Subgroups — #26134846

Order 12032.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241485	C₂	1	4	yes	maximal
4	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689680	—	2	1	no	—
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	#15689528	C₄	1	1	yes	characteristic
8	#15689528	C₄	1	1	yes	characteristic
9	#15689525	—	2	1	no	—
10	#15689525	C₄	1	2	yes	—
11	#15689521	—	2	1	no	—
12	#15689161	—	2	1	no	—
13	#15689161	C₂²	1	2	yes	—
14	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
16	C₂×#108×C₄₇	C₂²	1	2	yes	—
17	C₄×M₁₆×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#341	—	2	1	no	—
20	#13094024	—	2	1	no	—
21	#13094024	—	2	1	no	—
22	#13094016	—	2	2	no	—
23	#13094003	—	4	1	no	—
24	#13094003	—	4	1	no	—
25	C₄×Dic₉₄	—	4	1	no	—
26	C₄×Dic₉₄	—	4	1	no	—
27	#13093983	—	2	2	no	—
28	C₂×C₄₇⋊C₁₆	—	2	2	no	—
29	C₂×C₄₇⋊C₁₆	—	2	4	no	—
30	#13093958	—	2	1	no	—
31	#13093958	—	2	1	no	—
32	#13093958	C₂×C₄	1	2	yes	—
33	#13093958	C₂×C₄	1	2	yes	—
34	#13093956	—	2	2	no	—
35	#13093956	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#108×C₄₇	—	2	2	no	—
41	#108×C₄₇	Q₈	1	4	yes	complemented
42	#97×C₄₇	—	4	1	no	—
43	C₄×C₈×C₄₇	—	4	1	no	—
44	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₉₄	—	4	1	no	—
52	C₂×Dic₉₄	—	4	1	no	—
53	M₁₆×C₄₇	—	2	4	no	—
54	M₁₆×C₄₇	—	4	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₄₇	#41	1	2	yes	—
63	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₄₇	D₈	1	1	yes	characteristic
66	C₄²×C₄₇	D₈	1	1	yes	characteristic
67	C₂²⋊Dic₄₇	—	4	2	no	—
68	C₄⋊Dic₄₇	—	2	2	no	—
69	C₄⋊Dic₄₇	—	2	2	no	—
70	C₄⋊Dic₄₇	—	4	1	no	—
71	C₄⋊Dic₄₇	—	4	1	no	—
72	Dic₄₇⋊C₄	—	4	2	no	—
73	Dic₄₇⋊C₄	—	4	2	no	—
74	C₄×Dic₄₇	—	4	2	no	—
75	C₄₇⋊C₁₆	—	4	2	no	—
76	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	4	1	no	—
79	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#137	1	2	yes	—
87	C₂×Dic₄₇	—	4	2	no	—
88	C₂×Dic₄₇	—	4	2	no	—
89	Dic₉₄	—	8	2	no	—
90	Dic₉₄	—	8	2	no	—
91	C₈×C₄₇	—	2	1	no	—
92	C₈×C₄₇	—	2	1	no	—
93	C₈×C₄₇	—	4	4	no	—
94	C₈×C₄₇	C₄×Q₈	1	2	yes	—
95	C₈×C₄₇	#124	1	1	yes	characteristic
96	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
97	#13274	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	—	2	1	no	—
99	C₂²×C₄₇	—	2	1	no	—
100	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
101	C₂²×C₄₇	#475	1	2	yes	—
102	C₄×C₄₇	—	2	1	no	—
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	#359	1	2	yes	—
106	C₄×C₄₇	#391	1	1	yes	characteristic
107	C₄×C₄₇	#419	1	1	yes	characteristic
108	C₄×C₄₇	#474	1	2	yes	—
109	Dic₄₇	—	8	2	no	—
110	Dic₄₇	—	8	2	no	—
111	#3160	—	47	1	no	—
112	C₂×C₄⋊C₁₆	—	47	1	no	—
113	C₄×#108	D₄₇	1	1	yes	characteristic
114	#1382	—	47	4	no	—
115	C₂×C₄₇	—	2	2	no	—
116	C₂×C₄₇	#1988	1	1	yes	characteristic
117	C₂×C₄₇	C₂×#359	1	1	yes	characteristic
118	C₂×C₄₇	#2143	1	1	yes	characteristic
119	#533	—	94	1	no	—
120	C₂²×C₁₆	—	47	2	no	—
121	#500	—	47	1	no	—
122	#500	—	47	1	no	—
123	#499	—	47	2	no	—
124	#499	—	94	1	no	—
125	#498	—	94	1	no	—
