Subgroups — #27158545

Order 12416.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261786	C₂	1	2	yes	maximal, complemented
2	#20261770	C₂	1	2	yes	maximal, complemented
3	#20261766	C₂	1	4	yes	maximal, complemented
4	#20261752	C₂	1	2	yes	maximal, complemented
5	#20261751	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13095520	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#11715897	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×Dic₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13095520	C₂²	1	4	yes	—
10	C₂²×#11715897	C₂²	1	2	yes	—
11	#15690877	C₂²	1	4	yes	—
12	C₄⋊#11715897	C₂²	1	2	yes	—
13	C₂²⋊#11715897	C₂²	1	2	yes	—
14	C₂²⋊#11715897	C₂²	1	4	yes	—
15	#15690865	C₂²	1	4	yes	—
16	C₄×#11715897	C₂²	1	1	yes	characteristic
17	C₄×#11715897	C₂²	1	1	yes	characteristic
18	C₄×#11715897	C₂²	1	2	yes	—
19	C₄×#11715897	C₂²	1	2	yes	—
20	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
21	Q₈×Dic₉₇	C₂²	1	1	yes	characteristic
22	D₄×Dic₉₇	C₄	1	2	yes	—
23	D₄×Dic₉₇	C₄	1	1	yes	characteristic
24	D₄×Dic₉₇	C₂²	1	2	yes	—
25	D₄×Dic₉₇	C₂²	1	1	yes	characteristic
26	#15690769	C₄	1	2	yes	—
27	#15690769	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
30	C₂×#41×C₉₇	C₄	1	1	yes	characteristic
31	#13095520	—	2	8	no	—
32	C₂×#11715897	—	2	4	no	—
33	C₂×#11715897	C₂³	1	2	yes	—
34	C₂×#11715897	C₂³	1	4	yes	—
35	C₂×#11715897	C₂³	1	2	yes	—
36	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
37	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉₇	C₂³	1	2	yes	—
40	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
46	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
52	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
53	C₄×Dic₉₇	C₂³	1	2	yes	—
54	#41×C₉₇	—	2	4	no	—
55	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
56	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
57	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
59	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
61	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₉₇	—	2	2	no	—
63	D₄×C₉₇	—	2	4	no	—
64	D₄×C₉₇	—	2	2	no	—
65	C₂×C₄×C₉₇	—	2	4	no	—
66	C₂×C₄×C₉₇	M₁₆	1	4	yes	complemented
67	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₉₇	—	2	2	no	—
71	C₂×Dic₉₇	—	2	4	no	—
72	C₂×Dic₉₇	—	2	2	no	—
73	C₂×Dic₉₇	—	2	4	no	—
74	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
80	#11715897	—	2	2	no	—
81	#11715897	—	2	4	no	—
82	#11715897	#41	1	4	yes	complemented
83	C₂²×C₉₇	—	2	1	no	—
84	C₂²×C₉₇	—	2	1	no	—
85	C₂²×C₉₇	—	2	2	no	—
86	C₂²×C₉₇	—	2	2	no	—
87	C₂²×C₉₇	—	2	2	no	—
88	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
89	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₉₇	—	2	2	no	—
91	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
92	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
93	Dic₉₇	—	2	1	no	—
94	Dic₉₇	—	2	2	no	—
95	Dic₉₇	—	2	1	no	—
96	Dic₉₇	—	2	2	no	—
97	Dic₉₇	#131	1	2	yes	—
98	Dic₉₇	C₂×#41	1	2	yes	—
99	C₂×C₉₇	—	2	2	no	—
100	C₂×C₉₇	—	4	2	no	—
101	C₂×C₉₇	C₄×#41	1	1	yes	characteristic
102	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
104	#2965	—	97	1	no	maximal, Sylow
105	C₉₇	#2965	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	—	97	1	no	—
107	#446	—	97	2	no	—
