Subgroups — #27150687

Order 12416.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	#20260864	C₂	1	4	yes	maximal, complemented
3	#20260862	C₂	1	2	yes	maximal, complemented
4	#20260859	C₂	1	2	yes	maximal, complemented
5	#20260758	C₂	1	2	yes	maximal
6	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₉₇	C₂	1	2	yes	maximal
8	#423×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₉₄	C₄	1	1	yes	characteristic
10	C₂×#13095495	C₂²	1	1	yes	characteristic
11	C₂×#13095495	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₉₄	C₄	1	2	yes	—
13	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
14	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
15	C₄×Dic₁₉₄	C₂²	1	2	yes	—
16	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
19	C₂×#13095490	C₄	1	2	yes	—
20	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
21	#15690739	C₂²	1	4	yes	—
22	#15690739	C₂²	1	4	yes	—
23	#15690721	C₂²	1	2	yes	—
24	#15690720	C₂²	1	4	yes	—
25	C₄×#11715895	C₂²	1	2	yes	—
26	C₈×Dic₉₇	C₂²	1	4	yes	—
27	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
30	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
31	C₂×Dic₁₉₄	—	2	4	no	—
32	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
36	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
37	#13095495	—	2	2	no	—
38	#13095495	Q₈	1	4	yes	complemented
39	C₂×#11715895	C₂³	1	2	yes	—
40	C₂×#11715895	C₂³	1	2	yes	—
41	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
44	#13095490	C₂×C₄	1	2	yes	—
45	#13095490	C₂×C₄	1	2	yes	—
46	#13095490	C₂×C₄	1	2	yes	—
47	#13095490	C₂×C₄	1	2	yes	—
48	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₉₇	C₂³	1	2	yes	—
51	C₄×Dic₉₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
55	M₁₆×C₉₇	—	2	4	no	—
56	C₂×C₈×C₉₇	C₂³	1	4	yes	—
57	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₉₇	C₂³	1	1	yes	characteristic
60	C₄²×C₉₇	C₂³	1	1	yes	characteristic
61	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₉₇	—	2	2	no	—
63	C₂×C₄×C₉₇	—	4	1	no	—
64	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₉₇	#41	1	2	yes	—
71	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₉₇	—	2	4	no	—
73	C₂×Dic₉₇	—	4	2	no	—
74	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
78	Dic₁₉₄	—	2	2	no	—
79	Dic₁₉₄	—	2	4	no	—
80	Dic₁₉₄	M₁₆	1	4	yes	complemented
81	C₈×C₉₇	—	2	4	no	—
82	#11715895	—	2	2	no	—
83	#11715895	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₉₇	—	2	1	no	—
85	C₂²×C₉₇	—	2	1	no	—
86	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
87	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₉₇	—	2	1	no	—
89	C₄×C₉₇	—	2	1	no	—
90	C₄×C₉₇	—	2	1	no	—
91	C₄×C₉₇	C₄×Q₈	1	2	yes	—
92	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
93	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
95	Dic₉₇	—	2	2	no	—
96	Dic₉₇	—	2	2	no	—
97	Dic₉₇	—	2	2	no	—
98	Dic₉₇	C₂×M₁₆	1	4	yes	—
99	C₂×C₉₇	—	2	2	no	—
100	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₉₇	#568	1	1	yes	characteristic
103	Q₈×M₁₆	—	97	1	no	maximal, Sylow
104	C₉₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	97	1	no	—
106	#446	—	97	4	no	—
107	#446	—	97	2	no	—
108	C₈×Q₈	—	97	2	no	—
109	#423	—	97	2	no	—
110	#423	D₉₇	1	1	yes	characteristic
111	C₄×M₁₆	—	97	2	no	—
112	C₄×M₁₆	—	97	1	no	—
113	C₂²×Q₈	—	97	1	no	—
114	C₂×M₁₆	—	97	1	no	—
115	C₂×M₁₆	—	97	1	no	—
116	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
117	C₄×Q₈	—	97	1	no	—
118	C₄×Q₈	—	97	2	no	—
119	C₄×Q₈	—	97	1	no	—
120	C₄×Q₈	—	97	1	no	—
121	C₄×Q₈	—	97	2	no	—
122	C₄×Q₈	—	97	1	no	—
123	C₂×C₄⋊C₄	—	97	2	no	—
124	C₂×C₄⋊C₄	—	97	1	no	—
125	C₂×C₄²	—	97	2	no	—
126	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
127	C₄⋊C₈	—	97	4	no	—
128	C₄⋊C₈	—	97	4	no	—
129	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
130	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
131	#97	—	97	4	no	—
132	#97	—	97	2	no	—
133	C₄×C₈	—	97	4	no	—
134	C₄×C₈	—	97	2	no	—
135	C₂×Q₈	—	97	1	no	—
136	C₂×Q₈	—	97	2	no	—
137	C₂×Q₈	—	97	1	no	—
138	C₂×Q₈	—	194	4	no	—
139	C₂²×C₄	—	97	2	no	—
140	C₂²×C₄	—	97	2	no	—
141	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	97	4	no	—
146	M₁₆	—	194	2	no	—
147	C₂×C₈	—	97	2	no	—
148	C₂×C₈	—	97	2	no	—
149	C₂×C₈	C₂²×D₉₇	1	4	yes	—
150	C₄⋊C₄	—	97	2	no	—
151	C₄⋊C₄	—	97	2	no	—
152	C₄⋊C₄	—	97	2	no	—
153	C₄⋊C₄	—	97	2	no	—
154	C₄⋊C₄	—	97	2	no	—
155	C₄⋊C₄	—	97	1	no	—
156	C₄⋊C₄	—	97	1	no	—
157	C₄²	—	97	2	no	—
158	C₄²	—	97	2	no	—
159	C₄²	—	97	2	no	—
160	C₄²	—	97	2	no	—
161	C₄²	C₄×D₉₇	1	1	yes	characteristic
162	C₄²	C₄×D₉₇	1	1	yes	characteristic
163	C₄²	C₂²×D₉₇	1	1	yes	characteristic
164	C₄²	C₂²×D₉₇	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
166	Q₈	—	97	4	no	—
167	Q₈	—	194	4	no	—
168	Q₈	—	194	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	97	2	no	—
172	C₂×C₄	—	97	2	no	—
173	C₂×C₄	—	97	2	no	—
174	C₂×C₄	—	97	2	no	—
175	C₂×C₄	—	388	2	no	—
176	C₂×C₄	—	194	4	no	—
177	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
182	C₂×C₄	Q₈×D₉₇	1	2	yes	—
183	C₂×C₄	#13095510	1	2	yes	—
184	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	97	4	no	—
187	C₈	—	194	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#15690775	1	2	yes	—
191	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	97	4	no	—
196	C₄	—	194	2	no	—
197	C₄	—	194	2	no	—
198	C₄	—	194	2	no	—
199	C₄	#15690775	1	2	yes	—
200	C₄	M₁₆×D₉₇	1	2	yes	—
201	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
202	C₄	C₂×#13095510	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#15690775	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
206	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#423×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#423	D₉₇
hypocenter	C₉₇	Q₈×M₁₆
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #20260864	C₂	semidirect
3. #20260862	C₂	semidirect
4. #20260859	C₂	semidirect
38. #13095495	Q₈	semidirect
80. Dic₁₉₄	M₁₆	semidirect
83. #11715895	C₂×Q₈	semidirect
104. C₉₇	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₉₄	2
#20260864	2
#20260862	2
#20260859	2
#20260758	2
C₄×#13095495	2
M₁₆×Dic₉₇	2
#423×C₉₇	2
Q₈×M₁₆	97