Subgroups — #27150683

Order 12416.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261292	C₂	1	1	yes	maximal, characteristic
2	#20260865	C₂	1	2	yes	maximal, complemented
3	#20260863	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₉₄	C₂	1	2	yes	maximal, complemented
5	#20260763	C₂	1	2	yes	maximal
6	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
7	#20260693	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11715895	C₂²	1	2	yes	—
10	#15690810	C₄	1	1	yes	characteristic
11	#15690810	C₄	1	1	yes	characteristic
12	#15690802	C₄	1	2	yes	—
13	#15690789	C₄	1	2	yes	—
14	#15690789	C₄	1	2	yes	—
15	C₄×Dic₁₉₄	C₂²	1	2	yes	—
16	C₄×Dic₁₉₄	C₂²	1	2	yes	—
17	#15690769	C₂²	1	2	yes	—
18	C₄⋊#11715895	C₂²	1	1	yes	characteristic
19	C₄⋊#11715895	C₂²	1	1	yes	characteristic
20	C₄⋊#11715895	C₂²	1	2	yes	—
21	#15690739	C₂²	1	4	yes	—
22	#15690739	C₂²	1	4	yes	—
23	#15690720	C₂²	1	4	yes	—
24	C₈×Dic₉₇	C₂²	1	4	yes	—
25	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
28	#97×C₉₇	C₂²	1	2	yes	—
29	C₄×C₈×C₉₇	C₂²	1	2	yes	—
30	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
32	C₂×#11715895	—	2	4	no	—
33	C₂×#11715895	C₂³	1	2	yes	—
34	C₂×#11715895	C₂³	1	2	yes	—
35	C₄⋊Dic₉₇	C₂×C₄	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	2	yes	—
39	#13095490	C₂×C₄	1	4	yes	—
40	#13095490	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₉₇	C₂³	1	4	yes	—
43	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₉₇	—	2	2	no	—
47	M₁₆×C₉₇	Q₈	1	4	yes	complemented
48	C₂×C₈×C₉₇	C₂³	1	2	yes	—
49	C₂×C₈×C₉₇	C₂³	1	2	yes	—
50	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₉₇	C₂³	1	1	yes	characteristic
53	C₄²×C₉₇	C₂³	1	1	yes	characteristic
54	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	—	2	2	no	—
56	C₂×C₄×C₉₇	—	4	1	no	—
57	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₉₇	#41	1	2	yes	—
64	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
66	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
67	Dic₁₉₄	—	2	4	no	—
68	Dic₁₉₄	—	4	2	no	—
69	C₈×C₉₇	—	2	2	no	—
70	C₈×C₉₇	C₂×Q₈	1	4	yes	—
71	#11715895	—	2	2	no	—
72	#11715895	—	2	2	no	—
73	C₂²×C₉₇	—	2	1	no	—
74	C₂²×C₉₇	—	2	1	no	—
75	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
76	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₉₇	—	2	1	no	—
78	C₄×C₉₇	—	2	1	no	—
79	C₄×C₉₇	—	2	1	no	—
80	C₄×C₉₇	C₄×Q₈	1	2	yes	—
81	C₄×C₉₇	#131	1	2	yes	—
82	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
84	Dic₉₇	—	2	4	no	—
85	Dic₉₇	—	2	4	no	—
86	C₂×C₉₇	—	2	2	no	—
87	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
89	C₂×C₉₇	#568	1	1	yes	characteristic
90	#2962	—	97	1	no	maximal, Sylow
91	C₉₇	#2962	1	1	yes	characteristic, complemented, Sylow
92	#533	—	97	1	no	—
93	#446	—	97	4	no	—
94	#446	—	97	2	no	—
95	C₈×Q₈	—	97	2	no	—
96	#424	—	97	2	no	—
97	C₂×C₄⋊C₈	—	97	1	no	—
98	#405	—	97	2	no	—
99	C₄×M₁₆	D₉₇	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
101	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
102	C₂²×C₈	—	97	2	no	—
103	C₄⋊Q₈	—	97	1	no	—
104	C₄⋊Q₈	—	97	1	no	—
105	#125	—	97	2	no	—
106	C₂²⋊Q₈	—	97	2	no	—
107	C₂²⋊Q₈	—	97	2	no	—
108	C₄×Q₈	—	97	2	no	—
109	C₄×Q₈	—	97	2	no	—
110	#117	—	97	2	no	—
111	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
112	C₄⋊C₈	—	97	4	no	—
113	C₄⋊C₈	—	97	4	no	—
114	C₄⋊C₈	—	97	1	no	—
115	C₄⋊C₈	—	97	1	no	—
116	C₄⋊C₈	—	97	2	no	—
117	#97	—	97	4	no	—
118	#97	C₂×D₉₇	1	2	yes	—
119	C₄×C₈	—	97	4	no	—
120	C₄×C₈	C₂×D₉₇	1	2	yes	—
121	C₂×Q₈	—	97	2	no	—
122	C₂×Q₈	—	97	2	no	—
123	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₁₉₄	1	4	yes	complemented
128	C₂×C₈	—	97	2	no	—
129	C₂×C₈	—	97	2	no	—
130	C₂×C₈	—	194	4	no	—
131	C₂×C₈	C₂²×D₉₇	1	2	yes	—
132	C₂×C₈	C₂²×D₉₇	1	2	yes	—
133	C₄⋊C₄	—	97	4	no	—
134	C₄⋊C₄	—	97	4	no	—
135	C₄⋊C₄	—	97	2	no	—
136	C₄⋊C₄	—	97	2	no	—
137	C₄⋊C₄	—	97	2	no	—
138	C₄⋊C₄	—	97	2	no	—
139	C₂²⋊C₄	—	97	4	no	—
140	C₄²	—	97	4	no	—
141	C₄²	C₄×D₉₇	1	1	yes	characteristic
142	C₄²	C₄×D₉₇	1	1	yes	characteristic
143	C₄²	C₂²×D₉₇	1	1	yes	characteristic
144	C₄²	C₂²×D₉₇	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
146	Q₈	—	388	2	no	—
147	Q₈	—	194	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	97	4	no	—
151	C₂×C₄	—	97	4	no	—
152	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
158	C₂×C₄	#13095511	1	2	yes	—
159	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	194	2	no	—
162	C₈	—	194	2	no	—
163	C₈	C₂×Dic₁₉₄	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₁₉₄	1	2	yes	—
167	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	194	4	no	—
172	C₄	—	194	4	no	—
173	C₄	C₄×Dic₁₉₄	1	2	yes	—
174	C₄	#15690818	1	2	yes	—
175	C₄	C₂²×Dic₁₉₄	1	1	yes	characteristic
176	C₄	C₂×#13095511	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₁₉₄	1	1	yes	characteristic
179	C₂	C₂×#15690818	1	1	yes	characteristic
180	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₄×M₁₆×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	#2962
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20260865	C₂	semidirect
3. #20260863	C₂	semidirect
4. C₈×Dic₁₉₄	C₂	semidirect
47. M₁₆×C₉₇	Q₈	semidirect
91. C₉₇	#2962	semidirect
127. M₁₆	Dic₁₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20261292	2
#20260865	2
#20260863	2
C₈×Dic₁₉₄	2
#20260763	2
C₂×C₄⋊#11715895	2
#20260693	2
C₄×M₁₆×C₉₇	2
#2962	97