Subgroups — #26127565

Order 12032.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	#20241446	C₂	1	2	yes	maximal, complemented
4	#20241443	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
7	#15689176	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
10	#15688876	C₂²	1	2	yes	—
11	#15688874	—	2	1	no	—
12	C₄₇×#533	—	2	1	no	—
13	C₄₇×#500	C₂²	1	1	yes	characteristic
14	C₄₇×#500	C₂²	1	1	yes	characteristic
15	C₄₇×#499	—	2	1	no	—
16	C₄₇×#499	C₄	1	2	yes	—
17	C₄₇×#498	—	2	1	no	—
18	C₄₇×#427	—	2	1	no	—
19	C₄₇×#427	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	4	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
25	C₄×C₄₇⋊C₈	—	4	1	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₄₇	—	2	1	no	—
29	C₄⋊Q₈×C₄₇	—	2	1	no	—
30	#125×C₄₇	—	2	2	no	—
31	C₂²⋊Q₈×C₄₇	—	4	1	no	—
32	C₂²⋊Q₈×C₄₇	—	4	1	no	—
33	C₄×Q₈×C₄₇	—	4	1	no	—
34	C₄×Q₈×C₄₇	—	4	1	no	—
35	#117×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	#107×C₄₇	—	2	1	no	—
38	#107×C₄₇	—	2	1	no	—
39	#107×C₄₇	C₂×C₄	1	2	yes	—
40	#107×C₄₇	C₂×C₄	1	2	yes	—
41	#106×C₄₇	—	2	2	no	—
42	#106×C₄₇	—	2	2	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	—	4	1	no	—
48	C₂×Q₈×C₄₇	—	4	1	no	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₄⋊C₄×C₄₇	—	4	2	no	—
62	C₄⋊C₄×C₄₇	—	4	2	no	—
63	C₄⋊C₄×C₄₇	—	4	1	no	—
64	C₄⋊C₄×C₄₇	—	4	1	no	—
65	C₂²⋊C₄×C₄₇	—	4	2	no	—
66	C₄²×C₄₇	—	4	2	no	—
67	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₄₇	D₈	1	1	yes	characteristic
69	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₄₇⋊C₁₆	—	4	2	no	—
73	C₂³×C₄₇	#102	1	1	yes	characteristic
74	Q₈×C₄₇	—	8	2	no	—
75	Q₈×C₄₇	—	8	2	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	4	1	no	—
78	C₂×C₄×C₄₇	—	4	2	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	#101	1	2	yes	—
81	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#103	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₄₇	—	2	1	no	—
89	C₈×C₄₇	—	2	1	no	—
90	C₈×C₄₇	#103	1	2	yes	—
91	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
93	C₄₇⋊C₈	—	8	2	no	—
94	#11939	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	—	2	1	no	—
97	C₂²×C₄₇	#332	1	2	yes	—
98	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	1	no	—
102	C₄×C₄₇	—	8	2	no	—
103	C₄×C₄₇	—	8	2	no	—
104	C₄×C₄₇	#332	1	2	yes	—
105	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
106	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
107	C₄×C₄₇	#508	1	2	yes	—
108	#3160	D₄₇	1	1	yes	characteristic
109	C₂×C₄⋊C₁₆	—	47	1	no	—
110	#1564	—	47	1	no	—
111	#1365	—	47	2	no	—
112	#1364	—	47	2	no	—
113	C₂×C₄₇	—	2	2	no	—
114	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
115	C₂×C₄₇	#2138	1	1	yes	characteristic
116	C₂×C₄₇	#2148	1	1	yes	characteristic
117	#533	—	2	1	no	—
118	C₂²×C₁₆	—	94	1	no	—
119	#500	C₂×D₄₇	1	1	yes	characteristic
120	#500	C₂×D₄₇	1	1	yes	characteristic
121	#499	—	2	1	no	—
122	#499	Dic₄₇	1	2	yes	—
123	#498	—	2	1	no	—
124	#431	—	94	1	no	—
125	#427	—	2	1	no	—
126	#427	Dic₄₇	1	2	yes	—
127	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
128	C₄⋊C₁₆	—	47	2	no	—
129	C₄⋊C₁₆	—	94	1	no	—
130	#335	—	47	2	no	—
131	#334	—	94	1	no	—
132	C₄₇	#11939	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	1	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	#125	—	2	2	no	—
138	C₂²⋊Q₈	—	4	1	no	—
139	C₂²⋊Q₈	—	4	1	no	—
140	C₄×Q₈	—	4	1	no	—
141	C₄×Q₈	—	4	1	no	—
142	#117	—	2	2	no	—
143	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂×C₁₆	—	94	2	no	—
145	C₂×C₁₆	—	188	2	no	—
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	#107	C₂×Dic₄₇	1	2	yes	—
149	#107	C₂×Dic₄₇	1	2	yes	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	C₄⋊C₈	—	94	2	no	—
153	C₂²⋊C₈	—	188	1	no	—
154	C₄×C₈	—	188	1	no	—
155	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	#10625334	1	1	yes	characteristic
162	C₂²×C₄	#10625335	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	188	2	no	—
166	C₂×C₈	#10625336	1	2	yes	—
167	C₂×C₈	#10625337	1	2	yes	—
168	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	#10625334	1	1	yes	characteristic
181	C₄²	#10625335	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₁₆	—	188	2	no	—
185	C₂³	#13093945	1	1	yes	characteristic
186	Q₈	—	8	2	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	#13093943	1	2	yes	—
193	C₂×C₄	#13093945	1	1	yes	characteristic
194	C₂×C₄	#13093945	1	1	yes	characteristic
195	C₂×C₄	#13093945	1	1	yes	characteristic
196	C₂×C₄	#13093948	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
199	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	376	2	no	—
203	C₈	#13093948	1	2	yes	—
204	C₈	C₂×#10625336	1	1	yes	characteristic
205	C₈	C₂×#10625337	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#15688869	1	2	yes	—
209	C₂²	C₂×#13093945	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	2	no	—
214	C₄	—	8	2	no	—
215	C₄	#15688869	1	2	yes	—
216	C₄	C₂×#13093943	1	1	yes	characteristic
217	C₄	C₂×#13093948	1	1	yes	characteristic
218	C₄	#15689553	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	C₂×#15688869	1	1	yes	characteristic
221	C₂	#20245095	1	1	yes	characteristic
222	C₂	#20245137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#3160	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#11939
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #20241446	C₂	semidirect
4. #20241443	C₂	semidirect
132. C₄₇	#11939	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20242136	2
#20241446	2
#20241443	2
C₄₇×#3160	2
#11939	47