Finite Group Databasebeta

Subgroups — #20245389

Order 6016.

270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 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	#15689309	C₂	1	2	yes	maximal, complemented
2	#15689300	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688918	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688910	C₂	1	2	yes	maximal, complemented
5	C₄₇×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094069	C₂²	1	1	yes	characteristic
7	#13094055	—	2	1	no	—
8	#13093993	—	2	2	no	—
9	#13093952	—	2	1	no	—
10	#13093941	C₂²	1	1	yes	characteristic
11	#13093939	C₂²	1	2	yes	—
12	#13093938	—	2	2	no	—
13	#124×C₄₇	—	2	1	no	—
14	#124×C₄₇	—	2	2	no	—
15	#124×C₄₇	—	2	1	no	—
16	#124×C₄₇	C₂²	1	1	yes	characteristic
17	#120×C₄₇	—	2	2	no	—
18	#120×C₄₇	C₂²	1	2	yes	—
19	C₂⁴×C₄₇	D₄	1	2	yes	complemented
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic, complemented
22	C₂×D₄×C₄₇	—	2	1	no	—
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	2	no	—
25	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₄₇	—	2	2	no	—
27	C₂×C₂²⋊D₄₇	—	4	2	no	—
28	#10625352	—	4	1	no	—
29	D₄×D₄₇	—	4	1	no	—
30	#10625350	—	2	1	no	—
31	C₂²⋊C₄×C₄₇	—	2	2	no	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₂²⋊C₄×C₄₇	—	2	2	no	—
35	C₂²⋊C₄×C₄₇	D₄	1	2	yes	complemented
36	C₄²×C₄₇	—	2	1	no	—
37	C₄²×C₄₇	D₄	1	1	yes	characteristic, complemented
38	C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂²⋊Dic₄₇	—	4	2	no	—
40	#10625337	—	4	1	no	—
41	#10625336	—	4	1	no	—
42	#10625329	—	2	1	no	—
43	C₂³×C₄₇	—	2	2	no	—
44	C₂³×C₄₇	—	4	2	no	—
45	C₂³×C₄₇	—	8	2	no	—
46	C₂³×C₄₇	C₂×D₄	1	2	yes	—
47	C₂²×D₄₇	—	2	2	no	—
48	Q₈×C₄₇	—	4	1	no	—
49	Q₈×C₄₇	—	8	1	no	—
50	D₄×C₄₇	—	4	1	no	—
51	D₄×C₄₇	—	8	2	no	—
52	D₄×C₄₇	—	8	1	no	—
53	D₄×C₄₇	—	8	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₄₇	—	4	2	no	—
59	C₂²⋊D₄₇	—	4	1	no	—
60	C₂²⋊D₄₇	—	8	4	no	—
61	C₂×Dic₄₇	—	2	2	no	—
62	C₂×Dic₄₇	—	4	2	no	—
63	D₁₈₈	—	2	1	no	—
64	C₄×D₄₇	—	4	1	no	—
65	Dic₉₄	—	2	1	no	—
66	C₄₇⋊C₈	—	4	1	no	—
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	—	4	2	no	—
69	C₂²×C₄₇	—	4	2	no	—
70	C₂²×C₄₇	—	4	2	no	—
71	C₂²×C₄₇	—	4	4	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	8	2	no	—
74	C₂²×C₄₇	#120	1	1	yes	characteristic
75	C₂×D₄₇	—	4	1	no	—
76	C₂×D₄₇	—	8	2	no	—
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	4	2	no	—
79	C₄×C₄₇	—	4	1	no	—
80	C₄×C₄₇	—	4	1	no	—
81	Dic₄₇	—	4	1	no	—
82	Dic₄₇	—	8	2	no	—
83	#2199	—	47	1	no	maximal, Sylow
84	C₂×C₄₇	—	2	1	no	—
85	C₂×C₄₇	—	4	2	no	—
86	C₂×C₄₇	—	8	2	no	—
87	C₂×C₄₇	B(4,𝔽₂)	1	1	yes	characteristic
88	D₄₇	—	8	1	no	—
89	H(3,𝔽₄)	D₄₇	1	1	yes	characteristic, complemented
90	B(4,𝔽₂)	—	47	2	no	—
91	#455	—	47	1	no	—
92	#354	—	47	1	no	—
93	AΣL(1,𝔽₁₆)	—	47	2	no	—
94	C₄₇	#2199	1	1	yes	characteristic, complemented, Sylow
95	H(5,𝔽₂)	—	47	1	no	—
96	#137	—	94	1	no	—
97	#124	—	2	1	no	—
98	#124	—	2	2	no	—
99	#124	—	2	1	no	—
100	#124	C₂×D₄₇	1	1	yes	characteristic
101	#120	—	2	2	no	—
102	#120	—	94	2	no	—
103	#120	C₂×D₄₇	1	2	yes	—
104	#104	—	94	1	no	—
105	#100	—	47	1	no	—
106	#99	—	47	2	no	—
107	#99	—	94	2	no	—
108	C₂⁴	C₂²⋊D₄₇	1	2	yes	complemented
109	#41	—	94	1	no	—
110	#41	—	188	1	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic, complemented
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	94	2	no	—
117	C₂×D₄	—	188	1	no	—
118	C₂×D₄	—	188	2	no	—
119	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
120	Q₁₆	—	188	1	no	—
121	SD₁₆	—	188	1	no	—
122	M₁₆	—	94	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	94	2	no	—
128	C₂²⋊C₄	—	188	2	no	—
129	C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	complemented
130	C₄²	—	2	1	no	—
131	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic, complemented
132	C₂³	—	2	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	8	2	no	—
135	C₂³	—	94	2	no	—
136	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
137	Q₈	—	4	1	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	94	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	1	no	—
143	D₄	—	8	1	no	—
144	D₄	—	94	1	no	—
145	D₄	—	376	4	no	—
146	D₄	—	188	2	no	—
147	D₄	—	188	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	94	2	no	—
152	C₂×C₄	—	188	1	no	—
153	C₂×C₄	—	188	2	no	—
154	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	C₈	—	188	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	4	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	376	2	no	—
164	C₂²	—	188	1	no	—
165	C₂²	#13093993	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	376	2	no	—
171	C₄	—	188	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	8	2	no	—
175	C₂	—	376	1	no	—
176	C₂	#15689310	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15689310
commutator	C₂×D₄×C₄₇	C₂³
Frattini	C₂×D₄	C₂²×D₄₇
Fitting	C₄₇×H(3,𝔽₄)	C₂
socle	C₂×C₄₇	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₄₇
hypocenter	C₄₇	#2199
LCS	G ▷ C₂×D₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #15689309	C₂	semidirect
2. #15689300	C₂	semidirect
3. #15688918	C₂	semidirect
4. #15688910	C₂	semidirect
5. C₄₇×H(3,𝔽₄)	C₂	semidirect
19. C₂⁴×C₄₇	D₄	semidirect
21. C₂×Q₈×C₄₇	D₄	semidirect
35. C₂²⋊C₄×C₄₇	D₄	semidirect
37. C₄²×C₄₇	D₄	semidirect
89. H(3,𝔽₄)	D₄₇	semidirect
94. C₄₇	#2199	semidirect
108. C₂⁴	C₂²⋊D₄₇	semidirect
112. C₂×Q₈	C₂²⋊D₄₇	semidirect
129. C₂²⋊C₄	C₂²⋊D₄₇	semidirect
131. C₄²	C₂²⋊D₄₇	semidirect