Subgroups — C₂×#27161704 #38756106

Order 25216.

1296 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161704	C₂	1	16	yes	maximal, direct product factor
2	C₂×#20263306	C₂	1	2	yes	maximal, complemented
3	C₂×#20263305	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#20263304	C₂	1	2	yes	maximal, complemented
5	C₂×#20263304	C₂	1	2	yes	maximal, complemented
6	C₂×#20263303	C₂	1	2	yes	maximal, complemented
7	C₂×#20263302	C₂	1	2	yes	maximal, complemented
8	C₂²×#15691335	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊C₄×D₁₉₇	C₂	1	2	yes	maximal
10	C₂×C₂²⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₉₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₃₉₄	C₂²	1	2	yes	complemented
13	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic, complemented
14	C₂³×Dic₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
17	#20263306	C₂²	1	16	yes	complemented
18	#20263305	C₂²	1	8	yes	complemented
19	#20263304	C₂²	1	16	yes	complemented
20	#20263304	C₂²	1	16	yes	complemented
21	#20263303	C₂²	1	16	yes	complemented
22	#20263302	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	complemented
24	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×#15691337	C₂²	1	2	yes	complemented
26	C₂×#15691337	C₂²	1	2	yes	complemented
27	C₂×#15691337	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
29	C₂×#15691335	C₂²	1	8	yes	—
30	C₂×#15691335	C₂²	1	2	yes	—
31	C₂×#15691335	C₂²	1	2	yes	—
32	C₂×#15691335	C₂²	1	2	yes	—
33	C₂×#15691335	C₂²	1	2	yes	—
34	C₂²⋊C₄×D₁₉₇	C₂²	1	16	yes	—
35	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₁₉₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	2	yes	—
41	C₂³×D₁₉₇	—	4	2	no	—
42	C₂³×D₁₉₇	—	4	2	no	—
43	C₂³×D₁₉₇	Q₈	1	8	yes	complemented
44	C₂³×D₁₉₇	C₂³	1	2	yes	—
45	C₂³×D₁₉₇	C₂³	1	4	yes	—
46	C₂×Dic₃₉₄	—	4	2	no	—
47	C₂×Dic₃₉₄	—	4	4	no	—
48	C₂×Dic₃₉₄	C₂³	1	8	yes	complemented
49	C₂×Dic₃₉₄	C₂³	1	4	yes	complemented
50	C₂²×Dic₁₉₇	—	4	2	no	—
51	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
52	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
53	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
54	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
55	C₂×C₄×D₁₉₇	—	4	4	no	—
56	C₂×C₄×D₁₉₇	—	4	4	no	—
57	C₂×C₄×D₁₉₇	C₂³	1	8	yes	—
58	C₂×C₄×D₁₉₇	C₂³	1	8	yes	—
59	C₄⋊Dic₁₉₇	C₂³	1	8	yes	complemented
60	C₄⋊Dic₁₉₇	C₂³	1	4	yes	complemented
61	#15691337	C₂³	1	8	yes	complemented
62	#15691337	C₂³	1	8	yes	complemented
63	#15691337	C₂³	1	8	yes	complemented
64	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
65	#15691335	C₂³	1	8	yes	—
66	#15691335	C₂³	1	8	yes	—
67	#15691335	C₂³	1	8	yes	—
68	#15691335	C₂³	1	8	yes	—
69	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₁₉₇	—	4	2	no	—
