Subgroups — #20252331

Order 6016.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689680	C₂	1	1	yes	maximal, characteristic
2	#15689252	C₂	1	4	yes	maximal, complemented
3	#15689250	C₂	1	2	yes	maximal, complemented
4	#15689247	C₂	1	2	yes	maximal, complemented
5	#15689149	C₂	1	2	yes	maximal
6	#15689078	C₂	1	2	yes	maximal
7	C₄×#10625329	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	1	yes	characteristic
11	#13094024	C₄	1	1	yes	characteristic
12	#13094024	C₄	1	1	yes	characteristic
13	#13094016	C₄	1	2	yes	—
14	#13094003	C₄	1	2	yes	—
15	#13094003	C₄	1	2	yes	—
16	C₄×Dic₉₄	C₂²	1	2	yes	—
17	C₄×Dic₉₄	C₂²	1	2	yes	—
18	#13093983	C₂²	1	2	yes	—
19	Dic₄₇⋊C₈	C₂²	1	4	yes	—
20	Dic₄₇⋊C₈	C₂²	1	4	yes	—
21	#13093935	C₂²	1	2	yes	—
22	#13093934	C₂²	1	4	yes	—
23	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
24	C₈×Dic₄₇	C₂²	1	4	yes	—
25	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₄₇	—	2	4	no	—
36	C₂×C₈×C₄₇	C₂³	1	2	yes	—
37	C₂×C₈×C₄₇	C₂³	1	2	yes	—
38	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
47	Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
48	Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
49	C₄×Dic₄₇	C₂³	1	4	yes	—
50	#10625329	—	2	2	no	—
51	#10625329	Q₈	1	4	yes	complemented
52	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
53	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	#41	1	2	yes	—
64	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
66	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
67	Dic₉₄	—	2	4	no	—
68	Dic₉₄	—	4	2	no	—
69	C₈×C₄₇	—	2	2	no	—
70	C₈×C₄₇	—	2	2	no	—
71	C₄₇⋊C₈	—	2	2	no	—
72	C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	—	2	1	no	—
75	C₂²×C₄₇	C₄×Q₈	1	2	yes	—
76	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	C₄×Q₈	1	2	yes	—
81	C₄×C₄₇	#131	1	2	yes	—
82	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
84	Dic₄₇	—	2	4	no	—
85	Dic₄₇	—	2	4	no	—
86	#2962	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	2	2	no	—
88	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
90	C₂×C₄₇	#568	1	1	yes	characteristic
91	#533	—	47	1	no	—
92	#446	—	47	2	no	—
93	#446	—	47	4	no	—
94	C₈×Q₈	—	47	2	no	—
95	#424	—	47	2	no	—
96	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
97	#405	—	47	2	no	—
98	C₄×M₁₆	—	47	1	no	—
99	C₄₇	#2962	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	47	1	no	—
101	C₂×M₁₆	—	47	1	no	—
102	C₂²×C₈	C₂×D₄₇	1	2	yes	—
103	C₄⋊Q₈	—	47	1	no	—
104	C₄⋊Q₈	—	47	1	no	—
105	#125	—	47	2	no	—
106	C₂²⋊Q₈	—	47	2	no	—
107	C₂²⋊Q₈	—	47	2	no	—
108	C₄×Q₈	—	47	2	no	—
109	C₄×Q₈	—	47	2	no	—
110	#117	—	47	2	no	—
111	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	C₄⋊C₈	—	47	4	no	—
113	C₄⋊C₈	—	47	4	no	—
114	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
115	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
116	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
117	#97	—	47	4	no	—
118	#97	—	47	2	no	—
119	C₄×C₈	—	47	4	no	—
120	C₄×C₈	—	47	2	no	—
121	C₂×Q₈	—	47	2	no	—
122	C₂×Q₈	—	47	2	no	—
123	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
126	M₁₆	—	47	4	no	—
127	M₁₆	—	94	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	47	2	no	—
130	C₂×C₈	—	47	2	no	—
131	C₂×C₈	C₂²×D₄₇	1	2	yes	—
132	C₂×C₈	C₂²×D₄₇	1	2	yes	—
133	C₄⋊C₄	—	47	4	no	—
134	C₄⋊C₄	—	47	4	no	—
135	C₄⋊C₄	—	47	2	no	—
136	C₄⋊C₄	—	47	2	no	—
137	C₄⋊C₄	—	47	2	no	—
138	C₄⋊C₄	—	47	2	no	—
139	C₂²⋊C₄	—	47	4	no	—
140	C₄²	—	47	4	no	—
141	C₄²	C₄×D₄₇	1	1	yes	characteristic
142	C₄²	C₄×D₄₇	1	1	yes	characteristic
143	C₄²	C₂²×D₄₇	1	1	yes	characteristic
144	C₄²	C₂²×D₄₇	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
146	Q₈	—	94	4	no	—
147	Q₈	—	188	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	47	4	no	—
151	C₂×C₄	—	47	4	no	—
152	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	#10625352	1	2	yes	—
158	C₂×C₄	Q₈×D₄₇	1	2	yes	—
159	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	47	4	no	—
163	C₈	—	94	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#13093989	1	2	yes	—
167	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	94	4	no	—
172	C₄	—	94	4	no	—
173	C₄	#13093989	1	2	yes	—
174	C₄	#13094030	1	2	yes	—
175	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
176	C₄	C₂×#10625352	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×#13093989	1	1	yes	characteristic
179	C₂	C₂×#13094030	1	1	yes	characteristic
180	C₂	#15689889	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#2962
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #15689252	C₂	semidirect
3. #15689250	C₂	semidirect
4. #15689247	C₂	semidirect
51. #10625329	Q₈	semidirect
99. C₄₇	#2962	semidirect