Subgroups — #27150820

Order 12416.

356 conjugacy classes of subgroups.

The subgroup lattice has 192 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	#20260863	C₂	1	4	yes	maximal, complemented
4	#20260792	C₂	1	1	yes	maximal, characteristic
5	#20260789	C₂	1	2	yes	maximal
6	#20260785	C₂	1	2	yes	maximal
7	#432×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₉₄	C₄	1	1	yes	characteristic
9	C₄×Dic₁₉₄	C₄	1	2	yes	—
10	C₄×Dic₁₉₄	C₄	1	2	yes	—
11	C₄×Dic₁₉₄	C₂²	1	2	yes	—
12	C₄×Dic₁₉₄	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
14	C₂×#13095490	C₄	1	1	yes	characteristic
15	C₂×#13095490	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
18	C₄⋊#11715895	C₂²	1	2	yes	—
19	#15690739	C₂²	1	4	yes	—
20	#15690739	C₂²	1	4	yes	—
21	C₂²⋊#11715895	C₂²	1	2	yes	—
22	#15690721	C₂²	1	2	yes	—
23	#15690720	C₂²	1	4	yes	—
24	C₈×Dic₉₇	C₂²	1	4	yes	—
25	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
28	#97×C₉₇	C₂²	1	2	yes	—
29	C₂×Dic₁₉₄	—	2	4	no	—
30	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
32	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂×#11715895	C₂³	1	4	yes	—
37	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
39	#13095490	C₂×C₄	1	2	yes	—
40	#13095490	C₂×C₄	1	2	yes	—
41	#13095490	C₂×C₄	1	2	yes	—
42	#13095490	C₂×C₄	1	2	yes	—
43	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
44	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
45	C₄×Dic₉₇	C₂³	1	2	yes	—
46	C₄×Dic₉₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₉₇	C₂³	1	4	yes	—
51	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₉₇	C₂³	1	1	yes	characteristic
54	C₄²×C₉₇	C₂³	1	1	yes	characteristic
55	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	—	2	2	no	—
57	C₂×C₄×C₉₇	—	2	2	no	—
58	C₂×C₄×C₉₇	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₉₇	—	2	2	no	—
66	C₂×Dic₉₇	—	4	1	no	—
67	C₂×Dic₉₇	—	4	1	no	—
68	C₂×Dic₉₇	M₁₆	1	4	yes	—
69	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
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	Dic₁₉₄	—	2	4	no	—
74	Dic₁₉₄	—	2	4	no	—
75	C₈×C₉₇	—	2	4	no	—
76	#11715895	—	2	4	no	—
77	C₂²×C₉₇	—	2	1	no	—
78	C₂²×C₉₇	—	2	1	no	—
79	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
80	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
81	C₄×C₉₇	—	2	1	no	—
82	C₄×C₉₇	—	2	1	no	—
83	C₄×C₉₇	—	2	1	no	—
84	C₄×C₉₇	—	2	1	no	—
85	C₄×C₉₇	—	2	1	no	—
86	C₄×C₉₇	2₋¹⁺⁴	1	2	yes	—
87	Dic₉₇	—	2	2	no	—
88	Dic₉₇	—	2	2	no	—
89	Dic₉₇	—	2	2	no	—
90	Dic₉₇	C₂×M₁₆	1	4	yes	—
91	C₂×C₉₇	—	2	2	no	—
92	C₂×C₉₇	#519	1	1	yes	characteristic
93	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
94	C₂×C₉₇	#568	1	1	yes	characteristic
95	#2983	—	97	1	no	maximal, Sylow
96	C₉₇	#2983	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄×Q₈	—	97	1	no	—
98	#446	—	97	4	no	—
99	#446	—	97	4	no	—
100	#432	—	97	1	no	—
101	#432	—	97	2	no	—
102	#432	D₉₇	1	1	yes	characteristic
103	#431	—	97	2	no	—
104	C₂²×Q₈	—	97	1	no	—
105	C₄×Q₈	—	97	2	no	—
106	C₄×Q₈	—	97	2	no	—
107	C₄×Q₈	—	97	2	no	—
108	C₄×Q₈	—	97	2	no	—
109	C₂×C₄⋊C₄	—	97	1	no	—
110	C₂×C₄⋊C₄	—	97	1	no	—
111	C₂×C₄⋊C₄	—	97	1	no	—
112	C₂×C₄²	—	97	1	no	—
113	C₂×C₄²	—	97	1	no	—
114	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
115	C₄⋊C₈	—	97	4	no	—
116	C₄⋊C₈	—	97	4	no	—
117	C₄⋊C₈	—	97	2	no	—
118	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
119	C₂²⋊C₈	—	97	2	no	—
120	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
121	#97	—	97	4	no	—
122	#97	—	97	2	no	—
123	#97	C₂×D₉₇	1	2	yes	—
124	C₄×C₈	—	97	4	no	—
125	C₂×Q₈	—	97	2	no	—
126	C₂×Q₈	—	97	2	no	—
127	C₂×Q₈	—	194	4	no	—
128	C₂²×C₄	—	97	1	no	—
129	C₂²×C₄	—	97	1	no	—
130	C₂²×C₄	—	97	1	no	—
131	C₂²×C₄	—	97	1	no	—
132	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
135	C₂×C₈	—	97	4	no	—
136	C₂×C₈	C₂²×D₉₇	1	4	yes	—
137	C₄⋊C₄	—	97	2	no	—
138	C₄⋊C₄	—	97	2	no	—
139	C₄⋊C₄	—	97	2	no	—
140	C₄⋊C₄	—	97	2	no	—
141	C₄⋊C₄	—	97	2	no	—
142	C₄⋊C₄	—	97	2	no	—
143	C₄²	—	97	2	no	—
144	C₄²	—	97	2	no	—
145	C₄²	—	97	2	no	—
146	C₄²	—	97	2	no	—
147	C₄²	C₄×D₉₇	1	1	yes	characteristic
148	C₄²	C₄×D₉₇	1	1	yes	characteristic
149	C₄²	C₂²×D₉₇	1	1	yes	characteristic
150	C₄²	C₂²×D₉₇	1	1	yes	characteristic
151	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
152	Q₈	—	194	4	no	—
153	Q₈	—	194	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	97	4	no	—
158	C₂×C₄	—	97	2	no	—
159	C₂×C₄	—	97	2	no	—
160	C₂×C₄	—	97	2	no	—
161	C₂×C₄	—	97	2	no	—
162	C₂×C₄	—	388	1	no	—
163	C₂×C₄	—	388	1	no	—
164	C₂×C₄	—	194	2	no	—
165	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
170	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₈	—	194	4	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	M₁₆×D₉₇	1	2	yes	—
176	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	97	4	no	—
183	C₄	—	194	2	no	—
184	C₄	—	194	2	no	—
185	C₄	—	194	2	no	—
186	C₄	#15690857	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	#20261212	1	1	yes	characteristic
189	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
190	C₂	#20261501	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	#2983
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. #20260864	C₂	semidirect
3. #20260863	C₂	semidirect
96. C₉₇	#2983	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
#20260863	2
#20260792	2
#20260789	2
#20260785	2
#432×C₉₇	2
#2983	97