Subgroups — #26107410

Order 12032.

502 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688845	C₂	1	2	yes	maximal
4	C₄₇⋊#1273	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	#15688878	C₄	1	2	yes	—
13	#15688845	—	2	4	no	—
14	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#336	C₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄⋊(C₄₇⋊C₈)	C₈	1	8	yes	complemented
22	C₄×C₄₇⋊C₈	—	2	4	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	1	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	4	no	—
28	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₄₇	—	2	4	no	—
39	C₄×C₈×C₄₇	—	2	2	no	—
40	C₄×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	—	2	4	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	—	2	2	no	—
54	C₄²×C₄₇	C₂×C₈	1	4	yes	—
55	C₄²×C₄₇	M₁₆	1	4	yes	—
56	C₄²×C₄₇	M₁₆	1	4	yes	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₄₇	#41	1	1	yes	characteristic
60	C₄²×C₄₇	#41	1	1	yes	characteristic
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	2	2	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
68	C₂×C₄₇⋊C₈	C₂×C₈	1	8	yes	—
69	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
70	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
78	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₄₇	#97	1	2	yes	—
80	C₂×C₄×C₄₇	#97	1	2	yes	—
81	C₂×C₄×C₄₇	#97	1	2	yes	—
82	C₂×C₄×C₄₇	#97	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	#131	1	2	yes	—
94	C₂×C₄×C₄₇	#131	1	2	yes	—
95	C₈×C₄₇	—	4	4	no	—
96	C₄₇⋊C₈	—	2	4	no	—
97	C₄₇⋊C₈	—	2	4	no	—
98	C₄₇⋊C₈	—	2	8	no	—
99	#8261	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
104	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
105	C₂²×C₄₇	#405	1	1	yes	characteristic
106	C₂²×C₄₇	#405	1	1	yes	characteristic
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	#322	1	4	yes	complemented
111	C₄×C₄₇	#423	1	2	yes	—
112	C₄×C₄₇	#432	1	2	yes	—
113	C₄×C₄₇	C₈×D₄	1	2	yes	—
114	C₄×C₄₇	#436	1	2	yes	—
115	C₄×C₄₇	C₈×Q₈	1	2	yes	—
116	C₄×C₄₇	#446	1	2	yes	—
117	C₄×C₄⋊C₈	—	47	1	no	—
118	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
119	C₄²×C₈	—	47	1	no	—
120	C₂×#322	—	47	2	no	—
121	#1273	—	47	2	no	—
122	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
123	C₂×C₄₇	C₈×M₁₆	1	1	yes	characteristic
124	C₂×C₄₇	#1451	1	1	yes	characteristic
125	C₂×C₄₇	#1452	1	1	yes	characteristic
126	C₂×C₄₇	#1767	1	1	yes	characteristic
127	C₂×C₄₇	C₈×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄₇	#1770	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	47	2	no	—
130	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
131	C₂×C₄×C₈	—	47	1	no	—
132	C₂×C₄×C₈	—	47	2	no	—
133	C₂×C₄×C₈	—	47	2	no	—
134	C₂×C₄×C₈	—	47	2	no	—
135	C₂×C₄×C₈	—	47	1	no	—
136	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
137	C₄³	C₂×D₄₇	1	1	yes	characteristic
138	#336	—	47	2	no	—
139	#336	Dic₄₇	1	2	yes	—
140	#322	—	94	4	no	—
141	C₄₇	#8261	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₈	—	47	4	no	—
143	C₂²×C₈	—	47	2	no	—
144	C₂²×C₈	—	47	2	no	—
145	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
146	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	C₄⋊C₈	—	2	4	no	—
154	C₄⋊C₈	—	47	8	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	47	2	no	—
158	C₄×C₈	—	47	2	no	—
159	C₄×C₈	—	94	4	no	—
160	C₄×C₈	—	94	4	no	—
161	C₄×C₈	—	94	2	no	—
162	C₄×C₈	—	94	1	no	—
163	C₄×C₈	—	94	1	no	—
164	C₄×C₈	—	94	2	no	—
165	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
168	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	47	4	no	—
176	C₂×C₈	—	47	8	no	—
177	C₂×C₈	—	47	4	no	—
178	C₂×C₈	—	94	2	no	—
179	C₂×C₈	—	94	4	no	—
180	C₂×C₈	—	94	2	no	—
181	C₂×C₈	—	94	2	no	—
182	C₂×C₈	—	94	2	no	—
183	C₂×C₈	—	94	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	C₈×D₄₇	1	4	yes	—
188	C₄²	#10625324	1	4	yes	—
189	C₄²	#10625329	1	4	yes	—
190	C₄²	D₄×D₄₇	1	1	yes	characteristic
191	C₄²	#10625352	1	1	yes	characteristic
192	C₄²	Q₈×D₄₇	1	1	yes	characteristic
193	C₄²	#10625354	1	1	yes	characteristic
194	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
202	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
203	C₂×C₄	#13093934	1	2	yes	—
204	C₂×C₄	#13093934	1	2	yes	—
205	C₂×C₄	#13093935	1	2	yes	—
206	C₂×C₄	#13093935	1	2	yes	—
207	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
208	C₂×C₄	#13093982	1	1	yes	characteristic
209	C₂×C₄	#13093986	1	1	yes	characteristic
210	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	#13093989	1	1	yes	characteristic
212	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
214	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
215	C₂×C₄	C₂×#10625324	1	2	yes	—
216	C₂×C₄	C₂×#10625329	1	2	yes	—
217	C₂×C₄	#13094031	1	2	yes	—
218	C₂×C₄	#13094032	1	2	yes	—
219	C₈	—	4	4	no	—
220	C₈	—	94	4	no	—
221	C₈	—	94	4	no	—
222	C₈	—	94	8	no	—
223	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
224	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
225	C₂²	C₂×#13093934	1	1	yes	characteristic
226	C₂²	C₂×#13093935	1	1	yes	characteristic
227	C₂²	M₁₆×Dic₄₇	1	1	yes	characteristic
228	C₂²	#15689079	1	1	yes	characteristic
229	C₂²	#15689081	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	#15688845	1	4	yes	complemented
234	C₄	#15689145	1	2	yes	—
235	C₄	#15689179	1	2	yes	—
236	C₄	#15689188	1	2	yes	—
237	C₄	#15689201	1	2	yes	—
238	C₄	#15689247	1	2	yes	—
239	C₄	#15689254	1	2	yes	—
240	C₂	C₂×#15688845	1	1	yes	characteristic
241	C₂	#20241669	1	1	yes	characteristic
242	C₂	#20241675	1	1	yes	characteristic
243	C₂	#20241681	1	1	yes	characteristic
244	C₂	#20243243	1	1	yes	characteristic
245	C₂	#20243253	1	1	yes	characteristic
246	C₂	#20243257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#8261
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
21. C₄⋊(C₄₇⋊C₈)	C₈	semidirect
110. C₄×C₄₇	#322	semidirect
141. C₄₇	#8261	semidirect
233. C₄	#15688845	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄²×C₄₇⋊C₈	2
C₂×#15688845	2
C₄₇⋊#1273	2
C₄×C₄⋊C₈×C₄₇	2
#8261	47