Finite Group Databasebeta

← C₂³×D₄×D₉₇ #27156527

Subgroups — C₂³×D₄×D₉₇ #27156527

Order 12416.

10552 conjugacy classes of subgroups; only the 3557 normal subgroups are shown.

normal subgroups only

This group has 10552 conjugacy classes of subgroups — too many to draw. Only its 3557 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₉₇	C₂	1	2	yes	maximal, complemented
2	C₂³×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₂²⋊D₉₇	C₂	1	2	yes	maximal, complemented
4	C₂²×D₄×D₉₇	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₄×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₉₇	—	2	32	no	—
8	C₂⁴×D₉₇	C₂²	1	1	yes	characteristic
9	C₂⁴×D₉₇	C₂²	1	2	yes	complemented
10	C₂⁴×D₉₇	C₂²	1	56	yes	complemented
11	C₂²×D₃₈₈	C₂²	1	28	yes	complemented
12	C₂²×C₂²⋊D₉₇	C₂²	1	56	yes	complemented
13	C₂×D₄×D₉₇	C₂²	1	448	yes	direct product factor
14	C₂³×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₄×D₉₇	C₂²	1	28	yes	complemented
16	C₂⁵×C₉₇	C₂²	1	2	yes	complemented
17	C₂²×D₄×C₉₇	C₂²	1	28	yes	complemented
18	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₉₇	—	2	16	no	—
20	C₂³×D₉₇	—	2	448	no	—
21	C₂³×D₉₇	D₄	1	16	yes	direct product factor
22	C₂³×D₉₇	C₂³	1	14	yes	—
23	C₂³×D₉₇	C₂³	1	224	yes	complemented
24	C₂³×D₉₇	C₂³	1	28	yes	complemented
25	C₂×D₃₈₈	C₂³	1	112	yes	complemented
26	C₂×C₂²⋊D₉₇	C₂³	1	224	yes	complemented
27	D₄×D₉₇	C₂³	1	512	yes	direct product factor
28	C₂²×Dic₉₇	C₂³	1	14	yes	complemented
29	C₂×C₄×D₉₇	C₂³	1	112	yes	complemented
30	C₂⁴×C₉₇	—	2	16	no	—
31	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
32	C₂⁴×C₉₇	C₂³	1	28	yes	complemented
33	C₂×D₄×C₉₇	C₂³	1	112	yes	complemented
34	C₂²×C₄×C₉₇	C₂³	1	14	yes	complemented
35	C₂³×C₉₇	—	2	112	no	—
36	C₂³×C₉₇	C₂×D₄	1	8	yes	complemented
37	C₂³×C₉₇	C₂⁴	1	7	yes	—
38	C₂³×C₉₇	C₂⁴	1	56	yes	complemented
39	C₂²×D₉₇	—	2	112	no	—
40	C₂²×D₉₇	—	2	896	no	—
41	C₂²×D₉₇	C₂×D₄	1	112	yes	direct product factor
42	C₂²×D₉₇	C₂⁴	1	28	yes	—
43	C₂²×D₉₇	C₂⁴	1	128	yes	complemented
44	C₂²×D₉₇	C₂⁴	1	56	yes	complemented
45	D₄×C₉₇	C₂⁴	1	64	yes	complemented
46	C₂×C₄×C₉₇	C₂⁴	1	28	yes	complemented
47	C₂²⋊D₉₇	C₂⁴	1	128	yes	complemented
48	C₂×Dic₉₇	C₂⁴	1	28	yes	complemented
49	D₃₈₈	C₂⁴	1	64	yes	complemented
50	C₄×D₉₇	C₂⁴	1	64	yes	complemented
51	C₂²×C₉₇	—	2	112	no	—
52	C₂²×C₉₇	C₂²×D₄	1	28	yes	complemented
53	C₂²×C₉₇	C₂⁵	1	7	yes	—
54	C₂²×C₉₇	C₂⁵	1	16	yes	complemented