126	C₂×#108	C₂×D₄₇	1	2	yes	—
127	#427	—	47	2	no	—
128	#427	—	94	1	no	—
129	C₄×M₁₆	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₄⋊C₁₆	—	94	1	no	—
132	C₄⋊C₁₆	—	94	1	no	—
133	#341	—	2	1	no	—
134	C₄₇	#13274	1	1	yes	characteristic, complemented, Sylow
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	94	1	no	—
139	C₄⋊Q₈	—	94	1	no	—
140	#125	—	94	2	no	—
141	C₂²⋊Q₈	—	188	1	no	—
142	C₂²⋊Q₈	—	188	1	no	—
143	C₄×Q₈	—	188	1	no	—
144	C₄×Q₈	—	188	1	no	—
145	#117	—	94	2	no	—
146	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
147	C₂×C₁₆	—	94	2	no	—
148	C₂×C₁₆	—	94	4	no	—
149	#108	—	2	2	no	—
150	#108	Dic₉₄	1	4	yes	complemented
151	#107	—	47	2	no	—
152	#107	—	47	2	no	—
153	#107	—	94	1	no	—
154	#107	—	94	1	no	—
155	#106	—	94	2	no	—
156	#106	—	94	2	no	—
157	#97	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
160	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
161	C₄×C₈	D₁₈₈	1	1	yes	characteristic
162	C₄×C₈	D₁₈₈	1	1	yes	characteristic
163	C₂×Q₈	—	188	1	no	—
164	C₂×Q₈	—	188	1	no	—
165	C₂²×C₄	#10625336	1	1	yes	characteristic
166	C₂²×C₄	#10625336	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
176	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
177	C₂×C₈	#10625350	1	2	yes	—
178	C₄⋊C₄	—	94	2	no	—
179	C₄⋊C₄	—	94	2	no	—
180	C₄⋊C₄	—	188	2	no	—
181	C₄⋊C₄	—	188	2	no	—
182	C₄⋊C₄	—	188	1	no	—
183	C₄⋊C₄	—	188	1	no	—
184	C₂²⋊C₄	—	188	2	no	—
185	C₄²	—	188	2	no	—
186	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
187	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
188	C₄²	#10625334	1	1	yes	characteristic
189	C₄²	#10625334	1	1	yes	characteristic
190	C₁₆	—	188	2	no	—
191	C₂³	C₂×#10625336	1	1	yes	characteristic
192	Q₈	—	376	2	no	—
193	Q₈	—	376	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	188	2	no	—
197	C₂×C₄	—	188	2	no	—
198	C₂×C₄	#13093944	1	1	yes	characteristic
199	C₂×C₄	#13093944	1	1	yes	characteristic
200	C₂×C₄	#13093944	1	1	yes	characteristic
201	C₂×C₄	#13093944	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#13094003	1	2	yes	—
204	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
205	C₂×C₄	#13094054	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Dic₉₄	1	2	yes	—
210	C₈	#13094012	1	1	yes	characteristic
211	C₈	#13094024	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#13093944	1	1	yes	characteristic
215	C₂²	#15689408	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	376	2	no	—
220	C₄	—	376	2	no	—
221	C₄	#15688929	1	2	yes	—
222	C₄	#15689029	1	1	yes	characteristic
223	C₄	#15689131	1	1	yes	characteristic
224	C₄	#15689403	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#20244340	1	1	yes	characteristic
227	C₂	C₂×#15688929	1	1	yes	characteristic
228	C₂	#20245115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#13274
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
41. #108×C₄₇	Q₈	semidirect
134. C₄₇	#13274	semidirect
150. #108	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20241485	2
C₄×#108×C₄₇	2
#13274	47