Subgroups — #26115331

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688869	C₂	1	2	yes	maximal, complemented
4	C₂×#15688867	C₂	1	2	yes	maximal
5	C₄₇×#2858	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	2	no	—
11	#15688869	—	2	4	no	—
12	#15688867	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₄₇	—	2	1	no	—
16	C₂×#124×C₄₇	—	2	1	no	—
17	C₄₇×#390	—	2	2	no	—
18	C₄₇×#390	C₄	1	4	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
26	C₄×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₄₇	—	2	1	no	—
28	C₂²×D₄×C₄₇	—	2	1	no	—
29	C₂²×D₄×C₄₇	—	2	1	no	—
30	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₄₇	—	2	2	no	—
32	C₄⋊Q₈×C₄₇	—	2	2	no	—
33	#127×C₄₇	—	2	2	no	—
34	#127×C₄₇	—	2	2	no	—
35	#124×C₄₇	—	4	4	no	—
36	#124×C₄₇	—	4	2	no	—
37	#124×C₄₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂⁴×C₄₇	—	2	2	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂×Q₈×C₄₇	—	4	1	no	—
50	C₂×Q₈×C₄₇	—	4	1	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	2	4	no	—
53	C₂×D₄×C₄₇	—	4	4	no	—
54	C₂×D₄×C₄₇	—	4	2	no	—
55	C₂×D₄×C₄₇	—	4	1	no	—
56	C₂×D₄×C₄₇	—	4	1	no	—
57	C₂×D₄×C₄₇	—	4	1	no	—
58	C₂×D₄×C₄₇	—	4	1	no	—
59	C₂×D₄×C₄₇	—	4	4	no	—
60	C₂×D₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	4	1	no	—
69	C₄⋊C₄×C₄₇	—	4	1	no	—
70	C₂²⋊C₄×C₄₇	—	4	8	no	—
71	C₂²⋊C₄×C₄₇	—	4	8	no	—
72	C₄²×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	4	2	no	—
75	C₄²×C₄₇	D₈	1	2	yes	—
76	C₄²×C₄₇	SD₁₆	1	2	yes	—
77	C₄²×C₄₇	C₂×D₄	1	2	yes	—
78	C₄²×C₄₇	#41	1	2	yes	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	4	no	—
84	C₂³×C₄₇	—	4	2	no	—
85	C₂³×C₄₇	—	4	4	no	—
86	C₂³×C₄₇	—	4	2	no	—
87	C₂³×C₄₇	—	4	4	no	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₄₇	—	8	2	no	—
91	Q₈×C₄₇	—	8	2	no	—
92	D₄×C₄₇	—	8	4	no	—
93	D₄×C₄₇	—	8	2	no	—
94	D₄×C₄₇	—	8	2	no	—
95	D₄×C₄₇	—	8	2	no	—
96	D₄×C₄₇	—	8	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	#102	1	4	yes	—
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	#121	1	2	yes	—
109	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#135	1	4	yes	—
114	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
115	C₄₇⋊C₈	—	4	4	no	—
116	C₄₇⋊C₈	—	8	2	no	—
117	#9696	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	8	4	no	—
119	C₂²×C₄₇	—	8	4	no	—
120	C₂²×C₄₇	—	8	2	no	—
121	C₂²×C₄₇	—	8	2	no	—
122	C₂²×C₄₇	—	8	2	no	—
123	C₂²×C₄₇	#390	1	1	yes	characteristic
124	C₂²×C₄₇	#413	1	2	yes	—
125	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
126	C₂²×C₄₇	#416	1	2	yes	—
127	C₂²×C₄₇	#418	1	1	yes	characteristic
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	—	8	2	no	—
132	C₄×C₄₇	#459	1	2	yes	—
133	C₄×C₄₇	#460	1	2	yes	—
134	C₄×C₄₇	#488	1	2	yes	—
135	C₄×C₄₇	#495	1	2	yes	—
136	#2858	D₄₇	1	1	yes	characteristic
137	#1831	—	47	1	no	—
138	C₄×C₄⋊C₈	—	47	1	no	—
139	C₂×#332	—	47	2	no	—
140	C₂×#331	—	47	2	no	—
141	C₂×C₄₇	—	16	2	no	—
142	C₂×C₄₇	#1540	1	2	yes	—
143	C₂×C₄₇	#1553	1	2	yes	—
144	C₂×C₄₇	#1958	1	1	yes	characteristic
145	C₂×C₄₇	#1967	1	1	yes	characteristic