55	C₂×D₉₇	—	2	112	no	—
56	C₂×D₉₇	—	2	256	no	—
57	C₂×D₉₇	C₂²×D₄	1	112	yes	direct product factor
58	C₂×D₉₇	C₂⁵	1	8	yes	—
59	C₂×D₉₇	C₂⁵	1	16	yes	complemented
60	C₄×C₉₇	C₂⁵	1	8	yes	complemented
61	Dic₉₇	C₂⁵	1	8	yes	complemented
62	C₂×C₉₇	—	2	16	no	—
63	C₂×C₉₇	C₂³×D₄	1	14	yes	complemented
64	C₂×C₉₇	C₂⁶	1	1	yes	characteristic
65	D₉₇	—	2	16	no	—
66	D₉₇	C₂³×D₄	1	16	yes	direct product factor
67	C₂⁴×D₄	—	97	1	no	maximal, Sylow
68	C₉₇	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
69	C₂⁶	—	97	2	no	—
70	C₂³×D₄	—	97	2	no	—
71	C₂³×D₄	—	97	56	no	—
72	C₂³×D₄	—	97	1	no	—
73	C₂³×D₄	D₉₇	1	1	yes	characteristic, direct product factor
74	C₂⁴×C₄	—	97	1	no	—
75	C₂⁵	—	97	1	no	—
76	C₂⁵	—	97	2	no	—
77	C₂⁵	—	97	56	no	—
78	C₂⁵	—	194	32	no	—
79	C₂⁵	C₂×D₉₇	1	2	yes	complemented
80	C₂²×D₄	—	97	56	no	—
81	C₂²×D₄	—	97	448	no	—
82	C₂²×D₄	—	97	28	no	—
83	C₂²×D₄	C₂×D₉₇	1	28	yes	direct product factor
84	C₂³×C₄	—	97	1	no	—
85	C₂³×C₄	—	97	28	no	—
86	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
87	C₂⁴	—	2	16	no	—
88	C₂⁴	—	97	16	no	—
89	C₂⁴	—	97	14	no	—
90	C₂⁴	—	97	28	no	—
91	C₂⁴	—	97	224	no	—
92	C₂⁴	—	194	448	no	—
93	C₂⁴	—	194	16	no	—
94	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
95	C₂⁴	C₂²×D₉₇	1	28	yes	complemented
96	C₂×D₄	—	97	224	no	—
97	C₂×D₄	—	97	512	no	—
98	C₂×D₄	—	97	112	no	—
99	C₂×D₄	C₂²×D₉₇	1	112	yes	direct product factor
100	C₂²×C₄	—	97	14	no	—
101	C₂²×C₄	—	97	112	no	—
102	C₂²×C₄	C₂²×D₉₇	1	14	yes	complemented
103	C₂³	—	2	112	no	—
104	C₂³	—	97	112	no	—
105	C₂³	—	97	28	no	—
106	C₂³	—	97	56	no	—
107	C₂³	—	97	128	no	—
108	C₂³	—	194	896	no	—
109	C₂³	—	194	112	no	—
110	C₂³	D₄×D₉₇	1	8	yes	direct product factor
111	C₂³	C₂³×D₉₇	1	7	yes	—
112	C₂³	C₂³×D₉₇	1	56	yes	complemented
113	D₄	—	97	128	no	—
114	D₄	—	97	64	no	—
115	D₄	C₂³×D₉₇	1	64	yes	direct product factor
116	C₂×C₄	—	97	28	no	—
117	C₂×C₄	—	97	64	no	—
118	C₂×C₄	C₂³×D₉₇	1	28	yes	complemented
119	C₂²	—	2	112	no	—
120	C₂²	—	97	112	no	—
121	C₂²	—	97	8	no	—
122	C₂²	—	97	16	no	—
123	C₂²	—	194	256	no	—
124	C₂²	—	194	112	no	—
125	C₂²	C₂×D₄×D₉₇	1	28	yes	direct product factor
126	C₂²	C₂⁴×D₉₇	1	7	yes	—
127	C₂²	C₂⁴×D₉₇	1	16	yes	complemented
128	C₄	—	97	8	no	—
129	C₄	C₂⁴×D₉₇	1	8	yes	complemented
130	C₂	—	2	16	no	—
131	C₂	—	97	16	no	—
132	C₂	—	194	16	no	—
133	C₂	C₂²×D₄×D₉₇	1	14	yes	direct product factor
134	C₂	C₂⁵×D₉₇	1	1	yes	characteristic

