Finite Group Databasebeta

Subgroups — #20245366

Order 6016.

206 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 53 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	#15689563	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689398	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688939	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688930	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688907	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#578	C₂	1	1	yes	maximal, characteristic
8	#13094054	—	2	1	no	—
9	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	#13093970	—	2	2	no	—
11	#13093970	—	2	1	no	—
12	#13093968	—	2	1	no	—
13	#13093964	C₂²	1	1	yes	characteristic
14	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#13093960	C₂²	1	1	yes	characteristic
16	#13093942	—	2	1	no	—
17	2₋¹⁺⁴×C₄₇	—	2	1	no	—
18	#137×C₄₇	—	2	2	no	—
19	#137×C₄₇	—	2	1	no	—
20	#135×C₄₇	—	2	1	no	—
21	#135×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₄₇	—	2	1	no	—
23	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	#131×C₄₇	C₂²	1	1	yes	characteristic
25	#41×C₄₇	—	2	2	no	—
26	#41×C₄₇	—	2	1	no	—
27	#41×C₄₇	—	4	2	no	—
28	#41×C₄₇	—	4	1	no	—
29	#41×C₄₇	D₄	1	1	yes	characteristic
30	C₂×Q₈×C₄₇	—	2	2	no	—
31	C₂×Q₈×C₄₇	—	2	1	no	—
32	C₂×Q₈×C₄₇	—	4	1	no	—
33	C₂×Dic₉₄	—	2	1	no	—
34	Q₁₆×C₄₇	—	2	2	no	—
35	Q₁₆×C₄₇	—	2	1	no	—
36	Q₁₆×C₄₇	D₄	1	1	yes	characteristic
37	Q₁₆×C₄₇	D₄	1	2	yes	—
38	SD₁₆×C₄₇	—	2	2	no	—
39	SD₁₆×C₄₇	—	2	1	no	—
40	D₈×C₄₇	D₄	1	1	yes	characteristic
41	M₁₆×C₄₇	—	2	1	no	—
42	M₁₆×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	—	2	1	no	—
44	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	#10625337	—	4	1	no	—
46	#10625335	—	4	1	no	—
47	#10625329	—	2	1	no	—
48	Dic₁₈₈	—	2	1	no	—
49	Dic₁₈₈	—	2	1	no	—
50	C₄₇⋊C₁₆	—	2	1	no	—
51	C₄₇⋊C₁₆	—	2	1	no	—
52	Q₈×C₄₇	—	2	2	no	—
53	Q₈×C₄₇	—	2	1	no	—
54	Q₈×C₄₇	—	4	2	no	—
55	Q₈×C₄₇	—	4	1	no	—
56	Q₈×C₄₇	D₈	1	1	yes	characteristic
57	D₄×C₄₇	—	2	1	no	—
58	D₄×C₄₇	—	2	1	no	—
59	D₄×C₄₇	—	4	2	no	—
60	D₄×C₄₇	—	4	1	no	—
61	D₄×C₄₇	D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	4	2	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	1	no	—
66	C₂×C₄×C₄₇	—	4	1	no	—
67	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₄₇	—	4	1	no	—
69	Dic₉₄	—	4	1	no	—
70	Dic₉₄	—	4	1	no	—
71	C₈×C₄₇	—	2	1	no	—
72	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄₇⋊C₈	—	4	1	no	—
75	C₂²×C₄₇	—	2	1	no	—
76	C₂²×C₄₇	—	4	1	no	—
77	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	4	2	no	—
80	C₄×C₄₇	—	4	1	no	—
81	C₄×C₄₇	#120	1	1	yes	characteristic
82	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
83	Dic₄₇	—	8	1	no	—
84	#2195	—	47	1	no	maximal, Sylow
85	C₂×C₄₇	—	2	1	no	—
86	C₂×C₄₇	—	4	1	no	—
87	C₂×C₄₇	—	8	1	no	—
88	C₂×C₄₇	#447	1	1	yes	characteristic
89	#578	D₄₇	1	1	yes	characteristic
90	#510	—	47	1	no	—
91	C₂×Q₃₂	—	47	1	no	—
92	#473	—	47	1	no	—
93	#362	—	47	1	no	—
94	#359	—	47	1	no	—
95	#350	—	47	1	no	—
96	C₄₇	#2195	1	1	yes	characteristic, complemented, Sylow
97	2₋¹⁺⁴	—	2	1	no	—
98	#137	—	2	2	no	—
99	#137	—	2	1	no	—
100	#137	—	94	1	no	—
101	#135	—	2	1	no	—
102	#135	C₂×D₄₇	1	1	yes	characteristic
103	C₂×Q₁₆	—	2	1	no	—
104	C₂×Q₁₆	—	47	1	no	—
105	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
106	#131	C₂×D₄₇	1	1	yes	characteristic
107	Q₃₂	—	94	2	no	—
108	Q₃₂	—	94	1	no	—
109	SD₃₂	—	94	1	no	—
110	M₃₂	—	47	1	no	—
111	C₂×C₁₆	—	47	1	no	—
112	#108	—	47	1	no	—
113	#101	—	94	1	no	—
114	#41	—	2	2	no	—
115	#41	—	2	1	no	—
116	#41	—	4	2	no	—
117	#41	—	4	1	no	—
118	#41	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	4	1	no	—
122	C₂×Q₈	—	94	1	no	—
123	Q₁₆	—	2	2	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	94	1	no	—
126	Q₁₆	—	94	1	no	—
127	Q₁₆	—	188	1	no	—
128	Q₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
129	Q₁₆	C₂²⋊D₄₇	1	2	yes	—
130	SD₁₆	—	2	2	no	—
131	SD₁₆	—	2	1	no	—
132	SD₁₆	—	188	1	no	—
133	D₈	C₂²⋊D₄₇	1	1	yes	characteristic
134	M₁₆	—	2	1	no	—
135	M₁₆	—	94	1	no	—
136	M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
139	C₁₆	—	94	1	no	—
140	C₁₆	—	94	1	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	2	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	188	1	no	—
146	Q₈	—	188	1	no	—
147	Q₈	#10625334	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	1	no	—
152	D₄	#10625334	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	188	1	no	—
159	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	188	1	no	—
162	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
163	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#13094050	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	376	1	no	—
171	C₄	#13093993	1	1	yes	characteristic
172	C₄	C₂×#10625334	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	8	1	no	—
176	C₂	#15689257	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15689257
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#578	C₂
socle	C₂×C₄₇	#447
hypercenter	#578	D₄₇
hypocenter	C₄₇	#2195
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #578

Complements

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

N	H ≅ G/N	type
1. #15689563	C₂	semidirect
2. C₂×#13093970	C₂	semidirect
3. #15689398	C₂	semidirect
4. #15688939	C₂	semidirect
5. #15688930	C₂	semidirect
6. #15688907	C₂	semidirect
96. C₄₇	#2195	semidirect