Subgroups — #27150431

Order 12416.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690817	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260807	C₂	1	4	yes	maximal, complemented
4	D₄×#11715895	C₂	1	4	yes	maximal, complemented
5	#20260762	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260702	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260699	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260691	C₂	1	1	yes	maximal, characteristic, complemented
9	#525×C₉₇	C₂	1	1	yes	maximal, characteristic
10	#15690817	—	2	4	no	—
11	C₂×#13095495	C₂²	1	1	yes	characteristic
12	C₂×#13095495	C₂²	1	1	yes	characteristic
13	C₂×#13095495	C₂²	1	2	yes	complemented
14	C₂²×#11715895	—	2	4	no	—
15	C₂²×#11715895	C₂²	1	1	yes	characteristic
16	C₂²×#11715895	C₂²	1	2	yes	—
17	C₂²×#11715895	C₂²	1	2	yes	—
18	C₂²×#11715895	C₂²	1	1	yes	characteristic
19	C₂²×#11715895	C₂²	1	2	yes	complemented
20	C₄⋊#11715895	C₂²	1	4	yes	complemented
21	C₂²⋊#11715895	C₂²	1	4	yes	—
22	C₂²⋊#11715895	C₂²	1	4	yes	complemented
23	#15690721	C₂²	1	2	yes	complemented
24	C₄×#11715895	C₂²	1	2	yes	complemented
25	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
27	#123×C₉₇	C₄	1	2	yes	—
28	C₂²⋊Q₈×C₉₇	C₄	1	2	yes	—
29	#121×C₉₇	C₄	1	2	yes	—
30	#120×C₉₇	C₄	1	2	yes	—
31	C₄×D₄×C₉₇	C₂²	1	4	yes	—
32	#117×C₉₇	C₂²	1	1	yes	characteristic
33	#13095495	—	2	2	no	—
34	#13095495	—	2	2	no	—
35	#13095495	—	2	4	no	—
36	C₂×#11715895	—	2	2	no	—
37	C₂×#11715895	—	2	4	no	—
38	C₂×#11715895	—	2	4	no	—
39	C₂×#11715895	—	2	4	no	—
40	C₂×#11715895	D₄	1	4	yes	complemented
41	C₂×#11715895	C₂³	1	2	yes	—
42	C₂×#11715895	C₂³	1	2	yes	—
43	C₂×#11715895	C₂³	1	2	yes	—
44	C₂×#11715895	C₂³	1	1	yes	characteristic
45	C₂×#11715895	C₂³	1	1	yes	characteristic
46	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
47	#41×C₉₇	—	2	2	no	—
48	#41×C₉₇	—	2	2	no	—
49	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₉₇	C₂×C₄	1	4	yes	—
51	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
52	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₉₇	—	2	2	no	—
54	C₂²×C₄×C₉₇	—	2	2	no	—
55	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
65	C₄²×C₉₇	C₂³	1	2	yes	—
66	C₂³×C₉₇	—	2	2	no	—
67	C₂³×C₉₇	—	2	4	no	—
68	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
69	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
71	Q₈×C₉₇	—	2	2	no	—
72	D₄×C₉₇	—	2	4	no	—
73	D₄×C₉₇	—	2	8	no	—
74	D₄×C₉₇	—	2	2	no	—
75	C₂×C₄×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	—	2	4	no	—
77	C₂×C₄×C₉₇	—	2	2	no	—
78	C₂×C₄×C₉₇	—	2	2	no	—
79	C₂×C₄×C₉₇	—	2	2	no	—
80	C₂×C₄×C₉₇	—	2	2	no	—
81	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₉₇	#41	1	2	yes	—
87	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
88	#11715895	—	2	2	no	—
89	#11715895	—	2	2	no	—
90	#11715895	—	2	2	no	—
91	#11715895	C₂×D₄	1	2	yes	—
92	#11715895	C₂×D₄	1	2	yes	—
93	C₂²×C₉₇	—	2	8	no	—
94	C₂²×C₉₇	—	2	2	no	—
95	C₂²×C₉₇	—	2	2	no	—
96	C₂²×C₉₇	—	2	2	no	—
97	C₂²×C₉₇	—	2	2	no	—
98	C₂²×C₉₇	—	2	4	no	—
99	C₂²×C₉₇	C₄×D₄	1	2	yes	—
100	C₂²×C₉₇	#131	1	4	yes	complemented
101	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₉₇	—	2	2	no	—
103	C₄×C₉₇	—	2	2	no	—
104	C₄×C₉₇	—	2	4	no	—
105	C₄×C₉₇	C₄×D₄	1	2	yes	—
106	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₉₇	—	2	2	no	—
109	C₂×C₉₇	—	2	4	no	—
110	C₂×C₉₇	—	4	2	no	—
111	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₉₇	C₂×#131	1	2	yes	—
113	#2929	—	97	1	no	maximal, Sylow
114	C₉₇	#2929	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	97	1	no	—
