Subgroups — #27150396

Order 12416.

402 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690769	C₂	1	1	yes	maximal, characteristic
2	#20260793	C₂	1	4	yes	maximal, complemented
3	#20260789	C₂	1	2	yes	maximal, complemented
4	#20260785	C₂	1	2	yes	maximal, complemented
5	#20260763	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊#11715895	C₂	1	1	yes	maximal, characteristic
7	#20260693	C₂	1	2	yes	maximal, complemented
8	#407×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11715895	C₂²	1	2	yes	—
10	#15690769	C₄	1	2	yes	—
11	#15690769	C₄	1	1	yes	characteristic
12	#15690769	C₄	1	1	yes	characteristic
13	#15690769	C₂²	1	1	yes	characteristic
14	#15690769	C₂²	1	1	yes	characteristic
15	#15690769	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₉₇	C₄	1	2	yes	—
19	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
20	#15690739	C₂²	1	4	yes	complemented
21	#15690739	C₂²	1	4	yes	complemented
22	C₂²⋊#11715895	C₂²	1	1	yes	characteristic
23	C₂²⋊#11715895	C₂²	1	1	yes	characteristic
24	C₂²⋊#11715895	C₂²	1	2	yes	—
25	#15690720	C₂²	1	4	yes	complemented
26	C₈×Dic₉₇	C₂²	1	4	yes	complemented
27	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
30	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
31	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
32	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
33	C₂×#11715895	—	2	4	no	—
34	C₂×#11715895	C₂³	1	2	yes	—
35	C₂×#11715895	C₂³	1	2	yes	—
36	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
42	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
43	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
46	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₉₇	C₂³	1	2	yes	—
49	C₄×Dic₉₇	C₂³	1	2	yes	—
50	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₉₇	—	2	4	no	—
52	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
58	M₁₆×C₉₇	—	2	4	no	—
59	C₂×C₈×C₉₇	C₂³	1	4	yes	—
60	C₂³×C₉₇	—	2	2	no	—
61	C₂³×C₉₇	—	2	2	no	—
62	C₂³×C₉₇	—	4	1	no	—
63	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
64	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
65	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉₇	—	2	2	no	—
67	C₂×C₄×C₉₇	—	2	2	no	—
68	C₂×C₄×C₉₇	—	2	2	no	—
69	C₂×C₄×C₉₇	—	2	2	no	—
70	C₂×C₄×C₉₇	—	2	2	no	—
71	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₉₇	#41	1	4	yes	—
75	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₉₇	—	2	4	no	—
77	C₂×Dic₉₇	—	4	2	no	—
78	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
82	C₈×C₉₇	—	2	4	no	—
83	#11715895	—	2	2	no	—
84	#11715895	—	2	2	no	—
85	C₂²×C₉₇	—	2	1	no	—
86	C₂²×C₉₇	—	2	1	no	—
87	C₂²×C₉₇	—	2	1	no	—
88	C₂²×C₉₇	—	2	1	no	—
89	C₂²×C₉₇	—	2	1	no	—
90	C₂²×C₉₇	—	2	1	no	—
91	C₂²×C₉₇	—	2	1	no	—
92	C₂²×C₉₇	—	4	4	no	—
93	C₂²×C₉₇	#117	1	2	yes	—
94	C₂²×C₉₇	#131	1	2	yes	—
95	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₉₇	—	2	1	no	—
97	C₄×C₉₇	—	2	1	no	—
98	C₄×C₉₇	#117	1	2	yes	—
99	C₄×C₉₇	C₂×#41	1	2	yes	—
100	Dic₉₇	—	2	2	no	—
101	Dic₉₇	—	2	2	no	—
102	Dic₉₇	—	2	2	no	—
103	Dic₉₇	—	2	2	no	—
104	C₂×C₉₇	—	2	2	no	—
105	C₂×C₉₇	—	2	2	no	—
106	C₂×C₉₇	—	4	1	no	—
107	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
