Subgroups — C₂×#20261325 #27155079

Order 12416.

1336 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261325	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15690805	C₂	1	2	yes	maximal, complemented
3	C₂×#15690799	C₂	1	2	yes	maximal, complemented
4	C₂×#15690795	C₂	1	2	yes	maximal, complemented
5	C₂×#15690785	C₂	1	2	yes	maximal, complemented
6	C₂×#15690784	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15690780	C₂	1	2	yes	maximal, complemented
8	C₂×#15690770	C₂	1	2	yes	maximal, complemented
9	C₂²×#13095488	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₉₇	C₂²	1	1	yes	characteristic
12	C₂²×D₃₈₈	C₂²	1	2	yes	—
13	C₂²×D₃₈₈	C₂²	1	2	yes	—
14	C₂²×D₃₈₈	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₂²⋊D₉₇	C₂²	1	2	yes	—
16	C₂³×Dic₉₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₉₇	C₂²	1	2	yes	complemented
18	#15690805	C₂²	1	16	yes	complemented
19	#15690799	C₂²	1	16	yes	complemented
20	#15690795	C₂²	1	16	yes	complemented
21	#15690785	C₂²	1	16	yes	complemented
22	#15690784	C₂²	1	8	yes	complemented
23	#15690780	C₂²	1	16	yes	complemented
24	#15690770	C₂²	1	16	yes	complemented
25	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×#13095490	C₂²	1	2	yes	complemented
27	C₂×#13095488	C₂²	1	2	yes	—
28	C₂×#13095488	C₂²	1	2	yes	—
29	C₂×#13095488	C₂²	1	2	yes	—
30	C₂×#13095488	C₂²	1	2	yes	—
31	C₂×#13095488	C₂²	1	8	yes	complemented
32	C₂×#13095488	C₂²	1	2	yes	complemented
33	C₂×#13095488	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₉₇	C₂²	1	2	yes	complemented
35	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₉₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	complemented
39	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
41	C₂³×D₉₇	—	2	4	no	—
42	C₂³×D₉₇	—	4	2	no	—
43	C₂³×D₉₇	—	4	2	no	—
44	C₂³×D₉₇	C₂³	1	2	yes	—
45	C₂³×D₉₇	C₂³	1	2	yes	—
46	C₂³×D₉₇	C₂³	1	4	yes	—
47	C₂×D₃₈₈	—	4	4	no	—
48	C₂×D₃₈₈	—	4	4	no	—
49	C₂×D₃₈₈	—	4	2	no	—
50	C₂×D₃₈₈	C₂³	1	8	yes	—
51	C₂×D₃₈₈	C₂³	1	8	yes	—
52	C₂×D₃₈₈	C₂³	1	4	yes	complemented
53	C₂×C₂²⋊D₉₇	—	2	8	no	—
54	C₂×C₂²⋊D₉₇	C₂³	1	8	yes	—
55	C₂²×Dic₉₇	—	4	2	no	—
56	C₂²×Dic₉₇	C₂³	1	2	yes	—
57	C₂²×Dic₉₇	C₂³	1	2	yes	—
58	C₂²×Dic₉₇	C₂³	1	4	yes	—
59	C₂×C₄×D₉₇	—	4	4	no	—
60	C₂×C₄×D₉₇	C₂³	1	8	yes	complemented
61	C₄⋊Dic₉₇	C₂³	1	4	yes	complemented
62	#13095490	C₂³	1	8	yes	complemented
63	#13095488	C₂³	1	8	yes	—
64	#13095488	C₂³	1	8	yes	—
65	#13095488	C₂³	1	8	yes	—
66	#13095488	C₂³	1	8	yes	—
67	#13095488	C₂³	1	8	yes	complemented
68	#13095488	C₂³	1	8	yes	complemented
69	C₄×Dic₉₇	C₂³	1	8	yes	complemented
70	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
71	C₂×Q₈×C₉₇	—	4	2	no	—
72	C₂×Q₈×C₉₇	C₂³	1	4	yes	complemented
73	C₂²×C₄×C₉₇	—	4	2	no	—
74	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