71	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	complemented
72	C₂²×C₄×C₁₉₇	—	4	2	no	—
73	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
78	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	complemented
79	C₄⋊C₄×C₁₉₇	C₂³	1	8	yes	complemented
80	C₂²⋊C₄×C₁₉₇	C₂³	1	8	yes	—
81	C₂²×D₁₉₇	—	2	4	no	—
82	C₂²×D₁₉₇	—	2	4	no	—
83	C₂²×D₁₉₇	—	4	16	no	—
84	C₂²×D₁₉₇	—	4	4	no	—
85	C₂²×D₁₉₇	—	4	4	no	—
86	C₂²×D₁₉₇	C₂×Q₈	1	8	yes	—
87	C₂²×D₁₉₇	C₂×Q₈	1	16	yes	complemented
88	C₂²×D₁₉₇	C₂⁴	1	4	yes	—
89	Dic₃₉₄	—	4	4	no	—
90	Dic₃₉₄	—	4	8	no	—
91	C₂×Dic₁₉₇	—	2	4	no	—
92	C₂×Dic₁₉₇	—	2	4	no	—
93	C₂×Dic₁₉₇	—	2	4	no	—
94	C₂×Dic₁₉₇	—	4	4	no	—
95	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
96	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
97	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
98	C₄×D₁₉₇	—	4	8	no	—
99	C₄×D₁₉₇	—	4	8	no	—
100	C₂³×C₁₉₇	—	2	2	no	—
101	C₂³×C₁₉₇	—	2	2	no	—
102	C₂³×C₁₉₇	C₂×Q₈	1	4	yes	—
103	C₂³×C₁₉₇	C₂⁴	1	2	yes	—
104	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
105	Q₈×C₁₉₇	—	4	4	no	—
106	C₂×C₄×C₁₉₇	—	2	2	no	—
107	C₂×C₄×C₁₉₇	—	2	2	no	—
108	C₂×C₄×C₁₉₇	—	2	4	no	—
109	C₂×C₄×C₁₉₇	—	2	4	no	—
110	C₂×C₄×C₁₉₇	—	4	4	no	—
111	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
112	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
113	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
115	C₂²×C₁₉₇	—	2	8	no	—
116	C₂²×C₁₉₇	—	2	2	no	—
117	C₂²×C₁₉₇	—	2	2	no	—
118	C₂²×C₁₉₇	C₂²×Q₈	1	4	yes	—
119	C₂²×C₁₉₇	C₂²×Q₈	1	2	yes	—
120	C₂²×C₁₉₇	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₁₉₇	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
123	C₂×D₁₉₇	—	2	16	no	—
124	C₂×D₁₉₇	—	2	4	no	—
125	C₂×D₁₉₇	—	2	4	no	—
126	C₂×D₁₉₇	—	4	16	no	—
127	C₂×D₁₉₇	C₂²×Q₈	1	8	yes	—
128	C₄×C₁₉₇	—	2	2	no	—
129	C₄×C₁₉₇	—	2	2	no	—
130	C₄×C₁₉₇	—	2	4	no	—
131	C₄×C₁₉₇	—	2	4	no	—
132	Dic₁₉₇	—	2	4	no	—
133	Dic₁₉₇	—	2	4	no	—
134	Dic₁₉₇	—	2	4	no	—
135	C₂×C₁₉₇	—	2	4	no	—
136	C₂×C₁₉₇	#543	1	4	yes	complemented
137	C₂×C₁₉₇	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₁₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₁₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₁₉₇	—	2	8	no	—
141	C₁₉₇	C₂×#543	1	1	yes	characteristic, complemented, Sylow
142	C₂×#543	—	197	1	no	maximal, Sylow
143	#543	—	197	16	no	—
144	C₂×C₂²⋊Q₈	—	197	2	no	—
145	C₂×C₂²⋊Q₈	—	197	1	no	—
146	C₂×C₂²⋊Q₈	—	197	2	no	—
147	C₂×C₂²⋊Q₈	—	197	2	no	—
148	C₂×C₂²⋊Q₈	—	197	2	no	—
149	C₂×C₂²⋊Q₈	—	197	2	no	—
150	C₂×C₂²⋊Q₈	D₁₉₇	1	1	yes	characteristic, complemented
151	C₂²×C₂²⋊C₄	—	197	1	no	—
152	C₂²×C₂²⋊C₄	—	197	2	no	—
153	C₂⁵	—	197	1	no	—
154	C₂²×Q₈	—	197	1	no	—
155	C₂²×Q₈	—	197	2	no	—
156	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
