Subgroups — C₂×#27161703 #38756093

Order 25216.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161703	C₂	1	16	yes	maximal, direct product factor
2	C₂×#20263321	C₂	1	2	yes	maximal, complemented
3	C₂×#20263315	C₂	1	2	yes	maximal, complemented
4	C₂×#20263314	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#20263311	C₂	1	2	yes	maximal, complemented
6	C₂×#20263306	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#20263302	C₂	1	2	yes	maximal, complemented
8	C₂×#20263301	C₂	1	1	yes	maximal, characteristic
9	C₂×#20263291	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20263288	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#20263283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
14	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
15	C₂²×D₇₈₈	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
18	#20263321	C₂²	1	16	yes	complemented
19	#20263315	C₂²	1	16	yes	complemented
20	#20263314	C₂²	1	8	yes	complemented
21	#20263311	C₂²	1	16	yes	complemented
22	#20263306	C₂²	1	8	yes	complemented
23	#20263302	C₂²	1	16	yes	complemented
24	#20263301	C₂²	1	8	yes	—
25	#20263291	C₂²	1	8	yes	complemented
26	#20263288	C₂²	1	8	yes	complemented
27	#20263283	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
29	C₂×#15691337	C₂²	1	1	yes	characteristic
30	C₂×#15691337	C₂²	1	1	yes	characteristic
31	C₂×#15691337	C₂²	1	2	yes	—
32	C₂×#15691335	C₂²	1	2	yes	—
33	C₂×#15691335	C₂²	1	2	yes	—
34	C₂×#15691335	C₂²	1	2	yes	—
35	C₂×#15691335	C₂²	1	1	yes	characteristic
36	C₂×#15691335	C₂²	1	1	yes	characteristic
37	C₂×#15691335	C₂²	1	2	yes	—
38	C₂×#15691335	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₉₇	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₁₉₇	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
47	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
48	C₂³×D₁₉₇	C₂³	1	2	yes	—
49	C₂×Dic₃₉₄	—	2	4	no	—
50	C₂×Dic₃₉₄	C₂³	1	4	yes	—
51	C₂×D₇₈₈	—	2	4	no	—
52	C₂×D₇₈₈	—	4	4	no	—
53	C₂×D₇₈₈	C₂³	1	4	yes	—
54	C₂×D₇₈₈	C₂³	1	8	yes	—
55	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
57	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₉₇	—	2	4	no	—
60	C₂×C₄×D₁₉₇	—	4	4	no	—
61	C₂×C₄×D₁₉₇	C₂³	1	4	yes	—
62	C₂×C₄×D₁₉₇	C₂³	1	8	yes	—
63	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
64	#15691337	C₂³	1	4	yes	—
65	#15691337	C₂³	1	4	yes	—
66	#15691337	C₂³	1	8	yes	—
67	#15691335	C₂³	1	4	yes	—
68	#15691335	C₂³	1	4	yes	—
69	#15691335	C₂³	1	8	yes	—
70	#15691335	C₂³	1	8	yes	—
71	#15691335	C₂³	1	8	yes	—
72	#15691335	C₂³	1	8	yes	—
73	#15691335	C₂³	1	8	yes	—
74	C₄×Dic₁₉₇	C₂³	1	4	yes	—
75	C₄×Dic₁₉₇	C₂³	1	8	yes	—
76	C₂×Q₈×C₁₉₇	—	4	4	no	—
77	C₂×Q₈×C₁₉₇	C₂³	1	8	yes	—
78	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₉₇	C₂³	1	2	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	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
85	C₄⋊C₄×C₁₉₇	C₂³	1	8	yes	—
86	C₄²×C₁₉₇	C₂³	1	4	yes	—
87	C₂²×D₁₉₇	—	2	2	no	—
88	C₂²×D₁₉₇	—	2	4	no	—
89	C₂²×D₁₉₇	—	2	4	no	—
90	C₂²×D₁₉₇	—	2	2	no	—
91	C₂²×D₁₉₇	—	2	2	no	—
92	C₂²×D₁₉₇	—	2	4	no	—
93	C₂²×D₁₉₇	C₂⁴	1	2	yes	—
94	C₂²×D₁₉₇	C₂⁴	1	4	yes	—
95	Dic₃₉₄	—	2	8	no	—
96	D₇₈₈	—	2	8	no	—
97	D₇₈₈	—	4	8	no	—
98	C₂×Dic₁₉₇	—	2	2	no	—
99	C₂×Dic₁₉₇	—	2	4	no	—
100	C₂×Dic₁₉₇	—	2	2	no	—
101	C₂×Dic₁₉₇	—	2	2	no	—
102	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
103	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
104	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
