Subgroups — C₄×#15690717 #27139956

Order 12416.

752 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₉₇	C₂	1	2	yes	maximal
2	C₂×#15690717	C₂	1	4	yes	maximal
3	C₂²×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₉₇	C₂²	1	4	yes	—
5	C₂×C₄×Dic₉₇	—	2	4	no	—
6	C₂×C₄×Dic₉₇	—	2	4	no	—
7	C₂×C₄×Dic₉₇	C₄	1	8	yes	complemented
8	#15690717	C₄	1	16	yes	direct product factor
9	C₂³×C₄×C₉₇	C₂²	1	2	yes	—
10	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉₇	—	2	4	no	—
12	C₂×C₄²×C₉₇	C₄	1	4	yes	complemented
13	C₂²×Dic₉₇	—	2	4	no	—
14	C₂²×Dic₉₇	—	2	4	no	—
15	C₂²×Dic₉₇	—	2	8	no	—
16	C₂²×Dic₉₇	C₂×C₄	1	8	yes	—
17	C₂²×Dic₉₇	C₂×C₄	1	16	yes	—
18	C₄×Dic₉₇	—	2	16	no	—
19	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	—	2	4	no	—
21	C₂²×C₄×C₉₇	—	2	4	no	—
22	C₂²×C₄×C₉₇	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₉₇	D₄	1	2	yes	—
27	C₂²×C₄×C₉₇	D₄	1	4	yes	—
28	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
29	C₄²×C₉₇	—	2	8	no	—
30	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
31	C₂³×C₉₇	C₂²×C₄	1	4	yes	—
32	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂³×C₉₇	C₂×D₄	1	2	yes	—
34	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
36	C₂³×C₉₇	#41	1	4	yes	—
37	C₂×C₄×C₉₇	—	2	4	no	—
38	C₂×C₄×C₉₇	—	2	4	no	—
39	C₂×C₄×C₉₇	—	2	16	no	—
40	C₂×C₄×C₉₇	C₄²	1	4	yes	—
41	C₂×C₄×C₉₇	C₄²	1	8	yes	complemented
42	C₂×C₄×C₉₇	C₂²⋊C₄	1	8	yes	—
43	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
44	C₂×C₄×C₉₇	C₄⋊C₄	1	8	yes	—
45	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₉₇	—	2	8	no	—
47	C₂×Dic₉₇	—	2	8	no	—
48	C₂×Dic₉₇	—	2	16	no	—
49	C₂×Dic₉₇	—	2	16	no	—
50	C₂×Dic₉₇	C₄²	1	16	yes	complemented
51	C₂²×C₉₇	C₂×C₄²	1	4	yes	—
52	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
54	C₂²×C₉₇	C₂×C₂²⋊C₄	1	2	yes	—
55	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₉₇	#117	1	4	yes	—
59	C₂²×C₉₇	#117	1	2	yes	—
60	C₂²×C₉₇	C₄×D₄	1	8	yes	—
61	C₂²×C₉₇	C₄×D₄	1	4	yes	—
62	C₂²×C₉₇	C₄×Q₈	1	4	yes	—
63	C₄×C₉₇	—	2	8	no	—
64	C₄×C₉₇	#95	1	8	yes	complemented
65	Dic₉₇	—	2	16	no	—
66	C₂×C₉₇	C₄³	1	1	yes	characteristic
67	C₂×C₉₇	C₂×#95	1	1	yes	characteristic
68	C₂×C₉₇	#376	1	1	yes	characteristic
69	C₂×C₉₇	C₄×C₂²⋊C₄	1	2	yes	—
70	C₂×C₉₇	C₄×C₂²⋊C₄	1	4	yes	—
71	C₂×C₉₇	C₄×C₄⋊C₄	1	2	yes	—
72	C₂×C₉₇	C₄×C₄⋊C₄	1	4	yes	—
73	C₄×#95	—	97	1	no	maximal, Sylow
74	C₉₇	C₄×#95	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄²	—	97	2	no	—
76	C₂²×C₄²	D₉₇	1	1	yes	characteristic
77	C₂×#95	—	97	4	no	—
78	C₂³×C₄	—	97	4	no	—
79	C₂³×C₄	C₂×D₉₇	1	2	yes	—
80	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	—	97	8	no	—
83	C₂×C₄²	—	194	4	no	—
84	C₂×C₄²	—	194	4	no	—
85	C₂×C₄²	Dic₉₇	1	4	yes	complemented
86	#95	—	97	16	no	—
