Subgroups — #27141434

Order 12416.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11715895	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690769	C₂	1	1	yes	maximal, characteristic
3	#20260489	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	—	2	2	no	—
6	C₂²×#11715895	—	2	2	no	—
7	C₂²×#11715895	C₄	1	4	yes	complemented
8	C₂²×#11715895	C₂²	1	2	yes	—
9	#15690769	—	2	2	no	—
10	#15690769	—	2	2	no	—
11	C₂×C₄⋊Dic₉₇	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₉₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₉₇	—	2	2	no	—
16	C₂×M₁₆×C₉₇	—	2	2	no	—
17	C₂×M₁₆×C₉₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
19	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
20	C₂×#11715895	—	2	8	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₉₇	—	2	4	no	—
25	C₂²⋊Dic₉₇	—	2	4	no	—
26	C₄×Dic₉₇	—	2	2	no	—
27	C₄×Dic₉₇	—	2	2	no	—
28	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉₇	D₄	1	2	yes	—
33	C₂²×C₄×C₉₇	D₄	1	2	yes	—
34	C₂²×C₄×C₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
36	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₉₇	—	2	8	no	—
38	C₂×C₈×C₉₇	—	2	2	no	—
39	C₂×C₈×C₉₇	—	2	2	no	—
40	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
41	C₂³×C₉₇	—	2	4	no	—
42	C₂³×C₉₇	C₄²	1	2	yes	—
43	C₂³×C₉₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
45	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₄²	1	2	yes	—
47	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×Dic₉₇	—	2	4	no	—
60	C₂×Dic₉₇	—	2	8	no	—
61	C₈×C₉₇	—	2	4	no	—
62	#11715895	—	2	4	no	—
63	C₂²×C₉₇	—	2	4	no	—
64	C₂²×C₉₇	—	2	4	no	—
65	C₂²×C₉₇	—	2	4	no	—
66	C₂²×C₉₇	#95	1	4	yes	—
67	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₉₇	#131	1	4	yes	—
71	C₄×C₉₇	#95	1	4	yes	—
72	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	Dic₉₇	—	4	4	no	—
77	C₂×C₉₇	—	2	4	no	—
78	C₂×C₉₇	C₂×#95	1	1	yes	characteristic
79	C₂×C₉₇	#405	1	2	yes	—
80	C₂×C₉₇	#408	1	2	yes	—
81	C₂×C₉₇	#424	1	2	yes	—
82	#1729	—	97	1	no	maximal, Sylow
83	C₉₇	#1729	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	D₉₇	1	1	yes	characteristic
85	C₂³×C₈	—	97	1	no	—
86	C₂×#117	—	97	1	no	—
87	#336	—	97	4	no	—
88	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	Dic₉₇	1	4	yes	complemented
92	C₂²×C₈	—	97	4	no	—
93	C₂²×C₈	—	97	2	no	—
94	C₂²×C₈	—	194	2	no	—
95	C₂²×C₈	—	194	2	no	—
96	C₂²×C₈	C₂×D₉₇	1	2	yes	—
97	#117	—	194	2	no	—
98	#117	—	194	2	no	—
99	C₂×C₄⋊C₄	—	97	2	no	—
100	C₂×C₂²⋊C₄	—	97	2	no	—
101	C₂×C₄²	—	97	2	no	—
102	C₂⁴	C₄×D₉₇	1	1	yes	characteristic
103	C₂²×C₄	—	97	4	no	—
104	C₂²×C₄	Dic₁₉₄	1	2	yes	—
105	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₉₇	1	2	yes	—
107	C₂²×C₄	D₃₈₈	1	2	yes	—
108	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
109	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
112	M₁₆	—	2	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	97	4	no	—
116	C₂×C₈	—	194	8	no	—
117	C₂×C₈	—	194	2	no	—
118	C₂×C₈	—	194	2	no	—
119	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
120	C₄⋊C₄	—	194	4	no	—
121	C₂²⋊C₄	—	194	4	no	—
122	C₄²	—	194	2	no	—
123	C₄²	—	194	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₉₇	1	2	yes	—
126	C₂³	#13095488	1	2	yes	—
127	C₂³	#13095490	1	2	yes	—
128	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
129	C₂×C₄	—	194	4	no	—
130	C₂×C₄	—	194	8	no	—
131	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
132	C₂×C₄	#13095488	1	4	yes	—
133	C₂×C₄	#13095488	1	2	yes	—
134	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
135	C₂×C₄	#13095490	1	4	yes	—
136	C₂×C₄	#13095490	1	2	yes	—
137	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	194	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	#15690717	1	4	yes	—
150	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
151	C₂²	C₂×#13095488	1	1	yes	characteristic
152	C₂²	C₂×#13095490	1	1	yes	characteristic
153	C₂²	#15690818	1	4	yes	—
154	C₄	—	388	4	no	—
155	C₄	#15690717	1	4	yes	—
156	C₄	C₂×#13095488	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
158	C₄	C₂×#13095490	1	1	yes	characteristic
159	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×#15690717	1	1	yes	characteristic
162	C₂	#20260693	1	2	yes	—
163	C₂	#20260703	1	2	yes	—
164	C₂	#20260763	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₂²×M₁₆×C₉₇	C₂
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂²×M₁₆	D₉₇
hypocenter	C₉₇	#1729
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20260489	C₂	semidirect
7. C₂²×#11715895	C₄	semidirect
17. C₂×M₁₆×C₉₇	C₄	semidirect
83. C₉₇	#1729	semidirect
91. C₂×M₁₆	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11715895	2
C₂×#15690769	2
#20260489	2
C₂²×M₁₆×C₉₇	2
#1729	97