105	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
106	C₄×D₁₉₇	—	2	8	no	—
107	C₄×D₁₉₇	—	4	8	no	—
108	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
109	Q₈×C₁₉₇	—	4	8	no	—
110	C₂×C₄×C₁₉₇	—	2	2	no	—
111	C₂×C₄×C₁₉₇	—	2	4	no	—
112	C₂×C₄×C₁₉₇	—	2	4	no	—
113	C₂×C₄×C₁₉₇	—	2	2	no	—
114	C₂×C₄×C₁₉₇	#41	1	4	yes	—
115	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
116	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
117	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
118	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
119	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
120	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
121	C₂²×C₁₉₇	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₁₉₇	2₋¹⁺⁴	1	2	yes	—
123	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₉₇	—	2	2	no	—
125	C₂×D₁₉₇	—	2	4	no	—
126	C₂×D₁₉₇	—	2	4	no	—
127	C₂×D₁₉₇	—	2	2	no	—
128	C₂×D₁₉₇	—	2	2	no	—
129	C₂×D₁₉₇	—	2	4	no	—
130	C₂×D₁₉₇	—	4	4	no	—
131	C₂×D₁₉₇	—	4	8	no	—
132	C₄×C₁₉₇	—	2	2	no	—
133	C₄×C₁₉₇	—	2	4	no	—
134	C₄×C₁₉₇	—	2	4	no	—
135	C₄×C₁₉₇	—	2	2	no	—
136	C₄×C₁₉₇	C₂×#41	1	4	yes	—
137	Dic₁₉₇	—	2	2	no	—
138	Dic₁₉₇	—	2	4	no	—
139	Dic₁₉₇	—	2	2	no	—
140	Dic₁₉₇	—	2	2	no	—
141	C₂×C₁₉₇	#542	1	4	yes	complemented
142	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
143	C₂×C₁₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₁₉₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
145	D₁₉₇	—	4	2	no	—
146	D₁₉₇	—	4	4	no	—
147	C₁₉₇	C₂×#542	1	1	yes	characteristic, complemented, Sylow
148	C₂×#542	—	197	1	no	maximal, Sylow
149	#542	—	197	16	no	—
150	C₂×C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic, complemented
151	C₂×#126	—	197	2	no	—
152	C₂×#124	—	197	2	no	—
153	C₂×#124	—	197	1	no	—
154	C₂×#123	—	197	2	no	—
155	C₂×C₂²⋊Q₈	—	197	2	no	—
156	C₂×C₂²⋊Q₈	—	197	1	no	—
157	C₂×#121	—	197	1	no	—
158	C₂×C₄×Q₈	—	197	1	no	—
159	C₂×C₄×D₄	—	197	1	no	—
160	C₂×#117	—	197	1	no	—
161	C₂²×Q₈	—	197	1	no	—
162	C₂²×Q₈	C₂×D₁₉₇	1	2	yes	—
163	C₂²×D₄	—	197	2	no	—
164	C₂²×D₄	—	197	1	no	—
165	C₂³×C₄	—	197	2	no	—
166	C₂³×C₄	—	197	1	no	—
167	C₄⋊Q₈	C₂×D₁₉₇	1	8	yes	complemented
168	#126	—	197	16	no	—
169	#124	—	197	16	no	—
170	#124	—	197	8	no	—
171	#123	—	197	16	no	—
172	C₂²⋊Q₈	—	197	16	no	—
173	C₂²⋊Q₈	—	197	8	no	—
174	#121	—	197	8	no	—
175	C₄×Q₈	—	197	8	no	—
176	C₄×D₄	—	197	8	no	—
177	#117	—	197	8	no	—
178	C₂×C₄⋊C₄	—	197	1	no	—
179	C₂×C₄⋊C₄	—	197	1	no	—
180	C₂×C₄⋊C₄	—	197	2	no	—
181	C₂×C₄⋊C₄	—	197	2	no	—
182	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
185	C₂×C₂²⋊C₄	—	197	1	no	—
186	C₂×C₂²⋊C₄	—	197	1	no	—
187	C₂×C₂²⋊C₄	—	197	2	no	—
188	C₂×C₂²⋊C₄	—	197	2	no	—
189	C₂×C₂²⋊C₄	—	197	2	no	—
190	C₂×C₂²⋊C₄	—	197	2	no	—
191	C₂×C₂²⋊C₄	—	197	2	no	—
192	C₂×C₄²	—	197	1	no	—
193	C₂×C₄²	—	197	2	no	—
194	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
195	C₂⁴	—	197	1	no	—
196	C₂⁴	—	197	2	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	394	4	no	—
199	C₂×Q₈	—	197	4	no	—
200	C₂×Q₈	C₂²×D₁₉₇	1	8	yes	—
201	C₂×D₄	—	788	4	no	—
202	C₂×D₄	—	394	4	no	—
203	C₂×D₄	—	197	8	no	—
204	C₂×D₄	—	197	4	no	—
205	C₂²×C₄	—	788	4	no	—
206	C₂²×C₄	—	394	4	no	—
207	C₂²×C₄	—	197	8	no	—
208	C₂²×C₄	—	197	1	no	—
209	C₂²×C₄	—	197	2	no	—
210	C₂²×C₄	—	197	1	no	—
211	C₂²×C₄	—	197	1	no	—
212	C₂²×C₄	—	197	4	no	—
213	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
214	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
217	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
