Subgroups — Q₈×C₄⋊C₄×C₉₇ #27138308

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#389×C₉₇	C₂	1	6	yes	maximal
4	#384×C₉₇	C₂	1	3	yes	maximal
5	C₄×C₄⋊C₄×C₉₇	C₂	1	3	yes	maximal
6	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₄×Q₈×C₉₇	—	2	8	no	—
8	C₄×Q₈×C₉₇	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄×C₉₇	C₂²	1	6	yes	—
10	C₂×C₄⋊C₄×C₉₇	C₂²	1	3	yes	—
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	6	yes	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉₇	C₂²	1	6	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
16	#95×C₉₇	C₂²	1	6	yes	—
17	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	—
18	C₂×Q₈×C₉₇	D₄	1	4	yes	—
19	C₂×Q₈×C₉₇	Q₈	1	4	yes	—
20	C₂²×C₄×C₉₇	C₂³	1	6	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
24	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₉₇	—	2	12	no	—
26	C₄⋊C₄×C₉₇	—	2	6	no	—
27	C₄⋊C₄×C₉₇	—	4	6	no	—
28	C₄⋊C₄×C₉₇	C₂×C₄	1	12	yes	—
29	C₄⋊C₄×C₉₇	Q₈	1	4	yes	direct product factor
30	C₄²×C₉₇	—	2	12	no	—
31	C₄²×C₉₇	C₂×C₄	1	12	yes	—
32	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
33	Q₈×C₉₇	C₄⋊C₄	1	16	yes	direct product factor
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	3	no	—
36	C₂×C₄×C₉₇	—	2	3	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	6	no	—
39	C₂×C₄×C₉₇	—	2	6	no	—
40	C₂×C₄×C₉₇	—	2	6	no	—
41	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×D₄	1	6	yes	—
45	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₉₇	C₂×Q₈	1	6	yes	—
47	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₉₇	#41	1	2	yes	—
49	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
54	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
56	C₄×C₉₇	—	2	6	no	—
57	C₄×C₉₇	—	2	4	no	—
58	C₄×C₉₇	—	4	6	no	—
59	C₄×C₉₇	C₂×C₄⋊C₄	1	12	yes	—
60	C₄×C₉₇	C₄×Q₈	1	4	yes	complemented
61	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₉₇	#520	1	1	yes	characteristic
64	C₂×C₉₇	D₄×Q₈	1	1	yes	characteristic
65	C₂×C₉₇	#550	1	1	yes	characteristic
66	C₂×C₉₇	#557	1	1	yes	characteristic
67	C₂×C₉₇	Q₈²	1	1	yes	characteristic
68	Q₈×C₄⋊C₄	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₉₇	Q₈×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₄×Q₈	C₂×C₉₇	1	2	yes	—
71	C₂×C₄×Q₈	C₂×C₉₇	1	1	yes	characteristic
72	#389	C₂×C₉₇	1	6	yes	—
73	#384	C₂×C₉₇	1	3	yes	—
74	C₄×C₄⋊C₄	C₂×C₉₇	1	3	yes	—
75	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
76	C₄×Q₈	—	2	8	no	—
77	C₄×Q₈	C₄×C₉₇	1	8	yes	complemented
78	C₂×C₄⋊C₄	C₂²×C₉₇	1	6	yes	—
