Subgroups — C₂×#20261242 #27154974

Order 12416.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261242	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15690791	C₂	1	2	yes	maximal, complemented
4	C₂×#15690786	C₂	1	2	yes	maximal, complemented
5	C₂×#15690781	C₂	1	2	yes	maximal, complemented
6	C₂×#15690778	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₉₇	C₂	1	4	yes	maximal, complemented
8	C₂×#15690769	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₉₇	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₉₇	C₂²	1	4	yes	—
12	C₂³×Dic₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₉₇	—	2	8	no	—
14	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₉₇	C₂²	1	8	yes	complemented
17	#15690791	C₂²	1	16	yes	complemented
18	#15690786	C₂²	1	16	yes	complemented
19	#15690781	C₂²	1	16	yes	complemented
20	#15690778	C₂²	1	16	yes	complemented
21	C₄×C₂²⋊D₉₇	C₂²	1	32	yes	complemented
22	#15690769	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	complemented
24	C₂×#13095490	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊Dic₉₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₉₇	C₂²	1	4	yes	complemented
27	C₂×#13095488	C₂²	1	4	yes	—
28	C₂×C₄×Dic₉₇	C₂²	1	2	yes	complemented
29	C₂×#41×C₉₇	—	2	8	no	—
30	C₂×#41×C₉₇	C₂²	1	8	yes	complemented
31	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
32	C₂²×D₄×C₉₇	C₂²	1	2	yes	—
33	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
34	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₉₇	C₂²	1	2	yes	complemented
36	C₂³×D₉₇	—	2	4	no	—
37	C₂³×D₉₇	—	2	8	no	—
38	C₂³×D₉₇	C₂³	1	4	yes	—
39	C₂³×D₉₇	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₉₇	—	2	16	no	—
41	C₂×C₂²⋊D₉₇	C₂³	1	16	yes	—
42	C₂²×Dic₉₇	—	2	4	no	—
43	C₂²×Dic₉₇	C₂³	1	4	yes	—
44	C₂²×Dic₉₇	C₂³	1	2	yes	—
45	C₂²×Dic₉₇	C₂³	1	4	yes	—
46	C₂×C₄×D₉₇	—	2	8	no	—
47	C₂×C₄×D₉₇	—	2	8	no	—
48	C₂×C₄×D₉₇	—	2	32	no	—
49	C₂×C₄×D₉₇	C₂³	1	8	yes	—
50	C₂×C₄×D₉₇	C₂³	1	8	yes	—
51	C₄⋊Dic₉₇	C₂³	1	8	yes	complemented
52	#13095490	C₂³	1	16	yes	complemented
53	C₂²⋊Dic₉₇	C₂³	1	8	yes	—
54	C₂²⋊Dic₉₇	C₂³	1	16	yes	complemented
55	#13095488	C₂³	1	16	yes	—
56	C₄×Dic₉₇	C₂³	1	8	yes	complemented
57	C₂⁴×C₉₇	C₂³	1	2	yes	—
58	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
59	#41×C₉₇	—	2	32	no	—
60	C₂×Q₈×C₉₇	—	2	4	no	—
61	C₂×Q₈×C₉₇	C₂³	1	4	yes	complemented
62	C₂×D₄×C₉₇	—	2	8	no	—
63	C₂×D₄×C₉₇	—	2	4	no	—
64	C₂×D₄×C₉₇	C₂³	1	8	yes	—
65	C₂×D₄×C₉₇	C₂³	1	4	yes	complemented
66	C₂²×C₄×C₉₇	—	2	8	no	—
67	C₂²×C₄×C₉₇	D₄	1	8	yes	complemented
