Subgroups — #27150165

Order 12416.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690818	C₂	1	2	yes	maximal
2	C₄×#13095511	C₂	1	1	yes	maximal, characteristic
3	#20260693	C₂	1	2	yes	maximal, complemented
4	#20260689	C₂	1	4	yes	maximal, complemented
5	#20260687	C₂	1	2	yes	maximal, complemented
6	C₂×#15690721	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×D₉₇	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13095511	C₂²	1	1	yes	characteristic
10	#15690818	C₄	1	16	yes	complemented
11	C₂×#13095494	C₂²	1	4	yes	—
12	C₂²×#11715895	C₂²	1	2	yes	—
13	C₂×C₈×D₉₇	C₂²	1	4	yes	—
14	C₄×Dic₁₉₄	C₄	1	2	yes	—
15	C₄×D₃₈₈	C₄	1	2	yes	—
16	C₄×C₂²⋊D₉₇	C₄	1	4	yes	—
17	#15690769	C₂²	1	1	yes	characteristic
18	#15690767	C₂²	1	2	yes	—
19	C₄²×D₉₇	C₂²	1	2	yes	—
20	#15690721	C₂²	1	1	yes	characteristic
21	#15690721	C₂²	1	1	yes	characteristic
22	#15690721	C₂²	1	2	yes	—
23	#15690720	C₂²	1	4	yes	—
24	C₈×Dic₉₇	C₂²	1	4	yes	—
25	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
27	#97×C₉₇	C₂²	1	2	yes	—
28	C₄×C₈×C₉₇	C₂²	1	2	yes	—
29	#13095511	C₂×C₄	1	8	yes	—
30	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
31	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₉₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₉₇	C₂³	1	2	yes	—
35	#13095494	C₂×C₄	1	16	yes	complemented
36	C₂×#11715895	C₂×C₄	1	8	yes	complemented
37	C₂×#11715895	C₂³	1	2	yes	—
38	C₂×#11715895	C₂³	1	2	yes	—
39	C₈×D₉₇	C₂×C₄	1	16	yes	complemented
40	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
41	#13095490	C₂×C₄	1	4	yes	—
42	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
43	#13095488	C₂×C₄	1	4	yes	—
44	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
45	C₄×Dic₉₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
48	M₁₆×C₉₇	C₂×C₄	1	8	yes	complemented
49	C₂×C₈×C₉₇	C₂³	1	4	yes	—
50	C₄²×C₉₇	C₂×C₄	1	2	yes	—
51	C₄²×C₉₇	C₂³	1	1	yes	characteristic
52	C₄²×C₉₇	C₂³	1	1	yes	characteristic
53	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₉₇	—	4	2	no	—
56	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₉₇	C₄²	1	8	yes	—
62	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
63	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
64	D₃₈₈	C₄²	1	4	yes	—
65	C₄×D₉₇	—	4	4	no	—
66	C₄×D₉₇	C₂²×C₄	1	8	yes	—
67	Dic₁₉₄	C₄²	1	4	yes	—
68	C₈×C₉₇	C₂²×C₄	1	8	yes	—
69	#11715895	C₂²×C₄	1	4	yes	—
70	#11715895	C₂²×C₄	1	4	yes	—
71	C₂²×C₉₇	—	2	2	no	—
72	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
73	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₉₇	—	2	4	no	—
75	C₂×D₉₇	C₂×C₄²	1	4	yes	—
76	C₄×C₉₇	—	2	2	no	—
77	C₄×C₉₇	—	2	2	no	—
78	C₄×C₉₇	C₂×C₄²	1	2	yes	—
79	C₄×C₉₇	C₂³×C₄	1	2	yes	—
80	Dic₉₇	—	2	4	no	—
81	Dic₉₇	C₂×C₄²	1	4	yes	—
82	C₂×C₉₇	—	2	2	no	—
83	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
84	C₂×C₉₇	#568	1	2	yes	—
85	D₉₇	—	2	4	no	—
86	#2875	—	97	1	no	maximal, Sylow
87	C₉₇	#2875	1	1	yes	characteristic, complemented, Sylow
