Subgroups — #27150826

Order 12416.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#20260863	C₂	1	4	yes	maximal, complemented
3	#20260862	C₂	1	2	yes	maximal, complemented
4	#20260859	C₂	1	2	yes	maximal, complemented
5	#20260789	C₂	1	2	yes	maximal
6	#20260788	C₂	1	1	yes	maximal, characteristic
7	#20260785	C₂	1	2	yes	maximal
8	#432×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₉₄	C₄	1	1	yes	characteristic
10	C₄×Dic₁₉₄	C₄	1	2	yes	—
11	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
12	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
13	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₉₄	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
17	C₂×#13095490	C₄	1	2	yes	—
18	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
19	C₄⋊#11715895	C₂²	1	2	yes	—
20	#15690739	C₂²	1	4	yes	—
21	#15690739	C₂²	1	4	yes	—
22	C₂²⋊#11715895	C₂²	1	1	yes	characteristic
23	C₂²⋊#11715895	C₂²	1	1	yes	characteristic
24	#15690720	C₂²	1	4	yes	—
25	C₄×#11715895	C₂²	1	2	yes	—
26	C₈×Dic₉₇	C₂²	1	4	yes	—
27	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
30	#97×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	C₂×#11715895	C₂³	1	2	yes	—
38	C₂×#11715895	C₂³	1	2	yes	—
39	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
42	#13095490	C₂×C₄	1	2	yes	—
43	#13095490	C₂×C₄	1	2	yes	—
44	#13095490	C₂×C₄	1	2	yes	—
45	#13095490	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	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₉₇	C₂³	1	4	yes	—
54	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₉₇	C₂³	1	1	yes	characteristic
57	C₄²×C₉₇	C₂³	1	1	yes	characteristic
58	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₉₇	—	2	2	no	—
60	C₂×C₄×C₉₇	—	2	2	no	—
61	C₂×C₄×C₉₇	—	2	2	no	—
62	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₉₇	—	2	4	no	—
69	C₂×Dic₉₇	—	2	4	no	—
70	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
71	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
73	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
74	Dic₁₉₄	—	2	2	no	—
75	Dic₁₉₄	—	2	4	no	—
76	Dic₁₉₄	M₁₆	1	4	yes	—
77	C₈×C₉₇	—	2	4	no	—
78	#11715895	—	2	2	no	—
79	#11715895	—	2	2	no	—
80	C₂²×C₉₇	—	2	1	no	—
81	C₂²×C₉₇	—	2	1	no	—
82	C₂²×C₉₇	#131	1	2	yes	—
83	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₉₇	—	2	1	no	—
85	C₄×C₉₇	—	2	1	no	—
86	C₄×C₉₇	—	2	1	no	—
87	C₄×C₉₇	—	2	1	no	—
88	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
89	C₄×C₉₇	2₋¹⁺⁴	1	2	yes	—
90	Dic₉₇	—	2	2	no	—
91	Dic₉₇	—	2	2	no	—
92	Dic₉₇	—	2	2	no	—
93	Dic₉₇	C₂×M₁₆	1	4	yes	—
94	C₂×C₉₇	—	2	2	no	—
95	C₂×C₉₇	#519	1	1	yes	characteristic
96	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
97	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
98	#2984	—	97	1	no	maximal, Sylow
99	C₉₇	#2984	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×Q₈	—	97	1	no	—
101	#446	—	97	2	no	—
102	#446	—	97	4	no	—
103	C₈×Q₈	—	97	2	no	—
104	#432	—	97	2	no	—
105	#432	D₉₇	1	1	yes	characteristic
106	#431	—	97	1	no	—
107	#431	—	97	2	no	—
108	C₂²×Q₈	—	97	1	no	—
109	C₄×Q₈	—	97	1	no	—
110	C₄×Q₈	—	97	2	no	—
111	C₄×Q₈	—	97	1	no	—
112	C₄×Q₈	—	97	1	no	—
113	C₄×Q₈	—	97	2	no	—
114	C₄×Q₈	—	97	1	no	—
115	C₂×C₄⋊C₄	—	97	2	no	—
116	C₂×C₄⋊C₄	—	97	1	no	—
117	C₂×C₄²	—	97	2	no	—
118	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
119	C₄⋊C₈	—	97	4	no	—
120	C₄⋊C₈	—	97	4	no	—
121	C₄⋊C₈	—	97	2	no	—
122	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
123	C₂²⋊C₈	—	97	1	no	—
124	C₂²⋊C₈	—	97	1	no	—
125	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
126	#97	—	97	4	no	—
127	#97	C₂×D₉₇	1	2	yes	—
128	C₄×C₈	—	97	4	no	—
129	C₄×C₈	—	97	2	no	—
130	C₂×Q₈	—	97	1	no	—
131	C₂×Q₈	—	97	2	no	—
132	C₂×Q₈	—	97	1	no	—
133	C₂×Q₈	—	194	4	no	—
134	C₂²×C₄	—	97	2	no	—
135	C₂²×C₄	—	97	2	no	—
136	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
139	C₂×C₈	—	97	2	no	—
140	C₂×C₈	—	97	2	no	—
141	C₂×C₈	C₂²×D₉₇	1	4	yes	—
142	C₄⋊C₄	—	97	2	no	—
143	C₄⋊C₄	—	97	2	no	—
144	C₄⋊C₄	—	97	2	no	—
145	C₄⋊C₄	—	97	2	no	—
146	C₄⋊C₄	—	97	2	no	—
147	C₄⋊C₄	—	97	1	no	—
148	C₄⋊C₄	—	97	1	no	—
149	C₄²	—	97	2	no	—
150	C₄²	—	97	2	no	—
151	C₄²	—	97	2	no	—
152	C₄²	—	97	2	no	—
153	C₄²	C₄×D₉₇	1	1	yes	characteristic
154	C₄²	C₄×D₉₇	1	1	yes	characteristic
155	C₄²	C₂²×D₉₇	1	1	yes	characteristic
156	C₄²	C₂²×D₉₇	1	1	yes	characteristic
157	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
158	Q₈	—	97	4	no	—
159	Q₈	—	194	4	no	—
160	Q₈	—	194	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	97	2	no	—
165	C₂×C₄	—	97	2	no	—
166	C₂×C₄	—	97	2	no	—
167	C₂×C₄	—	97	2	no	—
168	C₂×C₄	—	194	4	no	—
169	C₂×C₄	—	194	4	no	—
170	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	—	194	2	no	—
178	C₈	—	194	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#15690818	1	2	yes	—
182	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	97	4	no	—
188	C₄	—	194	2	no	—
189	C₄	—	194	2	no	—
190	C₄	—	194	2	no	—
191	C₄	M₁₆×D₉₇	1	2	yes	—
192	C₄	#15690857	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#20261212	1	1	yes	characteristic
195	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
196	C₂	C₂×#15690818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#432×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#432	D₉₇
hypocenter	C₉₇	#2984
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #20260863	C₂	semidirect
3. #20260862	C₂	semidirect
4. #20260859	C₂	semidirect
99. C₉₇	#2984	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
#20260863	2
#20260862	2
#20260859	2
#20260789	2
#20260788	2
#20260785	2
#432×C₉₇	2
#2984	97