75	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
78	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
79	C₄⋊C₄×C₉₇	C₂³	1	8	yes	complemented
80	C₄⋊C₄×C₉₇	C₂³	1	4	yes	complemented
81	C₂²⋊C₄×C₉₇	C₂³	1	8	yes	—
82	C₂³×C₉₇	—	2	2	no	—
83	C₂³×C₉₇	—	2	2	no	—
84	C₂³×C₉₇	#41	1	4	yes	—
85	C₂³×C₉₇	C₂⁴	1	2	yes	—
86	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
87	C₂²×D₉₇	—	2	4	no	—
88	C₂²×D₉₇	—	2	8	no	—
89	C₂²×D₉₇	—	2	4	no	—
90	C₂²×D₉₇	—	2	4	no	—
91	C₂²×D₉₇	—	2	4	no	—
92	C₂²×D₉₇	—	2	4	no	—
93	C₂²×D₉₇	—	2	4	no	—
94	C₂²×D₉₇	—	4	4	no	—
95	C₂²×D₉₇	—	4	4	no	—
96	C₂²×D₉₇	—	4	16	no	—
97	C₂²×D₉₇	C₂⁴	1	4	yes	—
98	C₂²×D₉₇	C₂⁴	1	4	yes	—
99	Q₈×C₉₇	—	4	4	no	—
100	C₂×C₄×C₉₇	—	2	2	no	—
101	C₂×C₄×C₉₇	—	2	2	no	—
102	C₂×C₄×C₉₇	—	2	4	no	—
103	C₂×C₄×C₉₇	—	2	4	no	—
104	C₂×C₄×C₉₇	—	4	4	no	—
105	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
106	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
107	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
108	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
109	C₂²⋊D₉₇	—	2	16	no	—
110	C₂×Dic₉₇	—	2	4	no	—
111	C₂×Dic₉₇	—	2	4	no	—
112	C₂×Dic₉₇	—	4	4	no	—
113	C₂×Dic₉₇	C₂⁴	1	4	yes	—
114	C₂×Dic₉₇	C₂⁴	1	4	yes	—
115	D₃₈₈	—	4	8	no	—
116	D₃₈₈	—	4	8	no	—
117	D₃₈₈	—	4	4	no	—
118	C₄×D₉₇	—	4	8	no	—
119	C₂²×C₉₇	—	2	8	no	—
120	C₂²×C₉₇	—	2	2	no	—
121	C₂²×C₉₇	—	2	2	no	—
122	C₂²×C₉₇	C₂×#41	1	4	yes	—
123	C₂²×C₉₇	C₂×#41	1	2	yes	—
124	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
125	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
126	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
127	C₂×D₉₇	—	2	4	no	—
128	C₂×D₉₇	—	2	4	no	—
129	C₂×D₉₇	—	2	4	no	—
130	C₂×D₉₇	—	2	4	no	—
131	C₂×D₉₇	—	2	4	no	—
132	C₂×D₉₇	—	2	4	no	—
133	C₂×D₉₇	—	4	16	no	—
134	C₂×D₉₇	—	4	8	no	—
135	C₂×D₉₇	—	4	8	no	—
136	C₄×C₉₇	—	2	4	no	—
137	C₄×C₉₇	—	2	2	no	—
138	C₄×C₉₇	—	2	4	no	—
139	C₄×C₉₇	—	2	2	no	—
140	Dic₉₇	—	2	4	no	—
141	Dic₉₇	—	2	4	no	—
142	C₂×C₉₇	—	2	4	no	—
143	C₂×C₉₇	#538	1	4	yes	complemented
144	C₂×C₉₇	C₂²×#41	1	1	yes	characteristic
145	C₂×C₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	D₉₇	—	4	4	no	—
148	D₉₇	—	4	4	no	—
149	C₂×#538	—	97	1	no	maximal, Sylow
150	C₉₇	C₂×#538	1	1	yes	characteristic, complemented, Sylow
151	#538	—	97	16	no	—
152	C₂×#126	—	97	2	no	—
153	C₂×#124	—	97	2	no	—
154	C₂×#123	—	97	2	no	—
155	C₂×C₂²⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
156	C₂×#121	—	97	1	no	—
157	C₂×#121	—	97	2	no	—
158	C₂×#120	—	97	2	no	—
159	C₂×C₄×D₄	—	97	2	no	—
160	C₂²×C₂²⋊C₄	—	97	1	no	—
161	C₂⁵	—	97	1	no	—
162	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	97	2	no	—
164	C₂²×D₄	—	97	2	no	—
165	C₂²×D₄	—	97	2	no	—
166	C₂²×D₄	—	97	1	no	—
167	C₂³×C₄	—	97	1	no	—
168	C₂³×C₄	—	97	2	no	—