88	C₂×#131	—	97	2	no	—
89	C₄×#41	—	97	1	no	—
90	#405	—	97	2	no	—
91	#405	—	97	4	no	—
92	C₄×M₁₆	—	97	2	no	—
93	C₄×M₁₆	D₉₇	1	1	yes	characteristic, complemented
94	C₂×#97	—	97	1	no	—
95	C₂×#97	—	97	2	no	—
96	C₂×#41	—	97	1	no	—
97	#131	—	97	16	no	—
98	C₂×M₁₆	—	97	4	no	—
99	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
100	C₂²×C₈	—	97	2	no	—
101	C₂²×C₈	—	97	4	no	—
102	C₄×Q₈	—	97	2	no	—
103	C₄×D₄	—	97	4	no	—
104	C₄×D₄	—	97	2	no	—
105	#117	—	97	1	no	—
106	#117	—	97	2	no	—
107	C₂×C₄²	—	97	2	no	—
108	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
109	#97	—	97	4	no	—
110	#97	—	97	1	no	—
111	#97	—	97	1	no	—
112	#97	—	97	2	no	—
113	#97	C₂×D₉₇	1	2	yes	—
114	C₄×C₈	—	97	4	no	—
115	C₄×C₈	C₂×D₉₇	1	2	yes	—
116	#41	—	97	8	no	—
117	C₂×Q₈	—	97	1	no	—
118	C₂×D₄	—	97	2	no	—
119	C₂×D₄	—	97	1	no	—
120	C₂²×C₄	—	97	2	no	—
121	C₂²×C₄	—	97	4	no	—
122	C₂²×C₄	C₄×D₉₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
124	M₁₆	—	97	16	no	—
125	M₁₆	C₄×D₉₇	1	8	yes	complemented
126	C₂×C₈	—	97	8	no	—
127	C₂×C₈	—	97	2	no	—
128	C₂×C₈	—	97	2	no	—
129	C₂×C₈	—	97	16	no	—
130	C₂×C₈	C₂²×D₉₇	1	4	yes	—
131	C₄⋊C₄	—	97	4	no	—
132	C₄⋊C₄	—	97	2	no	—
133	C₂²⋊C₄	—	97	2	no	—
134	C₂²⋊C₄	—	97	4	no	—
135	C₄²	—	97	4	no	—
136	C₄²	—	97	2	no	—
137	C₄²	C₄×D₉₇	1	2	yes	—
138	C₄²	C₂²×D₉₇	1	1	yes	characteristic
139	C₄²	C₂²×D₉₇	1	1	yes	characteristic
140	C₂³	—	97	2	no	—
141	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
142	Q₈	—	97	4	no	—
143	D₄	—	97	8	no	—
144	D₄	—	97	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	97	2	no	—
147	C₂×C₄	—	97	8	no	—
148	C₂×C₄	—	97	4	no	—
149	C₂×C₄	—	388	4	no	—
150	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₉₇	1	4	yes	—
154	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
155	C₈	—	97	4	no	—
156	C₈	—	97	4	no	—
157	C₈	C₂×C₄×D₉₇	1	8	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	97	4	no	—
160	C₂²	—	194	4	no	—
161	C₂²	C₄²×D₉₇	1	2	yes	—
162	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	97	4	no	—
166	C₄	—	194	4	no	—
167	C₄	C₄²×D₉₇	1	2	yes	—
168	C₄	C₂²×C₄×D₉₇	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	194	4	no	—
171	C₂	C₂×C₄²×D₉₇	1	1	yes	characteristic
172	C₂	#20261501	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator	C₂×C₉₇	C₂²×C₄²
Fitting	C₄×M₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	#2875
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20260693	C₂	semidirect
4. #20260689	C₂	semidirect
5. #20260687	C₂	semidirect
6. C₂×#15690721	C₂	semidirect
7. #97×D₉₇	C₂	semidirect
8. C₄×M₁₆×C₉₇	C₂	semidirect
10. #15690818	C₄	semidirect
35. #13095494	C₂×C₄	semidirect
36. C₂×#11715895	C₂×C₄	semidirect
39. C₈×D₉₇	C₂×C₄	semidirect
48. M₁₆×C₉₇	C₂×C₄	semidirect
87. C₉₇	#2875	semidirect
93. C₄×M₁₆	D₉₇	semidirect
125. M₁₆	C₄×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690818	2
C₄×#13095511	2
#20260693	2
#20260689	2
#20260687	2
C₂×#15690721	2
#97×D₉₇	2
C₄×M₁₆×C₉₇	2
#2875	97