68	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
69	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
71	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₉₇	C₂³	1	8	yes	complemented
73	C₂³×C₉₇	—	2	4	no	—
74	C₂³×C₉₇	—	2	4	no	—
75	C₂³×C₉₇	—	2	8	no	—
76	C₂³×C₉₇	C₂×D₄	1	4	yes	—
77	C₂³×C₉₇	C₂⁴	1	4	yes	—
78	C₂³×C₉₇	C₂⁴	1	2	yes	—
79	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
80	C₂²×D₉₇	—	2	32	no	—
81	C₂²×D₉₇	—	2	8	no	—
82	C₂²×D₉₇	—	2	4	no	—
83	C₂²×D₉₇	—	2	4	no	—
84	C₂²×D₉₇	—	2	16	no	—
85	C₂²×D₉₇	#41	1	8	yes	complemented
86	C₂²×D₉₇	C₂⁴	1	4	yes	—
87	Q₈×C₉₇	—	2	8	no	—
88	D₄×C₉₇	—	2	16	no	—
89	D₄×C₉₇	—	2	8	no	—
90	C₂×C₄×C₉₇	—	2	4	no	—
91	C₂×C₄×C₉₇	—	2	16	no	—
92	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₉₇	C₂×D₄	1	16	yes	complemented
95	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
96	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
97	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
98	C₂²⋊D₉₇	—	2	32	no	—
99	C₂×Dic₉₇	—	2	8	no	—
100	C₂×Dic₉₇	—	2	4	no	—
101	C₂×Dic₉₇	—	2	8	no	—
102	C₂×Dic₉₇	C₂⁴	1	8	yes	—
103	C₂×Dic₉₇	C₂⁴	1	4	yes	—
104	C₄×D₉₇	—	2	16	no	—
105	C₄×D₉₇	—	2	16	no	—
106	C₂²×C₉₇	—	2	8	no	—
107	C₂²×C₉₇	—	2	4	no	—
108	C₂²×C₉₇	—	2	4	no	—
109	C₂²×C₉₇	—	2	8	no	—
110	C₂²×C₉₇	—	4	8	no	—
111	C₂²×C₉₇	C₂²×D₄	1	4	yes	—
112	C₂²×C₉₇	C₂²×D₄	1	2	yes	—
113	C₂²×C₉₇	C₂×#41	1	4	yes	—
114	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
115	C₂×D₉₇	—	2	32	no	—
116	C₂×D₉₇	—	2	16	no	—
117	C₂×D₉₇	—	2	4	no	—
118	C₂×D₉₇	—	2	4	no	—
119	C₂×D₉₇	C₂×#41	1	8	yes	complemented
120	C₄×C₉₇	—	2	4	no	—
121	C₄×C₉₇	C₂²×D₄	1	4	yes	—
122	C₄×C₉₇	C₂²×D₄	1	4	yes	—
123	Dic₉₇	—	2	4	no	—
124	Dic₉₇	—	2	8	no	—
125	C₂×C₉₇	—	2	4	no	—
126	C₂×C₉₇	—	4	4	no	—
127	C₂×C₉₇	#525	1	4	yes	complemented
128	C₂×C₉₇	C₂³×D₄	1	1	yes	characteristic
129	C₂×C₉₇	C₂²×#41	1	2	yes	—
130	D₉₇	—	2	8	no	—
131	C₂×#525	—	97	1	no	maximal, Sylow
132	C₉₇	C₂×#525	1	1	yes	characteristic, complemented, Sylow
133	C₂²×#41	D₉₇	1	1	yes	characteristic, complemented
134	C₂⁴×C₄	—	97	1	no	—
135	#525	—	97	16	no	—
136	C₂×#123	—	97	2	no	—
137	C₂×C₂²⋊Q₈	—	97	2	no	—
138	C₂×#121	—	97	2	no	—
139	C₂×#120	—	97	2	no	—
140	C₂×C₄×D₄	—	97	4	no	—
141	C₂×#117	—	97	1	no	—
142	C₂⁵	—	97	1	no	—
143	C₂×#41	—	2	8	no	—
144	C₂×#41	C₂×D₉₇	1	8	yes	complemented
145	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
146	C₂²×D₄	—	97	4	no	—
147	C₂²×D₄	C₂×D₉₇	1	2	yes	—
148	C₂²×D₄	C₂×D₉₇	1	1	yes	characteristic, complemented
149	C₂³×C₄	—	97	1	no	—
150	C₂³×C₄	—	97	2	no	—
