Subgroups — #29505497

Order 15296.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439958	C₂	1	2	yes	maximal
2	#21439951	C₂	1	2	yes	maximal
3	#21439950	C₂	1	4	yes	maximal
4	Q₈×Dic₂₃₉	C₂	1	2	yes	maximal
5	C₄×Dic₄₇₈	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₄₇₈	C₂	1	1	yes	maximal, characteristic
7	#21439925	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₄₇₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₇₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₂₃₉	C₂²	1	4	yes	—
14	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
15	Dic₂₃₉⋊C₄	C₂²	1	2	yes	—
16	Dic₂₃₉⋊C₄	C₂²	1	2	yes	—
17	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
18	C₄×Dic₂₃₉	C₂²	1	2	yes	—
19	C₄×Dic₂₃₉	C₂²	1	4	yes	—
20	C₄×Dic₂₃₉	C₂²	1	2	yes	—
21	C₂×Q₈×C₂₃₉	C₂²	1	2	yes	—
22	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
24	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
25	Dic₄₇₈	—	2	2	no	—
26	Dic₄₇₈	Q₈	1	4	yes	complemented
27	C₂×Dic₂₃₉	C₂³	1	2	yes	—
28	C₂×Dic₂₃₉	C₂³	1	2	yes	—
29	C₂×Dic₂₃₉	C₂³	1	2	yes	—
30	C₂×Dic₂₃₉	C₂³	1	2	yes	—
31	Q₈×C₂₃₉	—	2	4	no	—
32	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₂₃₉	C₂³	1	4	yes	—
34	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
36	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
37	C₄×C₂₃₉	—	2	4	no	—
38	C₄×C₂₃₉	—	2	1	no	—
39	C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
40	C₄×C₂₃₉	#41	1	2	yes	—
41	Dic₂₃₉	—	2	2	no	—
42	Dic₂₃₉	—	2	2	no	—
43	Dic₂₃₉	—	2	2	no	—
44	Dic₂₃₉	C₂×Q₈	1	4	yes	—
45	C₂×C₂₃₉	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₂₃₉	C₂×#41	1	1	yes	characteristic
47	C₂×C₂₃₉	2₋¹⁺⁴	1	1	yes	characteristic
48	C₂₃₉	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	239	1	no	maximal, Sylow
50	C₄⋊Q₈	—	239	2	no	—
51	C₄⋊Q₈	D₂₃₉	1	1	yes	characteristic
52	#125	—	239	4	no	—
53	#125	—	239	2	no	—
54	C₄×Q₈	—	239	2	no	—
55	C₄×Q₈	—	239	2	no	—
56	C₄×Q₈	—	239	1	no	—
57	C₄×Q₈	—	239	1	no	—
58	C₂×Q₈	—	239	1	no	—
59	C₂×Q₈	—	239	1	no	—
60	C₂×Q₈	C₂×D₂₃₉	1	2	yes	—
61	C₄⋊C₄	—	239	1	no	—
62	C₄⋊C₄	—	239	2	no	—
63	C₄⋊C₄	—	239	2	no	—
64	C₄⋊C₄	—	239	4	no	—
65	C₄⋊C₄	—	239	1	no	—
66	C₄⋊C₄	—	239	4	no	—
67	C₄⋊C₄	—	239	4	no	—
68	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
69	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
70	C₄²	—	239	2	no	—
71	C₄²	—	239	4	no	—
72	C₄²	—	239	2	no	—
73	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
74	Q₈	—	2	4	no	—
75	Q₈	—	478	2	no	—
76	Q₈	—	239	4	no	—
77	C₂×C₄	—	239	2	no	—
78	C₂×C₄	—	239	2	no	—
79	C₂×C₄	—	239	2	no	—
80	C₂×C₄	—	239	2	no	—
81	C₂×C₄	C₂²×D₂₃₉	1	4	yes	—
82	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
85	C₂²	C₂³×D₂₃₉	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	478	2	no	—
89	C₄	—	478	2	no	—
90	C₄	—	478	2	no	—
91	C₄	—	239	4	no	—
92	C₄	Q₈×D₂₃₉	1	2	yes	—
93	C₄	#16838547	1	2	yes	—
94	C₂	C₂×Q₈×D₂₃₉	1	1	yes	characteristic
95	C₂	C₂×#16838547	1	1	yes	characteristic
96	C₂	#21440008	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₉
commutator, socle, prime radical, cosocle	C₂²×C₂₃₉	C₂⁴
Fitting	C₄⋊Q₈×C₂₃₉	C₂
hypercenter	C₄⋊Q₈	D₂₃₉
hypocenter	C₂₃₉	#557
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. Dic₄₇₈	Q₈	semidirect
48. C₂₃₉	#557	semidirect

Maximal subgroups

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

subgroup	index
#21439958	2
#21439951	2
#21439950	2
Q₈×Dic₂₃₉	2
C₄×Dic₄₇₈	2
#21439925	2
C₄⋊Q₈×C₂₃₉	2
#557	239