157	C₂³×C₄	—	197	1	no	—
158	C₂³×C₄	—	197	2	no	—
159	C₂³×C₄	—	197	2	no	—
160	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
161	C₂²⋊Q₈	—	197	16	no	—
162	C₂²⋊Q₈	—	197	8	no	—
163	C₂²⋊Q₈	—	197	16	no	—
164	C₂²⋊Q₈	—	197	16	no	—
165	C₂²⋊Q₈	—	197	16	no	—
166	C₂²⋊Q₈	—	197	16	no	—
167	C₂²⋊Q₈	C₂×D₁₉₇	1	8	yes	complemented
168	C₂×C₄⋊C₄	—	197	2	no	—
169	C₂×C₄⋊C₄	—	197	2	no	—
170	C₂×C₄⋊C₄	—	197	2	no	—
171	C₂×C₄⋊C₄	—	197	1	no	—
172	C₂×C₄⋊C₄	—	197	2	no	—
173	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic, complemented
174	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	complemented
175	C₂×C₂²⋊C₄	—	197	2	no	—
176	C₂×C₂²⋊C₄	—	197	2	no	—
177	C₂×C₂²⋊C₄	—	197	2	no	—
178	C₂×C₂²⋊C₄	—	197	2	no	—
179	C₂×C₂²⋊C₄	—	197	2	no	—
180	C₂×C₂²⋊C₄	—	197	8	no	—
181	C₂×C₂²⋊C₄	—	197	16	no	—
182	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	2	yes	—
183	C₂⁴	—	788	2	no	—
184	C₂⁴	—	788	2	no	—
185	C₂⁴	—	197	8	no	—
186	C₂⁴	—	197	4	no	—
187	C₂⁴	—	197	2	no	—
188	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	788	2	no	—
191	C₂×Q₈	—	788	4	no	—
192	C₂×Q₈	—	197	4	no	—
193	C₂×Q₈	—	197	8	no	—
194	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	complemented
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	788	4	no	—
197	C₂²×C₄	—	788	4	no	—
198	C₂²×C₄	—	788	2	no	—
199	C₂²×C₄	—	197	4	no	—
200	C₂²×C₄	—	197	8	no	—
201	C₂²×C₄	—	197	8	no	—
202	C₂²×C₄	—	197	2	no	—
203	C₂²×C₄	—	197	2	no	—
204	C₂²×C₄	—	197	2	no	—
205	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
210	C₄⋊C₄	—	197	8	no	—
211	C₄⋊C₄	—	197	8	no	—
212	C₄⋊C₄	—	197	8	no	—
213	C₄⋊C₄	—	197	4	no	—
214	C₄⋊C₄	—	197	8	no	—
215	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	complemented
216	C₄⋊C₄	C₂²×D₁₉₇	1	8	yes	complemented
217	C₂²⋊C₄	—	197	8	no	—
218	C₂²⋊C₄	—	197	8	no	—
219	C₂²⋊C₄	—	197	8	no	—
220	C₂²⋊C₄	—	197	8	no	—
221	C₂²⋊C₄	—	197	8	no	—
222	C₂²⋊C₄	C₂²×D₁₉₇	1	8	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	788	4	no	—
226	C₂³	—	788	4	no	—
227	C₂³	—	788	16	no	—
228	C₂³	—	394	4	no	—
229	C₂³	—	394	4	no	—
230	C₂³	—	197	16	no	—
231	C₂³	—	197	8	no	—
232	C₂³	—	197	4	no	—
233	C₂³	Q₈×D₁₉₇	1	4	yes	—
234	C₂³	C₂³×D₁₉₇	1	2	yes	—
235	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	788	4	no	—
238	Q₈	—	788	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	788	8	no	—
245	C₂×C₄	—	788	8	no	—
246	C₂×C₄	—	788	4	no	—
247	C₂×C₄	—	394	4	no	—
248	C₂×C₄	—	394	4	no	—
249	C₂×C₄	—	394	4	no	—
250	C₂×C₄	—	197	4	no	—
251	C₂×C₄	—	197	4	no	—
252	C₂×C₄	—	197	4	no	—
253	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
254	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
255	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