108	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₉₇	#568	1	1	yes	characteristic
110	#2924	—	97	1	no	maximal, Sylow
111	C₉₇	#2924	1	1	yes	characteristic, complemented, Sylow
112	C₂×#117	—	97	1	no	—
113	#433	—	97	4	no	—
114	#432	—	97	2	no	—
115	#431	—	97	2	no	—
116	#424	—	97	2	no	—
117	#407	D₉₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	—	97	1	no	—
119	#405	—	97	2	no	—
120	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
121	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
122	C₂²×C₈	—	97	2	no	—
123	#117	—	97	1	no	—
124	#117	—	97	1	no	—
125	#117	—	97	2	no	—
126	#117	—	97	1	no	—
127	#117	—	97	1	no	—
128	#117	—	97	2	no	—
129	C₂×C₄⋊C₄	—	97	1	no	—
130	C₂×C₄⋊C₄	—	97	1	no	—
131	C₂×C₂²⋊C₄	—	97	2	no	—
132	C₂×C₄²	—	97	2	no	—
133	C₄⋊C₈	—	97	4	no	—
134	C₄⋊C₈	—	97	4	no	—
135	C₂²⋊C₈	—	97	1	no	—
136	C₂²⋊C₈	—	97	1	no	—
137	C₂²⋊C₈	—	97	2	no	—
138	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
139	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
140	#97	—	97	4	no	—
141	C₄×C₈	—	97	4	no	—
142	C₂⁴	C₄×D₉₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	97	2	no	—
145	C₂²×C₄	—	97	2	no	—
146	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
148	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
152	M₁₆	—	2	4	no	—
153	C₂×C₈	—	97	2	no	—
154	C₂×C₈	—	97	2	no	—
155	C₂×C₈	—	194	4	no	—
156	C₂×C₈	C₂²×D₉₇	1	4	yes	—
157	C₄⋊C₄	—	97	2	no	—
158	C₄⋊C₄	—	97	2	no	—
159	C₄⋊C₄	—	97	1	no	—
160	C₄⋊C₄	—	97	1	no	—
161	C₄⋊C₄	—	97	1	no	—
162	C₄⋊C₄	—	97	1	no	—
163	C₂²⋊C₄	—	97	2	no	—
164	C₂²⋊C₄	—	97	2	no	—
165	C₂²⋊C₄	—	97	2	no	—
166	C₂²⋊C₄	—	97	2	no	—
167	C₄²	—	97	2	no	—
168	C₄²	—	97	2	no	—
169	C₄²	—	97	2	no	—
170	C₄²	—	97	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
175	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
176	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	97	2	no	—
183	C₂×C₄	—	97	2	no	—
184	C₂×C₄	—	97	2	no	—
185	C₂×C₄	—	97	2	no	—
186	C₂×C₄	—	388	2	no	—
187	C₂×C₄	—	194	4	no	—
188	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
191	C₂×C₄	#13095510	1	4	yes	—
192	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	194	2	no	—
195	C₈	—	194	2	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#15690766	1	2	yes	—
205	C₂²	#15690818	1	2	yes	—
206	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	194	2	no	—
210	C₄	—	194	2	no	—
211	C₄	—	194	2	no	—
212	C₄	—	194	2	no	—
213	C₄	#15690766	1	2	yes	—
214	C₄	C₂×#13095510	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#15690766	1	1	yes	characteristic
219	C₂	C₂×#15690818	1	1	yes	characteristic
220	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#407×C₉₇	C₂
prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#407	D₉₇
hypocenter	C₉₇	#2924
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #20260793	C₂	semidirect
3. #20260789	C₂	semidirect
4. #20260785	C₂	semidirect
5. #20260763	C₂	semidirect
7. #20260693	C₂	semidirect
20. #15690739	C₂²	semidirect
21. #15690739	C₂²	semidirect
25. #15690720	C₂²	semidirect
26. C₈×Dic₉₇	C₂²	semidirect
111. C₉₇	#2924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690769	2
#20260793	2
#20260789	2
#20260785	2
#20260763	2
C₂×C₂²⋊#11715895	2
#20260693	2
#407×C₉₇	2
#2924	97