Subgroups — #26107578

Order 12032.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
2	#20241678	C₂	1	2	yes	maximal
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20241224	C₂	1	2	yes	maximal
5	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093960	C₄	1	2	yes	—
7	C₄×#10625329	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688888	C₄	1	2	yes	—
11	#15688845	—	2	2	no	—
12	#15688844	—	2	2	no	—
13	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
14	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#335	C₄	1	2	yes	—
18	C₄₇×#334	C₄	1	2	yes	—
19	C₂×#10625329	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
22	#13093960	—	2	4	no	—
23	#13093935	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	4	no	—
45	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
50	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₄₇	#41	1	1	yes	characteristic
54	#10625329	—	2	4	no	—
55	#10625329	—	2	4	no	—
56	C₂×C₄₇⋊C₈	—	2	4	no	—
57	C₂×C₄₇⋊C₈	—	2	2	no	—
58	C₂×C₄₇⋊C₈	—	2	4	no	—
59	C₂×C₄₇⋊C₈	—	2	4	no	—
60	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₄₇	#131	1	2	yes	—
73	C₈×C₄₇	—	2	1	no	—
74	C₈×C₄₇	—	2	1	no	—
75	C₈×C₄₇	—	4	4	no	—
76	C₈×C₄₇	#97	1	2	yes	—
77	C₈×C₄₇	#97	1	2	yes	—
78	C₄₇⋊C₈	—	2	4	no	—
79	C₄₇⋊C₈	—	2	2	no	—
80	C₄₇⋊C₈	—	4	4	no	—
81	C₄₇⋊C₈	#108	1	4	yes	complemented
82	#8293	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	1	no	—
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄₇	#446	1	2	yes	—
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
90	C₄×C₄₇	#405	1	1	yes	characteristic
91	C₄×C₄₇	C₂×#108	1	2	yes	—
92	C₄×C₄₇	#436	1	2	yes	—
93	C₄×C₄₇	#443	1	2	yes	—
94	C₄×#108	—	47	1	no	—
95	#1564	D₄₇	1	1	yes	characteristic
96	#1451	—	47	2	no	—
97	C₂×C₈²	—	47	1	no	—
98	#1278	—	47	2	no	—
99	C₂×C₄₇	—	2	2	no	—
100	C₂×C₄₇	#1770	1	1	yes	characteristic
101	C₂×C₄₇	#1772	1	1	yes	characteristic
102	C₂×C₄₇	C₄×#108	1	1	yes	characteristic
103	#431	C₂×D₄₇	1	2	yes	—
104	C₂×#108	—	47	2	no	—
105	C₄×M₁₆	—	47	2	no	—
106	C₂×C₄×C₈	—	47	1	no	—
107	C₂×C₄×C₈	—	47	1	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	#341	—	47	2	no	—
110	#335	Dic₄₇	1	2	yes	—
111	#334	Dic₄₇	1	2	yes	—
112	#322	—	94	2	no	—
113	#322	—	94	2	no	—
114	C₈²	—	47	2	no	—
115	C₈²	—	47	2	no	—
116	C₄₇	#8293	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	47	4	no	—
118	C₂²×C₈	—	47	2	no	—
119	C₂²×C₈	—	94	1	no	—
120	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	#108	—	94	4	no	—
124	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	94	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	47	2	no	—
129	C₄×C₈	—	47	2	no	—
130	C₄×C₈	—	47	1	no	—
131	C₄×C₈	—	47	1	no	—
132	C₄×C₈	—	94	2	no	—
133	C₄×C₈	—	94	1	no	—
134	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
135	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
139	C₂²×C₄	#10625352	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
141	M₁₆	—	94	4	no	—
142	M₁₆	—	94	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	47	2	no	—
146	C₂×C₈	—	47	2	no	—
147	C₂×C₈	—	94	4	no	—
148	C₂×C₈	—	94	4	no	—
149	C₂×C₈	—	94	2	no	—
150	C₂×C₈	—	94	4	no	—
151	C₂×C₈	#10625324	1	4	yes	—
152	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
157	C₄²	D₄×D₄₇	1	1	yes	characteristic
158	C₄²	#10625354	1	1	yes	characteristic
159	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
160	C₂³	#13093989	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄	#13093982	1	1	yes	characteristic
166	C₂×C₄	#13093986	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
169	C₂×C₄	C₂×#10625324	1	2	yes	—
170	C₂×C₄	#13094032	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	47	4	no	—
175	C₈	—	94	4	no	—
176	C₈	—	94	2	no	—
177	C₈	—	188	4	no	—
178	C₈	#13093934	1	2	yes	—
179	C₈	#13093934	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
183	C₂²	#15689250	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×#13093934	1	1	yes	characteristic
187	C₄	#15689081	1	1	yes	characteristic
188	C₄	#108×D₄₇	1	2	yes	—
189	C₄	#15689200	1	2	yes	—
190	C₄	#15689240	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#20243258	1	1	yes	characteristic
193	C₂	#20243269	1	1	yes	characteristic
194	C₂	#108×Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₄×C₄⋊C₄
prime radical	C₂³×C₄₇	C₄×Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#8293
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
81. C₄₇⋊C₈	#108	semidirect
116. C₄₇	#8293	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093960	2
#20241678	2
C₂×C₈×C₄₇⋊C₈	2
#20241224	2
C₄₇×#1564	2
#8293	47