Finite Group Databasebeta

Subgroups — #36374313

Order 22336.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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	#26013685	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013667	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013664	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013636	C₂	1	1	yes	maximal, characteristic, complemented
5	#26013567	C₂	1	1	yes	maximal, characteristic, complemented
6	#26013555	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19150234	—	2	1	no	—
9	#19150233	—	2	1	no	—
10	#19150233	C₂²	1	1	yes	characteristic
11	C₂×D₁₃₉₆	—	2	1	no	—
12	C₂×D₁₃₉₆	—	2	2	no	—
13	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic
14	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₃₄₉	—	2	2	no	—
16	D₄×D₃₄₉	—	2	1	no	—
17	#19150222	—	2	1	no	—
18	#19150220	—	2	1	no	—
19	D₂₇₉₂	—	2	1	no	—
20	#19150218	—	2	1	no	—
21	#19150217	C₂²	1	1	yes	characteristic, complemented
22	#41×C₃₄₉	C₂²	1	1	yes	characteristic
23	M₁₆×C₃₄₉	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₃₄₉	—	2	2	no	—
26	C₂²×D₃₄₉	—	2	1	no	—
27	C₂²×D₃₄₉	D₄	1	2	yes	—
28	Dic₆₉₈	D₄	1	1	yes	characteristic, complemented
29	D₁₃₉₆	—	2	2	no	—
30	D₁₃₉₆	—	2	1	no	—
31	D₁₃₉₆	—	2	1	no	—
32	D₁₃₉₆	—	2	1	no	—
33	D₁₃₉₆	—	2	1	no	—
34	D₁₃₉₆	—	4	4	no	—
35	D₁₃₉₆	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₃₄₉	—	2	2	no	—
37	C₂²⋊D₃₄₉	—	2	1	no	—
38	C₄×D₃₄₉	—	2	2	no	—
39	C₄×D₃₄₉	—	2	1	no	—
40	#14584095	—	2	1	no	—
41	Q₈×C₃₄₉	D₄	1	1	yes	characteristic, complemented
42	D₄×C₃₄₉	—	2	1	no	—
43	D₄×C₃₄₉	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₃₄₉	—	2	1	no	—
45	C₂×C₄×C₃₄₉	—	2	1	no	—
46	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
47	C₈×C₃₄₉	—	2	1	no	—
48	C₂²×C₃₄₉	—	2	1	no	—
49	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
50	C₂×D₃₄₉	—	2	2	no	—
51	C₂×D₃₄₉	—	2	1	no	—
52	C₂×D₃₄₉	—	4	2	no	—
53	C₂×D₃₄₉	—	4	2	no	—
54	C₂×D₃₄₉	—	4	1	no	—
55	C₂×D₃₄₉	—	8	2	no	—
56	C₄×C₃₄₉	—	2	2	no	—
57	C₄×C₃₄₉	—	2	1	no	—
58	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
59	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
60	Dic₃₄₉	—	2	1	no	—
61	C₂×C₃₄₉	—	2	1	no	—
62	C₂×C₃₄₉	—	4	1	no	—
63	C₂×C₃₄₉	#120	1	1	yes	characteristic
64	D₃₄₉	—	4	2	no	—
65	D₃₄₉	—	4	1	no	—
66	D₃₄₉	—	8	1	no	—
67	C₃₄₉	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	349	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	349	1	no	—
70	AGL(1,ℤ/8)	—	349	1	no	—
71	AGL(1,ℤ/8)	—	349	1	no	—
72	#127	—	349	1	no	—
73	#104	—	349	1	no	—
74	#104	D₃₄₉	1	1	yes	characteristic, complemented
75	#100	—	349	1	no	—
76	#41	—	349	1	no	—
77	#41	—	698	1	no	—
78	#41	—	698	1	no	—
79	#41	C₂×D₃₄₉	1	1	yes	characteristic
80	C₂×D₄	—	349	1	no	—
81	C₂×D₄	—	349	1	no	—
82	C₂×D₄	—	698	1	no	—
83	C₂×D₄	—	698	2	no	—
84	C₂×D₄	—	698	2	no	—
85	C₂×D₄	—	698	1	no	—
86	SD₁₆	—	698	1	no	—
87	SD₁₆	—	698	1	no	—
88	D₈	—	698	1	no	—
89	D₈	—	698	1	no	—
90	M₁₆	—	349	1	no	—
91	M₁₆	C₂×D₃₄₉	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₃₄₉	1	1	yes	characteristic, complemented
93	C₂³	—	349	2	no	—
94	C₂³	—	698	2	no	—
95	C₂³	—	698	1	no	—
96	Q₈	—	349	1	no	—
97	Q₈	D₁₃₉₆	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	349	1	no	—
100	D₄	—	1396	4	no	—
101	D₄	—	698	2	no	—
102	D₄	—	698	2	no	—
103	D₄	—	698	1	no	—
104	D₄	—	698	1	no	—
105	D₄	—	698	1	no	—
106	D₄	—	698	1	no	—
107	D₄	—	698	1	no	—
108	D₄	D₁₃₉₆	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	698	2	no	—
112	C₂×C₄	—	698	1	no	—
113	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	698	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	1396	2	no	—
118	C₂²	—	1396	2	no	—
119	C₂²	—	1396	1	no	—
120	C₂²	—	698	1	no	—
121	C₂²	—	698	2	no	—
122	C₂²	—	2792	2	no	—
123	C₂²	D₄×D₃₄₉	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	698	1	no	—
127	C₄	D₄×D₃₄₉	1	1	yes	characteristic
128	C₄	C₂×D₁₃₉₆	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	1396	2	no	—
132	C₂	—	1396	1	no	—
133	C₂	—	2792	1	no	—
134	C₂	#26013609	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013609
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#104×C₃₄₉	C₂
socle	C₂×C₃₄₉	#120
hypercenter	#104	D₃₄₉
hypocenter	C₃₄₉	#453
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #26013685	C₂	semidirect
2. #26013667	C₂	semidirect
3. #26013664	C₂	semidirect
4. #26013636	C₂	semidirect
5. #26013567	C₂	semidirect
6. #26013555	C₂	semidirect
7. #104×C₃₄₉	C₂	semidirect
14. C₂×D₁₃₉₆	C₂²	semidirect
21. #19150217	C₂²	semidirect
23. M₁₆×C₃₄₉	C₂²	semidirect
24. C₄²×C₃₄₉	C₂²	semidirect
28. Dic₆₉₈	D₄	semidirect
35. D₁₃₉₆	D₄	semidirect
41. Q₈×C₃₄₉	D₄	semidirect
43. D₄×C₃₄₉	D₄	semidirect
67. C₃₄₉	#453	semidirect
74. #104	D₃₄₉	semidirect
91. M₁₆	C₂×D₃₄₉	semidirect
92. C₄²	C₂×D₃₄₉	semidirect
97. Q₈	D₁₃₉₆	semidirect
108. D₄	D₁₃₉₆	semidirect

Maximal subgroups

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

subgroup	index
#26013685	2
#26013667	2
#26013664	2
#26013636	2
#26013567	2
#26013555	2
#104×C₃₄₉	2
#453	349