256	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	788	16	no	—
261	C₂²	—	394	4	no	—
262	C₂²	—	394	4	no	—
263	C₂²	—	394	16	no	—
264	C₂²	—	197	8	no	—
265	C₂²	C₂×Q₈×D₁₉₇	1	4	yes	—
266	C₂²	C₂×Q₈×D₁₉₇	1	2	yes	—
267	C₂²	#20263371	1	2	yes	—
268	C₂²	#20263372	1	2	yes	—
269	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	394	4	no	—
275	C₄	—	394	4	no	—
276	C₄	—	394	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	394	8	no	—
279	C₂	#27161704	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₉₇	1	1	yes	characteristic
281	C₂	C₂×#20263371	1	1	yes	characteristic
282	C₂	C₂×#20263372	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	C₂×C₂²⋊Q₈×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂⁴×D₁₉₇	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#543
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #27161704	C₂	direct (N × H)
2. C₂×#20263306	C₂	semidirect
3. C₂×#20263305	C₂	semidirect
4. C₂×#20263304	C₂	semidirect
5. C₂×#20263304	C₂	semidirect
6. C₂×#20263303	C₂	semidirect
7. C₂×#20263302	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₁₉₇	C₂	semidirect
12. C₂²×Dic₃₉₄	C₂²	semidirect
13. C₂²×Dic₃₉₄	C₂²	semidirect
17. #20263306	C₂²	semidirect
18. #20263305	C₂²	semidirect
19. #20263304	C₂²	semidirect
20. #20263304	C₂²	semidirect
21. #20263303	C₂²	semidirect
22. #20263302	C₂²	semidirect
23. C₂×C₄⋊Dic₁₉₇	C₂²	semidirect
24. C₂×C₄⋊Dic₁₉₇	C₂²	semidirect
25. C₂×#15691337	C₂²	semidirect
26. C₂×#15691337	C₂²	semidirect
27. C₂×#15691337	C₂²	semidirect
35. C₂²×Q₈×C₁₉₇	C₂²	semidirect
37. C₂²⋊Q₈×C₁₉₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₁₉₇	C₂²	semidirect
39. C₂×C₄⋊C₄×C₁₉₇	C₂²	semidirect
43. C₂³×D₁₉₇	Q₈	semidirect
48. C₂×Dic₃₉₄	C₂³	semidirect
49. C₂×Dic₃₉₄	C₂³	semidirect
59. C₄⋊Dic₁₉₇	C₂³	semidirect
60. C₄⋊Dic₁₉₇	C₂³	semidirect
61. #15691337	C₂³	semidirect
62. #15691337	C₂³	semidirect
63. #15691337	C₂³	semidirect
71. C₂×Q₈×C₁₉₇	C₂³	semidirect
78. C₄⋊C₄×C₁₉₇	C₂³	semidirect
79. C₄⋊C₄×C₁₉₇	C₂³	semidirect
87. C₂²×D₁₉₇	C₂×Q₈	semidirect
136. C₂×C₁₉₇	#543	semidirect
141. C₁₉₇	C₂×#543	semidirect
150. C₂×C₂²⋊Q₈	D₁₉₇	semidirect
156. C₂²×Q₈	C₂×D₁₉₇	semidirect
167. C₂²⋊Q₈	C₂×D₁₉₇	semidirect
173. C₂×C₄⋊C₄	C₂×D₁₉₇	semidirect
174. C₂×C₄⋊C₄	C₂×D₁₉₇	semidirect
194. C₂×Q₈	C₂²×D₁₉₇	semidirect
215. C₄⋊C₄	C₂²×D₁₉₇	semidirect
216. C₄⋊C₄	C₂²×D₁₉₇	semidirect
279. C₂	#27161704	direct (N × H)

Maximal subgroups

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

subgroup	index
#27161704	2
C₂×#20263306	2
C₂×#20263305	2
C₂×#20263304	2
C₂×#20263303	2
C₂×#20263302	2
C₂²×#15691335	2
C₂×C₂²⋊C₄×D₁₉₇	2
C₂×C₂²⋊Q₈×C₁₉₇	2
C₂×#543	197

Subgroups — C2×#27161704 #38756106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161704 #38756106