169	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
170	#126	—	97	16	no	—
171	#124	—	97	16	no	—
172	#123	—	97	16	no	—
173	C₂²⋊Q₈	C₂×D₉₇	1	8	yes	complemented
174	#121	—	97	8	no	—
175	#121	—	97	16	no	—
176	#120	—	97	16	no	—
177	C₄×D₄	—	97	16	no	—
178	C₂×C₄⋊C₄	—	97	2	no	—
179	C₂×C₄⋊C₄	—	97	1	no	—
180	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	complemented
181	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
182	C₂×C₂²⋊C₄	—	97	2	no	—
183	C₂×C₂²⋊C₄	—	97	2	no	—
184	C₂×C₂²⋊C₄	—	97	8	no	—
185	C₂×C₂²⋊C₄	—	97	2	no	—
186	C₂×C₂²⋊C₄	—	97	2	no	—
187	C₂×C₂²⋊C₄	—	97	2	no	—
188	C₂×C₂²⋊C₄	—	97	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
190	C₂×C₄²	—	97	2	no	—
191	C₂⁴	—	97	4	no	—
192	C₂⁴	—	97	2	no	—
193	C₂⁴	—	97	2	no	—
194	C₂⁴	—	388	2	no	—
195	C₂⁴	—	388	2	no	—
196	C₂⁴	—	194	4	no	—
197	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	C₂²×D₉₇	1	4	yes	complemented
200	C₂×D₄	—	97	8	no	—
201	C₂×D₄	—	97	8	no	—
202	C₂×D₄	—	97	8	no	—
203	C₂×D₄	—	97	4	no	—
204	C₂×D₄	—	388	2	no	—
205	C₂×D₄	—	388	4	no	—
206	C₂×D₄	—	388	4	no	—
207	C₂×D₄	—	194	8	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	97	4	no	—
210	C₂²×C₄	—	97	2	no	—
211	C₂²×C₄	—	97	2	no	—
212	C₂²×C₄	—	97	8	no	—
213	C₂²×C₄	—	388	4	no	—
214	C₂²×C₄	—	388	2	no	—
215	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
216	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
219	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
220	C₄⋊C₄	—	97	8	no	—
221	C₄⋊C₄	—	97	4	no	—
222	C₄⋊C₄	C₂²×D₉₇	1	8	yes	complemented
223	C₄⋊C₄	C₂²×D₉₇	1	4	yes	complemented
224	C₂²⋊C₄	—	97	8	no	—
225	C₂²⋊C₄	—	97	8	no	—
226	C₂²⋊C₄	—	97	8	no	—
227	C₂²⋊C₄	—	97	8	no	—
228	C₂²⋊C₄	—	97	8	no	—
229	C₂²⋊C₄	—	97	8	no	—
230	C₂²⋊C₄	C₂²×D₉₇	1	8	yes	—
231	C₄²	—	97	8	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	97	4	no	—
235	C₂³	—	97	4	no	—
236	C₂³	—	388	16	no	—
237	C₂³	—	388	4	no	—
238	C₂³	—	388	4	no	—
239	C₂³	—	194	4	no	—
240	C₂³	—	194	8	no	—
241	C₂³	—	194	4	no	—
242	C₂³	—	194	4	no	—
243	C₂³	—	194	4	no	—
244	C₂³	—	194	4	no	—
245	C₂³	—	194	4	no	—
246	C₂³	#13095512	1	4	yes	—
247	C₂³	C₂³×D₉₇	1	2	yes	—
248	C₂³	C₂³×D₉₇	1	1	yes	characteristic
249	Q₈	—	4	4	no	—
250	D₄	—	388	4	no	—
251	D₄	—	388	8	no	—
252	D₄	—	388	8	no	—
253	D₄	—	194	16	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	97	4	no	—
260	C₂×C₄	—	97	4	no	—
261	C₂×C₄	—	388	8	no	—
262	C₂×C₄	—	388	4	no	—
263	C₂×C₄	—	194	4	no	—
264	C₂×C₄	—	194	4	no	—
265	C₂×C₄	C₂³×D₉₇	1	4	yes	—
266	C₂×C₄	C₂³×D₉₇	1	2	yes	—
267	C₂×C₄	C₂³×D₉₇	1	2	yes	—
268	C₂×C₄	C₂³×D₉₇	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	388	16	no	—
273	C₂²	—	388	8	no	—
274	C₂²	—	388	8	no	—
275	C₂²	—	194	4	no	—
276	C₂²	—	194	4	no	—
277	C₂²	—	194	4	no	—