79	C₂×C₄⋊C₄	C₂²×C₉₇	1	3	yes	—
80	C₂×C₄⋊C₄	C₂²×C₉₇	1	6	yes	—
81	C₂×C₄⋊C₄	C₂²×C₉₇	1	3	yes	—
82	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
83	C₂×C₄²	C₂²×C₉₇	1	6	yes	—
84	C₂×C₄²	C₂²×C₉₇	1	3	yes	—
85	#95	C₂²×C₉₇	1	6	yes	—
86	C₂×Q₈	C₂×C₄×C₉₇	1	4	yes	—
87	C₂×Q₈	D₄×C₉₇	1	4	yes	—
88	C₂×Q₈	Q₈×C₉₇	1	4	yes	—
89	C₂²×C₄	C₂³×C₉₇	1	6	yes	—
90	C₂²×C₄	C₂³×C₉₇	1	3	yes	—
91	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
92	C₂²×C₄	C₂³×C₉₇	1	3	yes	—
93	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	—	4	6	no	—
97	C₄⋊C₄	C₂×C₄×C₉₇	1	12	yes	—
98	C₄⋊C₄	Q₈×C₉₇	1	4	yes	direct product factor
99	C₄²	—	2	12	no	—
100	C₄²	C₂×C₄×C₉₇	1	12	yes	—
101	C₂³	C₂⁴×C₉₇	1	1	yes	characteristic
102	Q₈	C₄⋊C₄×C₉₇	1	16	yes	direct product factor
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	6	no	—
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	C₂²×C₄×C₉₇	1	6	yes	—
111	C₂×C₄	C₂²×C₄×C₉₇	1	6	yes	—
112	C₂×C₄	C₂²×C₄×C₉₇	1	2	yes	—
113	C₂×C₄	C₂×D₄×C₉₇	1	6	yes	—
114	C₂×C₄	C₂×Q₈×C₉₇	1	4	yes	—
115	C₂×C₄	C₂×Q₈×C₉₇	1	6	yes	—
116	C₂×C₄	C₂×Q₈×C₉₇	1	2	yes	—
117	C₂×C₄	#41×C₉₇	1	2	yes	—
118	C₂²	C₂³×C₄×C₉₇	1	1	yes	characteristic
119	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
120	C₂²	C₂²×Q₈×C₉₇	1	1	yes	characteristic
121	C₂²	C₂²×Q₈×C₉₇	1	1	yes	characteristic
122	C₂²	C₂×#41×C₉₇	1	1	yes	characteristic
123	C₂²	H(5,𝔽₂)×C₉₇	1	1	yes	characteristic
124	C₂²	2₋¹⁺⁴×C₉₇	1	1	yes	characteristic
125	C₄	—	2	6	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	6	no	—
128	C₄	C₂×C₄⋊C₄×C₉₇	1	12	yes	—
129	C₄	C₄×Q₈×C₉₇	1	4	yes	complemented
130	C₂	C₂²×C₄⋊C₄×C₉₇	1	1	yes	characteristic
131	C₂	C₂×C₄×Q₈×C₉₇	1	1	yes	characteristic
132	C₂	#520×C₉₇	1	1	yes	characteristic
133	C₂	D₄×Q₈×C₉₇	1	1	yes	characteristic
134	C₂	#550×C₉₇	1	1	yes	characteristic
135	C₂	#557×C₉₇	1	1	yes	characteristic
136	C₂	Q₈²×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₉₇
Frattini, cosocle	C₂³	C₂⁴×C₉₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₄×Q₈×C₉₇	C₄	semidirect
29. C₄⋊C₄×C₉₇	Q₈	direct (N × H)
33. Q₈×C₉₇	C₄⋊C₄	direct (N × H)
60. C₄×C₉₇	C₄×Q₈	semidirect
68. Q₈×C₄⋊C₄	C₉₇	direct (N × H)
69. C₉₇	Q₈×C₄⋊C₄	direct (N × H)
77. C₄×Q₈	C₄×C₉₇	semidirect
98. C₄⋊C₄	Q₈×C₉₇	direct (N × H)
102. Q₈	C₄⋊C₄×C₉₇	direct (N × H)
129. C₄	C₄×Q₈×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉₇	2
#389×C₉₇	2
#384×C₉₇	2
C₄×C₄⋊C₄×C₉₇	2
Q₈×C₄⋊C₄	97

Subgroups — Q8×C4⋊C4×C97 #27138308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₄⋊C₄×C₉₇ #27138308