151	C₂³×C₄	—	97	8	no	—
152	C₂³×C₄	—	97	2	no	—
153	C₂³×C₄	—	194	8	no	—
154	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₉₇	1	2	yes	complemented
156	#123	—	97	16	no	—
157	C₂²⋊Q₈	—	97	16	no	—
158	#121	—	97	16	no	—
159	#120	—	97	16	no	—
160	C₄×D₄	—	97	32	no	—
161	#117	—	97	8	no	—
162	C₂×C₄⋊C₄	—	97	2	no	—
163	C₂×C₄⋊C₄	—	97	4	no	—
164	C₂×C₂²⋊C₄	—	97	2	no	—
165	C₂×C₂²⋊C₄	—	97	4	no	—
166	C₂×C₂²⋊C₄	—	97	4	no	—
167	C₂×C₄²	—	97	2	no	—
168	C₂⁴	—	97	2	no	—
169	C₂⁴	—	97	4	no	—
170	C₂⁴	—	194	8	no	—
171	C₂⁴	—	194	4	no	—
172	C₂⁴	C₂²×D₉₇	1	2	yes	—
173	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
174	#41	—	2	32	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	C₂²×D₉₇	1	4	yes	complemented
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	97	16	no	—
180	C₂×D₄	—	194	16	no	—
181	C₂×D₄	C₂²×D₉₇	1	8	yes	—
182	C₂×D₄	C₂²×D₉₇	1	4	yes	complemented
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	97	4	no	—
185	C₂²×C₄	—	97	2	no	—
186	C₂²×C₄	—	97	4	no	—
187	C₂²×C₄	—	97	8	no	—
188	C₂²×C₄	—	97	8	no	—
189	C₂²×C₄	—	194	4	no	—
190	C₂²×C₄	—	194	8	no	—
191	C₂²×C₄	—	194	8	no	—
192	C₂²×C₄	—	194	32	no	—
193	C₂²×C₄	C₂²⋊D₉₇	1	8	yes	complemented
194	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
195	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
197	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₉₇	1	8	yes	complemented
199	C₄⋊C₄	—	97	16	no	—
200	C₄⋊C₄	—	97	8	no	—
201	C₂²⋊C₄	—	97	8	no	—
202	C₂²⋊C₄	—	97	16	no	—
203	C₂²⋊C₄	—	97	16	no	—
204	C₄²	—	97	8	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	97	8	no	—
209	C₂³	—	97	4	no	—
210	C₂³	—	194	16	no	—
211	C₂³	—	194	32	no	—
212	C₂³	—	194	8	no	—
213	C₂³	—	194	4	no	—
214	C₂³	—	194	4	no	—
215	C₂³	C₂×C₂²⋊D₉₇	1	4	yes	—
216	C₂³	C₂³×D₉₇	1	4	yes	—
217	C₂³	C₂³×D₉₇	1	2	yes	—
218	C₂³	C₂³×D₉₇	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	D₄	—	2	16	no	—
221	D₄	—	2	8	no	—
222	D₄	—	194	32	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	97	8	no	—
226	C₂×C₄	—	97	4	no	—
227	C₂×C₄	—	194	8	no	—
228	C₂×C₄	—	194	4	no	—
229	C₂×C₄	—	194	8	no	—
230	C₂×C₄	—	194	16	no	—
231	C₂×C₄	—	194	16	no	—
232	C₂×C₄	C₂×C₂²⋊D₉₇	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₉₇	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₉₇	1	16	yes	complemented
235	C₂×C₄	C₂³×D₉₇	1	4	yes	—
236	C₂×C₄	C₂³×D₉₇	1	2	yes	—
237	C₂×C₄	C₂³×D₉₇	1	2	yes	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	8	no	—
243	C₂²	—	97	8	no	—
244	C₂²	—	194	32	no	—
245	C₂²	—	194	16	no	—
246	C₂²	—	194	4	no	—
