Subgroups — #20245866

Order 6016.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689030	C₂	1	2	yes	maximal
2	#15689022	C₂	1	2	yes	maximal
3	#15689009	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
6	#15688969	C₂	1	2	yes	maximal
7	C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₄₇⋊#95	C₂²	1	4	yes	—
18	C₄₇⋊#95	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₇	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
26	C₂×Q₈×C₄₇	—	2	4	no	—
27	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	C₂³	1	4	yes	—
34	C₂²×Dic₄₇	C₂³	1	2	yes	—
35	C₂²×Dic₄₇	C₂³	1	2	yes	—
36	C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
37	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₄²×C₄₇	C₂×C₄	1	8	yes	—
39	C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₄₇	—	2	4	no	—
41	C₄⋊Dic₄₇	—	2	8	no	—
42	C₄×Dic₄₇	—	2	4	no	—
43	C₄×Dic₄₇	—	2	2	no	—
44	C₄×Dic₄₇	—	2	4	no	—
45	C₄×Dic₄₇	—	2	4	no	—
46	C₄×Dic₄₇	Q₈	1	4	yes	complemented
47	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
48	Q₈×C₄₇	—	2	8	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	—	2	4	no	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₄₇	#41	1	4	yes	—
59	C₂×C₄×C₄₇	#41	1	2	yes	—
60	C₂×Dic₄₇	—	2	2	no	—
61	C₂×Dic₄₇	—	2	2	no	—
62	C₂×Dic₄₇	—	2	2	no	—
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
68	C₂×Dic₄₇	#41	1	4	yes	—
69	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₄₇	C₂×#41	1	2	yes	—
72	C₂²×C₄₇	C₂×#41	1	2	yes	—
73	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
74	C₄×C₄₇	—	2	4	no	—
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	4	no	—
77	C₄×C₄₇	#117	1	4	yes	—
78	C₄×C₄₇	C₄×Q₈	1	4	yes	—
79	Dic₄₇	—	2	4	no	—
80	Dic₄₇	—	2	4	no	—
81	Dic₄₇	—	4	4	no	—
82	#2295	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
84	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
86	C₂×C₄₇	#529	1	2	yes	—
87	C₂×C₄₇	#533	1	2	yes	—
88	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
89	#391	—	47	2	no	—
90	#389	—	47	2	no	—
91	#387	—	47	4	no	—
92	C₄×C₄⋊C₄	—	47	1	no	—
93	C₄×C₄⋊C₄	—	47	2	no	—
94	#376	—	47	2	no	—
95	C₄³	—	47	1	no	—
96	C₄₇	#2295	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄×Q₈	Dic₄₇	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	47	4	no	—
100	C₂×C₄⋊C₄	—	47	2	no	—
101	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
103	C₂×C₄²	—	47	1	no	—
104	C₂×C₄²	—	47	2	no	—
105	C₂×C₄²	—	47	2	no	—
106	C₂×C₄²	—	47	1	no	—
107	C₂×C₄²	—	47	2	no	—
108	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄₇	1	2	yes	—
110	#95	—	47	4	no	—
111	#95	—	47	4	no	—
112	#95	—	47	2	no	—
113	#95	—	47	4	no	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×Dic₄₇	1	4	yes	—
116	C₂²×C₄	—	47	2	no	—
117	C₂²×C₄	—	47	2	no	—
118	C₂²×C₄	—	47	4	no	—
119	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
122	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₄	—	94	8	no	—
125	C₄⋊C₄	—	94	4	no	—
126	C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
127	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
128	C₄²	—	47	4	no	—
129	C₄²	—	94	4	no	—
130	C₄²	—	94	4	no	—
131	C₄²	—	94	2	no	—
132	C₄²	—	94	4	no	—
133	C₄²	C₂×Dic₄₇	1	8	yes	—
134	C₄²	C₂×Dic₄₇	1	4	yes	—
135	C₂³	C₂³×D₄₇	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	47	4	no	—
141	C₂×C₄	—	47	4	no	—
142	C₂×C₄	—	94	2	no	—
143	C₂×C₄	—	94	2	no	—
144	C₂×C₄	—	94	2	no	—
145	C₂×C₄	—	94	2	no	—
146	C₂×C₄	—	94	4	no	—
147	C₂×C₄	—	94	4	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	#10625350	1	4	yes	—
150	C₂×C₄	Q₈×D₄₇	1	2	yes	—
151	C₂×C₄	#10625354	1	2	yes	—
152	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
153	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
154	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
155	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
156	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
157	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
158	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
159	C₂²	#41×D₄₇	1	2	yes	—
160	C₂²	C₂×#10625350	1	2	yes	—
161	C₂²	C₂×#10625354	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	94	4	no	—
166	C₄	—	94	4	no	—
167	C₄	—	188	4	no	—
168	C₄	#13093983	1	4	yes	—
169	C₄	Q₈×Dic₄₇	1	4	yes	—
170	C₂	C₂×#13093983	1	1	yes	characteristic
171	C₂	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
172	C₂	#41×Dic₄₇	1	1	yes	characteristic
173	C₂	#15689660	1	2	yes	—
174	C₂	#15689677	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Frattini	C₂³	C₂³×D₄₇
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#2295
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×Q₈×C₄₇	C₄	semidirect
46. C₄×Dic₄₇	Q₈	semidirect
96. C₄₇	#2295	semidirect
98. C₄×Q₈	Dic₄₇	semidirect