Finite Group Databasebeta

Subgroups — #14580937

Order 2624.

240 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13050160	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050155	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050147	C₂	1	2	yes	maximal, complemented
4	#13050146	C₂	1	2	yes	maximal, complemented
5	#13050139	C₂	1	1	yes	maximal, characteristic, complemented
6	#13050138	C₂	1	2	yes	maximal, complemented
7	#13050137	C₂	1	2	yes	maximal, complemented
8	#13050135	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#13050121	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
17	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
18	C₂×D₁₆₄	C₂²	1	2	yes	—
19	C₂×C₄×D₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄₁	C₂²	1	2	yes	—
21	C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₄²×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
25	#10620281	C₂²	1	1	yes	characteristic
26	#10620281	C₂²	1	1	yes	characteristic
27	#10620281	C₂²	1	2	yes	—
28	#10620281	C₂²	1	2	yes	—
29	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₄₁	C₂²	1	2	yes	—
31	#10620279	C₂²	1	1	yes	characteristic
32	#10620279	C₂²	1	1	yes	characteristic
33	#10620279	C₂²	1	2	yes	—
34	C₄×Dic₄₁	C₂²	1	1	yes	characteristic
35	C₂³×C₄₁	C₂³	1	2	yes	—
36	C₂²×D₄₁	C₂³	1	1	yes	characteristic
37	C₂²×D₄₁	C₂³	1	2	yes	—
38	D₄×C₄₁	—	2	2	no	—
39	C₂×C₄×C₄₁	—	4	2	no	—
40	C₂×C₄×C₄₁	C₂³	1	2	yes	—
41	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
44	C₂²⋊D₄₁	—	4	2	no	—
45	C₂²⋊D₄₁	—	4	2	no	—
46	C₂²⋊D₄₁	—	4	2	no	—
47	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
49	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
50	C₂×Dic₄₁	C₂³	1	2	yes	—
51	D₁₆₄	—	2	2	no	—
52	D₁₆₄	—	4	2	no	—
53	C₄×D₄₁	—	2	2	no	—
54	C₄×D₄₁	—	4	2	no	—
55	C₂²×C₄₁	—	2	2	no	—
56	C₂²×C₄₁	—	2	2	no	—
57	C₂²×C₄₁	—	2	2	no	—
58	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
59	C₂×D₄₁	—	2	1	no	—
60	C₂×D₄₁	—	2	2	no	—
61	C₂×D₄₁	—	2	2	no	—
62	C₂×D₄₁	—	2	1	no	—
63	C₂×D₄₁	—	2	2	no	—
64	C₂×D₄₁	—	2	1	no	—
65	C₄×C₄₁	—	2	1	no	—
66	C₄×C₄₁	—	2	2	no	—
67	C₄×C₄₁	—	2	1	no	—
68	C₄×C₄₁	#41	1	2	yes	—
69	Dic₄₁	—	2	1	no	—
70	Dic₄₁	—	2	2	no	—
71	Dic₄₁	—	2	1	no	—
72	Dic₄₁	—	2	1	no	—
73	C₂×C₄₁	—	4	2	no	—
74	C₂×C₄₁	C₂×#41	1	1	yes	characteristic
75	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
77	D₄₁	—	4	1	no	—
78	D₄₁	—	4	2	no	—
79	#540	—	41	1	no	maximal, Sylow
80	C₄₁	#540	1	1	yes	characteristic, complemented, Sylow
81	#127	—	41	1	no	—
82	#125	—	41	1	no	—
83	#123	—	41	2	no	—
84	#123	—	41	2	no	—
85	#121	—	41	1	no	—
86	#121	—	41	2	no	—
87	#121	—	41	1	no	—
88	#121	—	41	2	no	—
89	C₄×D₄	—	41	1	no	—
90	C₄×D₄	D₄₁	1	1	yes	characteristic, complemented
91	#117	—	41	1	no	—
92	C₂×D₄	—	41	2	no	—
93	C₂×D₄	—	41	2	no	—
94	C₂×D₄	—	41	2	no	—
95	C₂×D₄	—	41	1	no	—
96	C₂×D₄	—	41	2	no	—
97	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
98	C₂²×C₄	—	41	1	no	—
99	C₂²×C₄	—	41	2	no	—
100	C₂²×C₄	C₂×D₄₁	1	2	yes	—
101	C₄⋊C₄	—	41	1	no	—
102	C₄⋊C₄	—	41	1	no	—
103	C₄⋊C₄	—	41	1	no	—
104	C₄⋊C₄	—	41	2	no	—
105	C₄⋊C₄	—	41	2	no	—
106	C₄⋊C₄	C₂×D₄₁	1	1	yes	characteristic
107	C₂²⋊C₄	—	41	2	no	—
108	C₂²⋊C₄	—	41	1	no	—
109	C₂²⋊C₄	—	41	1	no	—
110	C₂²⋊C₄	—	41	2	no	—
111	C₂²⋊C₄	—	41	2	no	—
112	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
113	C₄²	—	41	1	no	—
114	C₄²	C₂×D₄₁	1	1	yes	characteristic
115	C₂³	—	41	1	no	—
116	C₂³	—	41	2	no	—
117	C₂³	C₂²×D₄₁	1	2	yes	—
118	D₄	—	2	2	no	—
119	D₄	—	82	2	no	—
120	D₄	—	164	2	no	—
121	D₄	—	164	2	no	—
122	D₄	—	164	2	no	—
123	D₄	—	164	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	41	1	no	—
126	C₂×C₄	—	41	1	no	—
127	C₂×C₄	—	41	1	no	—
128	C₂×C₄	—	41	2	no	—
129	C₂×C₄	—	82	2	no	—
130	C₂×C₄	—	164	2	no	—
131	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
132	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₄₁	1	2	yes	—
134	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	82	1	no	—
139	C₂²	—	82	2	no	—
140	C₂²	—	82	1	no	—
141	C₂²	—	82	2	no	—
142	C₂²	—	82	2	no	—
143	C₂²	—	82	1	no	—
144	C₂²	C₂³×D₄₁	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	82	1	no	—
149	C₄	—	82	1	no	—
150	C₄	—	82	2	no	—
151	C₄	—	82	1	no	—
152	C₄	#10620306	1	2	yes	—
153	C₂	—	4	2	no	—
154	C₂	—	164	1	no	—
155	C₂	—	164	2	no	—
156	C₂	C₂×#10620306	1	1	yes	characteristic
157	C₂	#13050209	1	1	yes	characteristic
158	C₂	#13050210	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₄×D₄×C₄₁	C₂
hypercenter	C₄×D₄	D₄₁
hypocenter	C₄₁	#540
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13050160	C₂	semidirect
2. #13050155	C₂	semidirect
3. #13050147	C₂	semidirect
4. #13050146	C₂	semidirect
5. #13050139	C₂	semidirect
6. #13050138	C₂	semidirect
7. #13050137	C₂	semidirect
8. #13050135	C₂	semidirect
9. C₄×D₁₆₄	C₂	semidirect
10. #13050121	C₂	semidirect
11. C₄×D₄×C₄₁	C₂	semidirect
80. C₄₁	#540	semidirect
90. C₄×D₄	D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050160	2
#13050155	2
#13050147	2
#13050146	2
#13050139	2
#13050138	2
#13050137	2
#13050135	2
C₄×D₁₆₄	2
#13050121	2
C₄×D₄×C₄₁	2
#540	41