218	C₄⋊C₄	—	197	4	no	—
219	C₄⋊C₄	—	197	4	no	—
220	C₄⋊C₄	—	197	8	no	—
221	C₄⋊C₄	—	197	8	no	—
222	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
223	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
224	C₄⋊C₄	C₂²×D₁₉₇	1	8	yes	—
225	C₂²⋊C₄	—	197	4	no	—
226	C₂²⋊C₄	—	197	4	no	—
227	C₂²⋊C₄	—	197	8	no	—
228	C₂²⋊C₄	—	197	8	no	—
229	C₂²⋊C₄	—	197	8	no	—
230	C₂²⋊C₄	—	197	8	no	—
231	C₂²⋊C₄	—	197	8	no	—
232	C₄²	—	197	4	no	—
233	C₄²	—	197	8	no	—
234	C₄²	C₂²×D₁₉₇	1	4	yes	—
235	C₂³	—	394	2	no	—
236	C₂³	—	394	4	no	—
237	C₂³	—	394	4	no	—
238	C₂³	—	394	2	no	—
239	C₂³	—	394	2	no	—
240	C₂³	—	394	4	no	—
241	C₂³	—	197	2	no	—
242	C₂³	—	197	4	no	—
243	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
244	Q₈	—	4	8	no	—
245	Q₈	—	394	8	no	—
246	D₄	—	788	8	no	—
247	D₄	—	394	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	788	8	no	—
253	C₂×C₄	—	394	2	no	—
254	C₂×C₄	—	394	4	no	—
255	C₂×C₄	—	394	2	no	—
256	C₂×C₄	—	394	2	no	—
257	C₂×C₄	—	394	8	no	—
258	C₂×C₄	—	197	2	no	—
259	C₂×C₄	—	197	4	no	—
260	C₂×C₄	—	197	2	no	—
261	C₂×C₄	—	197	2	no	—
262	C₂×C₄	#15691359	1	4	yes	—
263	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
264	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
265	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
266	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
267	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
268	C₂²	—	788	4	no	—
269	C₂²	—	788	8	no	—
270	C₂²	—	394	2	no	—
271	C₂²	—	394	4	no	—
272	C₂²	—	394	4	no	—
273	C₂²	—	394	2	no	—
274	C₂²	—	394	2	no	—
275	C₂²	—	394	4	no	—
276	C₂²	C₂×#15691359	1	2	yes	—
277	C₂²	#20263371	1	2	yes	—
278	C₂²	#20263374	1	2	yes	—
279	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	394	2	no	—
285	C₄	—	394	4	no	—
286	C₄	—	394	2	no	—
287	C₄	—	394	2	no	—
288	C₄	C₂×#15691359	1	4	yes	—
289	C₂	—	788	2	no	—
290	C₂	—	788	4	no	—
291	C₂	#27161703	1	4	yes	direct product factor
292	C₂	C₂²×#15691359	1	1	yes	characteristic
293	C₂	C₂×#20263371	1	1	yes	characteristic
294	C₂	C₂×#20263374	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	C₂×C₄⋊Q₈×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂×#20263301	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#542
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #27161703	C₂	direct (N × H)
2. C₂×#20263321	C₂	semidirect
3. C₂×#20263315	C₂	semidirect
4. C₂×#20263314	C₂	semidirect
5. C₂×#20263311	C₂	semidirect
6. C₂×#20263306	C₂	semidirect
7. C₂×#20263302	C₂	semidirect
9. C₂×#20263291	C₂	semidirect
10. C₂×#20263288	C₂	semidirect
11. C₂×#20263283	C₂	semidirect
12. C₂×C₄⋊Q₈×C₁₉₇	C₂	semidirect
18. #20263321	C₂²	semidirect
19. #20263315	C₂²	semidirect
20. #20263314	C₂²	semidirect
21. #20263311	C₂²	semidirect
22. #20263306	C₂²	semidirect
23. #20263302	C₂²	semidirect
25. #20263291	C₂²	semidirect
26. #20263288	C₂²	semidirect
27. #20263283	C₂²	semidirect
42. C₄⋊Q₈×C₁₉₇	C₂²	semidirect
141. C₂×C₁₉₇	#542	semidirect
147. C₁₉₇	C₂×#542	semidirect
150. C₂×C₄⋊Q₈	D₁₉₇	semidirect
167. C₄⋊Q₈	C₂×D₁₉₇	semidirect
291. C₂	#27161703	direct (N × H)

Maximal subgroups

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

subgroup	index
#27161703	2
C₂×#20263321	2
C₂×#20263315	2
C₂×#20263314	2
C₂×#20263311	2
C₂×#20263306	2
C₂×#20263302	2
C₂×#20263301	2
C₂×#20263291	2
C₂×#20263288	2
C₂×#20263283	2
C₂×C₄⋊Q₈×C₁₉₇	2
C₂×#542	197

Subgroups — C2×#27161703 #38756093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161703 #38756093