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₉₇	C₂	semidirect
2. C₂³×D₃₈₈	C₂	semidirect
3. C₂³×C₂²⋊D₉₇	C₂	semidirect
4. C₂²×D₄×D₉₇	C₂	direct (N × H)
5. C₂³×C₄×D₉₇	C₂	semidirect
6. C₂³×D₄×C₉₇	C₂	semidirect
9. C₂⁴×D₉₇	C₂²	semidirect
10. C₂⁴×D₉₇	C₂²	semidirect
11. C₂²×D₃₈₈	C₂²	semidirect
12. C₂²×C₂²⋊D₉₇	C₂²	semidirect
13. C₂×D₄×D₉₇	C₂²	direct (N × H)
14. C₂³×Dic₉₇	C₂²	semidirect
15. C₂²×C₄×D₉₇	C₂²	semidirect
16. C₂⁵×C₉₇	C₂²	semidirect
17. C₂²×D₄×C₉₇	C₂²	semidirect
18. C₂³×C₄×C₉₇	C₂²	semidirect
21. C₂³×D₉₇	D₄	direct (N × H)
23. C₂³×D₉₇	C₂³	semidirect
24. C₂³×D₉₇	C₂³	semidirect
25. C₂×D₃₈₈	C₂³	semidirect
26. C₂×C₂²⋊D₉₇	C₂³	semidirect
27. D₄×D₉₇	C₂³	direct (N × H)
28. C₂²×Dic₉₇	C₂³	semidirect
29. C₂×C₄×D₉₇	C₂³	semidirect
32. C₂⁴×C₉₇	C₂³	semidirect
33. C₂×D₄×C₉₇	C₂³	semidirect
34. C₂²×C₄×C₉₇	C₂³	semidirect
36. C₂³×C₉₇	C₂×D₄	semidirect
38. C₂³×C₉₇	C₂⁴	semidirect
41. C₂²×D₉₇	C₂×D₄	direct (N × H)
43. C₂²×D₉₇	C₂⁴	semidirect
44. C₂²×D₉₇	C₂⁴	semidirect
45. D₄×C₉₇	C₂⁴	semidirect
46. C₂×C₄×C₉₇	C₂⁴	semidirect
47. C₂²⋊D₉₇	C₂⁴	semidirect
48. C₂×Dic₉₇	C₂⁴	semidirect
49. D₃₈₈	C₂⁴	semidirect
50. C₄×D₉₇	C₂⁴	semidirect
52. C₂²×C₉₇	C₂²×D₄	semidirect
54. C₂²×C₉₇	C₂⁵	semidirect
57. C₂×D₉₇	C₂²×D₄	direct (N × H)
59. C₂×D₉₇	C₂⁵	semidirect
60. C₄×C₉₇	C₂⁵	semidirect
61. Dic₉₇	C₂⁵	semidirect
63. C₂×C₉₇	C₂³×D₄	semidirect
66. D₉₇	C₂³×D₄	direct (N × H)
68. C₉₇	C₂⁴×D₄	semidirect
73. C₂³×D₄	D₉₇	direct (N × H)
79. C₂⁵	C₂×D₉₇	semidirect
83. C₂²×D₄	C₂×D₉₇	direct (N × H)
86. C₂³×C₄	C₂×D₉₇	semidirect
95. C₂⁴	C₂²×D₉₇	semidirect
99. C₂×D₄	C₂²×D₉₇	direct (N × H)
102. C₂²×C₄	C₂²×D₉₇	semidirect
110. C₂³	D₄×D₉₇	direct (N × H)
112. C₂³	C₂³×D₉₇	semidirect
115. D₄	C₂³×D₉₇	direct (N × H)
118. C₂×C₄	C₂³×D₉₇	semidirect
125. C₂²	C₂×D₄×D₉₇	direct (N × H)
127. C₂²	C₂⁴×D₉₇	semidirect
129. C₄	C₂⁴×D₉₇	semidirect
133. C₂	C₂²×D₄×D₉₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₉₇	2
C₂³×D₃₈₈	2
C₂³×C₂²⋊D₉₇	2
C₂²×D₄×D₉₇	2
C₂³×C₄×D₉₇	2
C₂³×D₄×C₉₇	2
C₂⁴×D₄	97