×Q₈	—	197	4	no	—
171	C₄×Q₈	—	197	4	no	—
172	C₄×D₄	—	197	8	no	—
173	C₄×D₄	—	197	4	no	—
174	C₄×D₄	—	197	8	no	—
175	C₄×D₄	C₂×D₁₉₇	1	4	yes	complemented
176	#117	—	197	2	no	—
177	#117	—	197	4	no	—
178	#117	—	197	4	no	—
179	#117	—	197	2	no	—
180	#117	—	197	2	no	—
181	C₂×C₄⋊C₄	—	197	2	no	—
182	C₂×C₂²⋊C₄	—	197	2	no	—
183	C₂×C₄²	—	197	1	no	—
184	C₂×C₄²	—	197	4	no	—
185	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
186	C₂⁴	—	197	1	no	—
187	#41	—	2	8	no	—
188	C₂×Q₈	—	197	4	no	—
189	C₂×Q₈	C₂²×D₁₉₇	1	2	yes	—
190	C₂×D₄	—	197	8	no	—
191	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
192	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
193	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
194	C₂²×C₄	—	394	4	no	—
195	C₂²×C₄	—	197	2	no	—
196	C₂²×C₄	—	197	8	no	—
197	C₂²×C₄	—	197	1	no	—
198	C₂²×C₄	—	197	2	no	—
199	C₂²×C₄	—	197	2	no	—
200	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
201	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
203	C₄⋊C₄	—	197	4	no	—
204	C₄⋊C₄	—	197	4	no	—
205	C₄⋊C₄	—	197	8	no	—
206	C₄⋊C₄	—	197	8	no	—
207	C₄⋊C₄	—	197	4	no	—
208	C₄⋊C₄	—	197	8	no	—
209	C₄⋊C₄	—	197	4	no	—
210	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
211	C₂²⋊C₄	—	197	4	no	—
212	C₂²⋊C₄	—	197	8	no	—
213	C₂²⋊C₄	—	197	8	no	—
214	C₂²⋊C₄	—	197	4	no	—
215	C₂²⋊C₄	—	197	4	no	—
216	C₂²⋊C₄	—	197	8	no	—
217	C₂²⋊C₄	C₂²×D₁₉₇	1	8	yes	—
218	C₄²	—	197	2	no	—
219	C₄²	—	197	8	no	—
220	C₄²	—	197	2	no	—
221	C₄²	—	197	4	no	—
222	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
223	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
224	C₄²	C₂²×D₁₉₇	1	2	yes	—
225	C₂³	—	788	2	no	—
226	C₂³	—	394	2	no	—
227	C₂³	—	197	2	no	—
228	C₂³	C₂³×D₁₉₇	1	4	yes	—
229	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	788	4	no	—
232	D₄	—	2	8	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	788	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	788	2	no	—
239	C₂×C₄	—	788	8	no	—
240	C₂×C₄	—	394	2	no	—
241	C₂×C₄	—	394	4	no	—
242	C₂×C₄	—	394	4	no	—
243	C₂×C₄	—	197	4	no	—
244	C₂×C₄	—	197	8	no	—
245	C₂×C₄	—	197	2	no	—
246	C₂×C₄	#15691357	1	4	yes	—
247	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
248	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
249	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
250	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	788	4	no	—
256	C₂²	—	394	2	no	—
257	C₂²	—	394	4	no	—
258	C₂²	C₂×#15691357	1	2	yes	—
259	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	394	2	no	—
264	C₄	—	394	4	no	—
265	C₄	—	394	8	no	—
266	C₄	C₂×#15691357	1	2	yes	—
267	C₄	C₂×#15691357	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	—	788	2	no	—
271	C₂	C₂²×#15691357	1	1	yes	characteristic
272	C₂	#27161915	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉₇
commutator, socle, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	#532×C₁₉₇	C₂
hypercenter	#532	D₁₉₇
hypocenter	C₁₉₇	#3455
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #27161698	C₂	semidirect
2. #27161690	C₂	semidirect
3. #27161683	C₂	semidirect
4. #27161674	C₂	semidirect
5. #27161666	C₂	semidirect
6. #27161636	C₂	semidirect
7. #27161609	C₂	semidirect
8. #27161586	C₂	semidirect
9. #41×Dic₁₉₇	C₂	semidirect
10. C₂×#20263283	C₂	semidirect
11. #532×C₁₉₇	C₂	semidirect
13. #20263326	C₂²	semidirect
15. #20263319	C₂²	semidirect
16. #20263318	C₂²	semidirect
17. #20263316	C₂²	semidirect
21. #20263307	C₂²	semidirect
23. #20263303	C₂²	semidirect
28. Q₈×Dic₁₉₇	C₂²	semidirect
29. C₄×Dic₃₉₄	C₂²	semidirect
31. D₄×Dic₁₉₇	C₂²	semidirect
34. #20263285	C₂²	semidirect
41. C₂×C₄×Dic₁₉₇	C₂²	semidirect
43. C₄⋊Q₈×C₁₉₇	C₂²	semidirect
44. #127×C₁₉₇	C₂²	semidirect
45. #124×C₁₉₇	C₂²	semidirect
47. C₄×D₄×C₁₉₇	C₂²	semidirect
136. C₁₉₇	#3455	semidirect
144. #532	D₁₉₇	semidirect
152. C₄⋊Q₈	C₂×D₁₉₇	semidirect
153. #127	C₂×D₁₉₇	semidirect
159. #124	C₂×D₁₉₇	semidirect
175. C₄×D₄	C₂×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161698	2
#27161690	2
#27161683	2
#27161674	2
#27161666	2
#27161636	2
#27161609	2
#27161586	2
#41×Dic₁₉₇	2
C₂×#20263283	2
#532×C₁₉₇	2
#3455	197