278	C₂²	—	194	4	no	—
279	C₂²	—	194	4	no	—
280	C₂²	—	194	4	no	—
281	C₂²	C₂×#13095512	1	4	yes	—
282	C₂²	C₂×#13095512	1	2	yes	—
283	C₂²	#15690855	1	2	yes	—
284	C₂²	#15690856	1	2	yes	—
285	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	194	4	no	—
291	C₄	—	194	4	no	—
292	C₂	—	2	4	no	—
293	C₂	—	388	4	no	—
294	C₂	—	388	4	no	—
295	C₂	#20261325	1	4	yes	direct product factor
296	C₂	C₂²×#13095512	1	1	yes	characteristic
297	C₂	C₂×#15690855	1	1	yes	characteristic
298	C₂	C₂×#15690856	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉₇
commutator, cosocle	C₂²×C₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₉₇
Fitting	C₂×C₂²⋊Q₈×C₉₇	C₂
socle	C₂³×C₉₇	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₉₇
hypocenter	C₉₇	C₂×#538
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20261325	C₂	direct (N × H)
2. C₂×#15690805	C₂	semidirect
3. C₂×#15690799	C₂	semidirect
4. C₂×#15690795	C₂	semidirect
5. C₂×#15690785	C₂	semidirect
6. C₂×#15690784	C₂	semidirect
7. C₂×#15690780	C₂	semidirect
8. C₂×#15690770	C₂	semidirect
9. C₂²×#13095488	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₉₇	C₂	semidirect
14. C₂²×D₃₈₈	C₂²	semidirect
17. C₂²×C₄×D₉₇	C₂²	semidirect
18. #15690805	C₂²	semidirect
19. #15690799	C₂²	semidirect
20. #15690795	C₂²	semidirect
21. #15690785	C₂²	semidirect
22. #15690784	C₂²	semidirect
23. #15690780	C₂²	semidirect
24. #15690770	C₂²	semidirect
25. C₂×C₄⋊Dic₉₇	C₂²	semidirect
26. C₂×#13095490	C₂²	semidirect
31. C₂×#13095488	C₂²	semidirect
32. C₂×#13095488	C₂²	semidirect
33. C₂×#13095488	C₂²	semidirect
34. C₂×C₄×Dic₉₇	C₂²	semidirect
35. C₂²×Q₈×C₉₇	C₂²	semidirect
37. C₂²⋊Q₈×C₉₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₉₇	C₂²	semidirect
39. C₂×C₄⋊C₄×C₉₇	C₂²	semidirect
52. C₂×D₃₈₈	C₂³	semidirect
60. C₂×C₄×D₉₇	C₂³	semidirect
61. C₄⋊Dic₉₇	C₂³	semidirect
62. #13095490	C₂³	semidirect
67. #13095488	C₂³	semidirect
68. #13095488	C₂³	semidirect
69. C₄×Dic₉₇	C₂³	semidirect
72. C₂×Q₈×C₉₇	C₂³	semidirect
79. C₄⋊C₄×C₉₇	C₂³	semidirect
80. C₄⋊C₄×C₉₇	C₂³	semidirect
143. C₂×C₉₇	#538	semidirect
150. C₉₇	C₂×#538	semidirect
155. C₂×C₂²⋊Q₈	D₉₇	semidirect
162. C₂²×Q₈	C₂×D₉₇	semidirect
173. C₂²⋊Q₈	C₂×D₉₇	semidirect
180. C₂×C₄⋊C₄	C₂×D₉₇	semidirect
181. C₂×C₄⋊C₄	C₂×D₉₇	semidirect
199. C₂×Q₈	C₂²×D₉₇	semidirect
222. C₄⋊C₄	C₂²×D₉₇	semidirect
223. C₄⋊C₄	C₂²×D₉₇	semidirect
295. C₂	#20261325	direct (N × H)

Maximal subgroups

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

subgroup	index
#20261325	2
C₂×#15690805	2
C₂×#15690799	2
C₂×#15690795	2
C₂×#15690785	2
C₂×#15690784	2
C₂×#15690780	2
C₂×#15690770	2
C₂²×#13095488	2
C₂×C₂²⋊Q₈×C₉₇	2
C₂×#538	97

Subgroups — C2×#20261325 #27155079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261325 #27155079