146	C₂×C₄₇	#1982	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
149	C₂×#124	—	2	1	no	—
150	C₂×#124	—	2	1	no	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×C₄⋊C₈	—	47	2	no	—
153	C₂×C₄×C₈	—	94	1	no	—
154	#390	—	2	2	no	—
155	#390	Dic₄₇	1	4	yes	—
156	C₄³	C₂×D₄₇	1	1	yes	characteristic
157	#336	—	94	2	no	—
158	#336	—	94	1	no	—
159	#332	—	94	4	no	—
160	#331	—	94	4	no	—
161	C₄₇	#9696	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	2	1	no	—
165	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
166	C₂²×C₈	—	94	2	no	—
167	C₂²×C₈	—	94	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	#127	—	2	2	no	—
171	#127	—	2	2	no	—
172	#124	—	4	4	no	—
173	#124	—	4	2	no	—
174	#124	—	4	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
182	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₄⋊C₈	—	47	4	no	—
185	C₄⋊C₈	—	94	2	no	—
186	C₄⋊C₈	—	94	2	no	—
187	C₄⋊C₈	—	94	2	no	—
188	C₄×C₈	—	188	2	no	—
189	C₂⁴	—	2	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
205	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	188	2	no	—
211	C₂×C₈	—	188	4	no	—
212	C₂×C₈	—	188	2	no	—
213	C₂×C₈	—	188	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₂²⋊C₄	—	4	8	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	#10625334	1	2	yes	—
223	C₄²	#10625335	1	2	yes	—
224	C₄²	D₄×D₄₇	1	2	yes	—
225	C₄²	#10625352	1	2	yes	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
232	Q₈	—	8	2	no	—
233	Q₈	—	8	2	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#13093945	1	4	yes	—
249	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
250	C₂×C₄	#13093995	1	2	yes	—
251	C₂×C₄	#13094011	1	1	yes	characteristic
252	C₂×C₄	#13094020	1	1	yes	characteristic
253	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
255	C₂×C₄	#13094044	1	4	yes	—
256	C₂×C₄	#13094050	1	2	yes	—
257	C₈	—	376	2	no	—
258	C₈	—	188	4	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	4	no	—
262	C₂²	—	8	4	no	—
263	C₂²	—	8	2	no	—
264	C₂²	#15689026	1	1	yes	characteristic
265	C₂²	#15689107	1	2	yes	—
266	C₂²	C₂×#13093945	1	1	yes	characteristic
267	C₂²	#15689119	1	2	yes	—
268	C₂²	#15689128	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	2	no	—
272	C₄	—	8	2	no	—
273	C₄	#15689315	1	2	yes	—
274	C₄	#15689320	1	2	yes	—
275	C₄	#15689477	1	2	yes	—
276	C₄	#15689505	1	2	yes	—
277	C₂	—	16	2	no	—
278	C₂	#20242037	1	2	yes	—
279	C₂	#20242093	1	2	yes	—
280	C₂	#20244200	1	1	yes	characteristic
281	C₂	#20244238	1	1	yes	characteristic
282	C₂	#20244316	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2858	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2858	D₄₇
hypocenter	C₄₇	#9696
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2858

Complements

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

N	H ≅ G/N	type
1. #20243505	C₂	semidirect
2. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. C₂×#15688869	C₂	semidirect
161. C₄₇	#9696	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688869	2
C₂×#15688867	2
C₄₇×#2858	2
#9696	47