87	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	97	16	no	—
91	C₂²×C₄	—	97	8	no	—
92	C₂²×C₄	—	194	4	no	—
93	C₂²×C₄	—	194	4	no	—
94	C₂²×C₄	—	194	8	no	—
95	C₂²×C₄	Dic₁₉₄	1	2	yes	—
96	C₂²×C₄	C₄×D₉₇	1	8	yes	—
97	C₂²×C₄	C₄×D₉₇	1	4	yes	—
98	C₂²×C₄	D₃₈₈	1	2	yes	—
99	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
100	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₉₇	1	4	yes	—
102	C₄²	—	2	8	no	—
103	C₄²	—	194	16	no	—
104	C₂³	C₂×C₄×D₉₇	1	2	yes	—
105	C₂³	C₂×C₄×D₉₇	1	4	yes	—
106	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₉₇	1	2	yes	—
108	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
109	C₂³	C₂×Dic₁₉₄	1	1	yes	characteristic
110	C₂³	#13095511	1	4	yes	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	16	no	—
114	C₂×C₄	—	97	16	no	—
115	C₂×C₄	—	194	8	no	—
116	C₂×C₄	—	194	8	no	—
117	C₂×C₄	—	194	16	no	—
118	C₂×C₄	—	194	16	no	—
119	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
120	C₂×C₄	C₄×Dic₉₇	1	8	yes	complemented
121	C₂×C₄	#13095488	1	8	yes	—
122	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
123	C₂×C₄	#13095490	1	8	yes	—
124	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
125	C₂²	C₄²×D₉₇	1	4	yes	—
126	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
127	C₂²	C₂×C₄×Dic₉₇	1	2	yes	—
128	C₂²	C₂×#13095488	1	2	yes	—
129	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
130	C₂²	C₂×#13095490	1	2	yes	—
131	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
132	C₂²	#15690767	1	4	yes	—
133	C₂²	#15690769	1	2	yes	—
134	C₂²	C₄×C₂²⋊D₉₇	1	8	yes	—
135	C₂²	C₄×D₃₈₈	1	4	yes	—
136	C₂²	C₄×Dic₁₉₄	1	4	yes	—
137	C₄	—	2	8	no	—
138	C₄	—	194	16	no	—
139	C₄	#15690717	1	8	yes	direct product factor
140	C₂	C₄²×Dic₉₇	1	1	yes	characteristic
141	C₂	C₂×#15690717	1	1	yes	characteristic
142	C₂	#20260581	1	1	yes	characteristic
143	C₂	C₄×C₂²⋊Dic₉₇	1	2	yes	—
144	C₂	C₄×#13095488	1	4	yes	—
145	C₂	C₄×C₄⋊Dic₉₇	1	2	yes	—
146	C₂	C₄×#13095490	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₉₇
commutator	C₂×C₉₇	C₄³
Frattini	C₂⁴	C₂²×D₉₇
Fitting	C₂²×C₄²×C₉₇	C₂
socle, prime radical, cosocle	C₂⁴×C₉₇	C₂³
hypercenter	C₂²×C₄²	D₉₇
hypocenter	C₉₇	C₄×#95
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×C₄×Dic₉₇	C₄	semidirect
8. #15690717	C₄	direct (N × H)
12. C₂×C₄²×C₉₇	C₄	semidirect
41. C₂×C₄×C₉₇	C₄²	semidirect
50. C₂×Dic₉₇	C₄²	semidirect
64. C₄×C₉₇	#95	semidirect
74. C₉₇	C₄×#95	semidirect
85. C₂×C₄²	Dic₉₇	semidirect
120. C₂×C₄	C₄×Dic₉₇	semidirect
139. C₄	#15690717	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Dic₉₇	2
C₂×#15690717	2
C₂²×C₄²×C₉₇	2
C₄×#95	97

Subgroups — C4×#15690717 #27139956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690717 #27139956