108	#436	—	97	2	no	—
109	#435	—	97	4	no	—
110	#431	—	97	1	no	—
111	#431	—	97	2	no	—
112	C₄×M₁₆	—	97	2	no	—
113	C₂×C₄×C₈	—	97	1	no	—
114	C₂×#41	#87604	1	1	yes	characteristic
115	C₂×M₁₆	—	97	4	no	—
116	C₂²×C₈	—	97	2	no	—
117	C₄×Q₈	—	97	1	no	—
118	C₄×Q₈	—	97	1	no	—
119	C₄×D₄	—	97	2	no	—
120	C₄×D₄	—	97	1	no	—
121	C₄×D₄	—	97	2	no	—
122	C₄×D₄	—	97	1	no	—
123	#117	—	97	2	no	—
124	#117	—	97	1	no	—
125	C₂×C₄²	—	97	1	no	—
126	C₂×C₄²	—	97	2	no	—
127	C₄⋊C₈	—	97	2	no	—
128	C₄⋊C₈	—	97	4	no	—
129	C₂²⋊C₈	—	97	2	no	—
130	C₂²⋊C₈	—	97	4	no	—
131	#97	—	97	4	no	—
132	C₄×C₈	—	97	1	no	—
133	C₄×C₈	—	97	1	no	—
134	C₄×C₈	—	97	2	no	—
135	C₄×C₈	—	97	2	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×#87604	1	1	yes	characteristic
138	C₂×D₄	C₂×#87604	1	2	yes	—
139	C₂×D₄	C₂×#87604	1	1	yes	characteristic
140	C₂²×C₄	—	97	2	no	—
141	C₂²×C₄	—	97	1	no	—
142	C₂²×C₄	—	97	1	no	—
143	C₂²×C₄	—	97	2	no	—
144	C₂²×C₄	C₂×#87604	1	2	yes	—
145	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
146	M₁₆	—	194	8	no	—
147	C₂×C₈	—	97	2	no	—
148	C₂×C₈	—	97	2	no	—
149	C₂×C₈	—	97	4	no	—
150	C₂×C₈	—	194	4	no	—
151	C₄⋊C₄	—	97	2	no	—
152	C₄⋊C₄	—	97	2	no	—
153	C₄⋊C₄	—	97	1	no	—
154	C₄⋊C₄	—	97	1	no	—
155	C₂²⋊C₄	—	97	2	no	—
156	C₂²⋊C₄	—	97	2	no	—
157	C₂²⋊C₄	—	97	2	no	—
158	C₄²	—	97	2	no	—
159	C₄²	—	97	1	no	—
160	C₄²	—	97	1	no	—
161	C₄²	—	97	2	no	—
162	C₄²	—	97	1	no	—
163	C₄²	—	97	2	no	—
164	C₄²	—	97	1	no	—
165	C₂³	C₂²×#87604	1	2	yes	—
166	C₂³	C₂²×#87604	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	97	1	no	—
172	C₂×C₄	—	97	1	no	—
173	C₂×C₄	—	97	1	no	—
174	C₂×C₄	—	97	2	no	—
175	C₂×C₄	—	97	1	no	—
176	C₂×C₄	—	97	2	no	—
177	C₂×C₄	—	194	2	no	—
178	C₂×C₄	—	194	4	no	—
179	C₂×C₄	—	194	2	no	—
180	C₂×C₄	—	194	4	no	—
181	C₂×C₄	#13095520	1	4	yes	complemented
182	C₂×C₄	C₂²×#87604	1	2	yes	—
183	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
184	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
185	C₈	—	97	4	no	—
186	C₈	—	194	2	no	—
187	C₈	—	194	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₂×#13095520	1	2	yes	—
194	C₂²	C₂³×#87604	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	97	2	no	—
197	C₄	—	97	2	no	—
198	C₄	—	194	1	no	—
199	C₄	—	194	2	no	—
200	C₄	—	194	1	no	—
201	C₄	—	194	2	no	—
202	C₄	C₂×#13095520	1	2	yes	—
203	C₄	C₂×#13095520	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	4	2	no	—
206	C₂	#41×#87604	1	1	yes	characteristic
207	C₂	C₂²×#13095520	1	1	yes	characteristic
208	C₂	D₉₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Frattini	C₂²	C₂³×#87604
Fitting, prime radical	C₂×#41×C₉₇	C₄
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	C₂×#41	#87604
hypocenter	C₉₇	#2965
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20261786	C₂	semidirect
2. #20261770	C₂	semidirect
3. #20261766	C₂	semidirect
4. #20261752	C₂	semidirect
5. #20261751	C₂	semidirect
6. C₄×#13095520	C₂	semidirect
7. C₂×C₄×#11715897	C₂	semidirect
66. C₂×C₄×C₉₇	M₁₆	semidirect
82. #11715897	#41	semidirect
105. C₉₇	#2965	semidirect
181. C₂×C₄	#13095520	semidirect

Maximal subgroups

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

subgroup	index
#20261786	2
#20261770	2
#20261766	2
#20261752	2
#20261751	2
C₄×#13095520	2
C₂×C₄×#11715897	2
#41×Dic₉₇	2
#2965	97