Subgroups — C₉₇×#1947 #27137905

Order 12416.

256 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107×C₉₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#340×C₉₇	C₂	1	2	yes	maximal
6	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
8	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
9	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉₇	—	2	1	no	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#107×C₉₇	—	2	4	no	—
15	C₄⋊C₈×C₉₇	C₄	1	4	yes	complemented
16	#95×C₉₇	—	2	1	no	—
17	C₂²×C₄×C₉₇	—	2	1	no	—
18	C₂²×C₄×C₉₇	—	2	1	no	—
19	C₂²×C₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
26	C₄⋊C₄×C₉₇	—	2	1	no	—
27	C₄⋊C₄×C₉₇	—	2	1	no	—
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	—	4	2	no	—
31	C₄⋊C₄×C₉₇	—	4	2	no	—
32	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₉₇	Q₈	1	2	yes	—
34	C₄²×C₉₇	—	4	2	no	—
35	C₄²×C₉₇	C₂×C₄	1	2	yes	—
36	C₄²×C₉₇	Q₈	1	2	yes	—
37	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	—	2	1	no	—
39	C₂×C₄×C₉₇	—	2	1	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	4	1	no	—
43	C₂×C₄×C₉₇	—	4	1	no	—
44	C₂×C₄×C₉₇	—	4	1	no	—
45	C₂×C₄×C₉₇	—	4	1	no	—
46	C₂×C₄×C₉₇	—	4	2	no	—
47	C₂×C₄×C₉₇	—	4	2	no	—
48	C₂×C₄×C₉₇	—	4	2	no	—
49	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₉₇	D₈	1	2	yes	—
52	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
58	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
59	C₈×C₉₇	—	2	4	no	—
60	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₉₇	#121	1	1	yes	characteristic
62	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₉₇	#137	1	1	yes	characteristic
67	C₄×C₉₇	—	2	2	no	—
68	C₄×C₉₇	—	4	2	no	—
69	C₄×C₉₇	—	4	2	no	—
70	C₄×C₉₇	—	8	2	no	—
71	C₄×C₉₇	#107	1	4	yes	complemented
72	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
73	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₉₇	#125	1	1	yes	characteristic
75	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₉₇	#389	1	1	yes	characteristic
77	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
78	C₂×C₉₇	#428	1	1	yes	characteristic
79	C₂×C₉₇	#459	1	1	yes	characteristic
80	C₂×C₉₇	#462	1	1	yes	characteristic
81	C₂×C₉₇	#474	1	1	yes	characteristic
82	C₂×C₉₇	#477	1	1	yes	characteristic
83	#1947	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₉₇	#1947	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×#107	C₂×C₉₇	1	2	yes	—
86	C₂×C₄⋊C₈	C₂×C₉₇	1	1	yes	characteristic
87	#384	C₂×C₉₇	1	1	yes	characteristic
88	C₄×C₄⋊C₄	C₂×C₉₇	1	1	yes	characteristic
89	#340	C₂×C₉₇	1	2	yes	—
90	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
98	#107	—	2	4	no	—
99	C₄⋊C₈	C₄×C₉₇	1	4	yes	complemented
100	#95	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
105	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×C₄×C₉₇	1	4	yes	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	D₄×C₉₇	1	2	yes	—
117	C₄⋊C₄	Q₈×C₉₇	1	2	yes	—
118	C₄²	—	4	2	no	—
119	C₄²	C₂×C₄×C₉₇	1	2	yes	—
120	C₄²	Q₈×C₉₇	1	2	yes	—
121	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
134	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
135	C₂×C₄	D₈×C₉₇	1	2	yes	—
136	C₂×C₄	Q₁₆×C₉₇	1	2	yes	—
137	C₂×C₄	C₂²×C₄×C₉₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
139	C₂×C₄	C₂×Q₈×C₉₇	1	1	yes	characteristic
140	C₂×C₄	C₂×Q₈×C₉₇	1	1	yes	characteristic
141	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
142	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₂²	C₂×C₄⋊C₄×C₉₇	1	1	yes	characteristic
145	C₂²	#121×C₉₇	1	1	yes	characteristic
146	C₂²	C₂²⋊Q₈×C₉₇	1	1	yes	characteristic
147	C₂²	C₂×D₈×C₉₇	1	1	yes	characteristic
148	C₂²	C₂×Q₁₆×C₉₇	1	1	yes	characteristic
149	C₂²	AGL(1,ℤ/8)×C₉₇	1	1	yes	characteristic
150	C₂²	#137×C₉₇	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	8	2	no	—
155	C₄	#107×C₉₇	1	4	yes	complemented
156	C₄	C₄×D₄×C₉₇	1	1	yes	characteristic
157	C₄	C₄×Q₈×C₉₇	1	1	yes	characteristic
158	C₄	#125×C₉₇	1	1	yes	characteristic
159	C₄	C₄⋊Q₈×C₉₇	1	1	yes	characteristic
160	C₂	#389×C₉₇	1	1	yes	characteristic
161	C₂	C₂×#107×C₉₇	1	1	yes	characteristic
162	C₂	#428×C₉₇	1	1	yes	characteristic
163	C₂	#459×C₉₇	1	1	yes	characteristic
164	C₂	#462×C₉₇	1	1	yes	characteristic
165	C₂	#474×C₉₇	1	1	yes	characteristic
166	C₂	#477×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₄⋊C₈×C₉₇	C₄	semidirect
71. C₄×C₉₇	#107	semidirect
83. #1947	C₉₇	direct (N × H)
84. C₉₇	#1947	direct (N × H)
99. C₄⋊C₈	C₄×C₉₇	semidirect
155. C₄	#107×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107×C₉₇	2
C₂×C₄⋊C₈×C₉₇	2
#384×C₉₇	2
C₄×C₄⋊C₄×C₉₇	2
#340×C₉₇	2
#1947	97

Subgroups — C97×#1947 #27137905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1947 #27137905