247	C₂²	—	194	4	no	—
248	C₂²	C₂²×C₂²⋊D₉₇	1	4	yes	—
249	C₂²	C₂²×C₂²⋊D₉₇	1	2	yes	—
250	C₂²	#41×D₉₇	1	4	yes	—
251	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	194	4	no	—
254	C₄	—	194	8	no	—
255	C₄	C₂²×C₂²⋊D₉₇	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₉₇	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	4	4	no	—
259	C₂	—	194	8	no	—
260	C₂	#20261242	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₉₇	1	1	yes	characteristic
262	C₂	C₂×#41×D₉₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉₇
commutator, cosocle	C₂²×C₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₉₇
Fitting	C₂²×#41×C₉₇	C₂
socle	C₂³×C₉₇	C₂⁴
hypercenter	C₂²×#41	D₉₇
hypocenter	C₉₇	C₂×#525
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄×D₉₇	C₂	semidirect
2. #20261242	C₂	direct (N × H)
3. C₂×#15690791	C₂	semidirect
4. C₂×#15690786	C₂	semidirect
5. C₂×#15690781	C₂	semidirect
6. C₂×#15690778	C₂	semidirect
7. C₂×C₄×C₂²⋊D₉₇	C₂	semidirect
8. C₂×#15690769	C₂	semidirect
9. C₂²×#41×C₉₇	C₂	semidirect
16. C₂²×C₄×D₉₇	C₂²	semidirect
17. #15690791	C₂²	semidirect
18. #15690786	C₂²	semidirect
19. #15690781	C₂²	semidirect
20. #15690778	C₂²	semidirect
21. C₄×C₂²⋊D₉₇	C₂²	semidirect
22. #15690769	C₂²	semidirect
23. C₂×C₄⋊Dic₉₇	C₂²	semidirect
24. C₂×#13095490	C₂²	semidirect
26. C₂×C₂²⋊Dic₉₇	C₂²	semidirect
28. C₂×C₄×Dic₉₇	C₂²	semidirect
30. C₂×#41×C₉₇	C₂²	semidirect
31. C₂²×Q₈×C₉₇	C₂²	semidirect
33. C₂²×D₄×C₉₇	C₂²	semidirect
35. C₂³×C₄×C₉₇	C₂²	semidirect
51. C₄⋊Dic₉₇	C₂³	semidirect
52. #13095490	C₂³	semidirect
54. C₂²⋊Dic₉₇	C₂³	semidirect
56. C₄×Dic₉₇	C₂³	semidirect
61. C₂×Q₈×C₉₇	C₂³	semidirect
65. C₂×D₄×C₉₇	C₂³	semidirect
67. C₂²×C₄×C₉₇	D₄	semidirect
72. C₂²×C₄×C₉₇	C₂³	semidirect
85. C₂²×D₉₇	#41	semidirect
94. C₂×C₄×C₉₇	C₂×D₄	semidirect
119. C₂×D₉₇	C₂×#41	semidirect
127. C₂×C₉₇	#525	semidirect
132. C₉₇	C₂×#525	semidirect
133. C₂²×#41	D₉₇	semidirect
144. C₂×#41	C₂×D₉₇	semidirect
145. C₂²×Q₈	C₂×D₉₇	semidirect
148. C₂²×D₄	C₂×D₉₇	semidirect
155. C₂³×C₄	C₂×D₉₇	semidirect
176. C₂×Q₈	C₂²×D₉₇	semidirect
182. C₂×D₄	C₂²×D₉₇	semidirect
193. C₂²×C₄	C₂²⋊D₉₇	semidirect
198. C₂²×C₄	C₂²×D₉₇	semidirect
234. C₂×C₄	C₂×C₂²⋊D₉₇	semidirect
260. C₂	#20261242	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₉₇	2
#20261242	2
C₂×#15690791	2
C₂×#15690786	2
C₂×#15690781	2
C₂×#15690778	2
C₂×C₄×C₂²⋊D₉₇	2
C₂×#15690769	2
C₂²×#41×C₉₇	2
C₂×#525	97

Subgroups — C2×#20261242 #27154974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261242 #27154974