116	C₂³×C₈	—	97	1	no	—
117	#525	D₉₇	1	1	yes	characteristic
118	#435	—	97	4	no	—
119	C₈×D₄	—	97	4	no	—
120	#424	—	97	1	no	—
121	#408	—	97	1	no	—
122	#407	—	97	1	no	—
123	#405	—	97	1	no	—
124	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
126	#131	—	194	4	no	—
127	C₂×M₁₆	—	97	1	no	—
128	C₂×M₁₆	—	97	2	no	—
129	C₂×M₁₆	—	97	1	no	—
130	C₂²×C₈	—	97	1	no	—
131	C₂²×C₈	—	97	1	no	—
132	C₂²×C₈	—	97	2	no	—
133	C₂²×C₈	—	97	2	no	—
134	C₂²×C₈	—	97	2	no	—
135	C₂²×C₈	—	194	4	no	—
136	#123	Dic₉₇	1	2	yes	—
137	C₂²⋊Q₈	Dic₉₇	1	2	yes	—
138	#121	Dic₉₇	1	2	yes	—
139	#120	Dic₉₇	1	2	yes	—
140	C₄×D₄	C₂×D₉₇	1	4	yes	—
141	#117	C₂×D₉₇	1	1	yes	characteristic
142	C₄⋊C₈	—	97	4	no	—
143	C₂²⋊C₈	—	97	4	no	—
144	C₂²⋊C₈	—	97	4	no	—
145	#97	—	97	2	no	—
146	C₄×C₈	—	97	2	no	—
147	C₂⁴	C₂×Dic₉₇	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₉₇	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₉₇	1	4	yes	—
152	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
153	C₂×D₄	C₂×Dic₉₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
157	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
161	M₁₆	—	194	4	no	—
162	M₁₆	—	194	2	no	—
163	M₁₆	—	194	2	no	—
164	C₂×C₈	—	97	4	no	—
165	C₂×C₈	—	97	1	no	—
166	C₂×C₈	—	97	1	no	—
167	C₂×C₈	—	97	2	no	—
168	C₂×C₈	—	97	2	no	—
169	C₂×C₈	—	97	2	no	—
170	C₂×C₈	—	194	4	no	—
171	C₂×C₈	—	194	4	no	—
172	C₂×C₈	—	194	4	no	—
173	C₂×C₈	—	194	2	no	—
174	C₄⋊C₄	C₂×Dic₉₇	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₉₇	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₉₇	1	4	yes	—
179	C₄²	C₂²×D₉₇	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂²×Dic₉₇	1	2	yes	—
183	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
184	C₂³	C₂²×Dic₉₇	1	2	yes	—
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	8	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
196	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
197	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
198	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
199	C₂×C₄	D₄×D₉₇	1	2	yes	—
200	C₂×C₄	#13095510	1	2	yes	—
201	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
202	C₈	—	97	2	no	—
203	C₈	—	97	2	no	—
204	C₈	—	194	2	no	—
205	C₈	—	194	2	no	—
206	C₈	—	194	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	D₄×Dic₉₇	1	2	yes	—
214	C₂²	#15690817	1	4	yes	complemented
215	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₉₇	1	2	yes	—
220	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
221	C₄	C₂×#13095510	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₉₇	1	1	yes	characteristic
226	C₂	C₂×#15690817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting, prime radical	#525×C₉₇	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#525	D₉₇
hypocenter	C₉₇	#2929
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#15690817	C₂	semidirect
2. C₂³×#11715895	C₂	semidirect
3. #20260807	C₂	semidirect
4. D₄×#11715895	C₂	semidirect
5. #20260762	C₂	semidirect
6. #20260702	C₂	semidirect
7. #20260699	C₂	semidirect
8. #20260691	C₂	semidirect
13. C₂×#13095495	C₂²	semidirect
19. C₂²×#11715895	C₂²	semidirect
20. C₄⋊#11715895	C₂²	semidirect
22. C₂²⋊#11715895	C₂²	semidirect
23. #15690721	C₂²	semidirect
24. C₄×#11715895	C₂²	semidirect
40. C₂×#11715895	D₄	semidirect
100. C₂²×C₉₇	#131	semidirect
114. C₉₇	#2929	semidirect
214. C₂²	#15690817	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690817	2
C₂³×#11715895	2
#20260807	2
D₄×#11715895	2
#20260762	2
#20260702	2
#20260699	2
#20260691	2
#525×C₉₇	2
#2929	97