Subgroups — #26115382

Order 12032.

610 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	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688872	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)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#13093935	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	#15688872	—	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	C₄	1	4	yes	—
16	C₄₇×#390	—	2	1	no	—
17	C₄₇×#390	—	2	1	no	—
18	C₄₇×#390	—	2	2	no	—
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)	—	4	4	no	—
26	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
27	#13093935	—	4	2	no	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×D₄×C₄₇	—	2	1	no	—
30	C₂²×D₄×C₄₇	—	2	1	no	—
31	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₄₇	—	2	2	no	—
33	C₄⋊Q₈×C₄₇	—	2	2	no	—
34	#127×C₄₇	—	2	2	no	—
35	#127×C₄₇	—	2	2	no	—
36	#124×C₄₇	—	4	8	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂⁴×C₄₇	—	2	2	no	—
47	C₂×Q₈×C₄₇	—	2	4	no	—
48	C₂×Q₈×C₄₇	—	4	4	no	—
49	C₂×D₄×C₄₇	—	2	4	no	—
50	C₂×D₄×C₄₇	—	4	4	no	—
51	C₂×D₄×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	4	1	no	—
53	C₂×D₄×C₄₇	—	4	1	no	—
54	C₂×D₄×C₄₇	—	4	2	no	—
55	C₂×D₄×C₄₇	—	4	1	no	—
56	C₂×D₄×C₄₇	—	4	2	no	—
57	C₂×D₄×C₄₇	—	4	1	no	—
58	C₂×D₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	—	4	2	no	—
66	C₄⋊C₄×C₄₇	—	4	1	no	—
67	C₄⋊C₄×C₄₇	—	4	1	no	—
68	C₂²⋊C₄×C₄₇	—	4	8	no	—
69	C₂²⋊C₄×C₄₇	—	4	8	no	—
70	C₄²×C₄₇	—	2	4	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	1	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	1	no	—
76	C₄²×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	#41	1	2	yes	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	4	no	—
83	C₂³×C₄₇	—	4	2	no	—
84	C₂³×C₄₇	—	4	4	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	—	4	4	no	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₄₇	—	8	4	no	—
90	D₄×C₄₇	—	8	4	no	—
91	D₄×C₄₇	—	8	2	no	—
92	D₄×C₄₇	—	8	2	no	—
93	D₄×C₄₇	—	8	2	no	—
94	D₄×C₄₇	—	8	1	no	—
95	D₄×C₄₇	—	8	1	no	—
96	C₂×C₄×C₄₇	—	2	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	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	#100	1	4	yes	—
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	#121	1	2	yes	—
107	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#135	1	2	yes	—
110	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#137	1	2	yes	—
114	C₄₇⋊C₈	—	4	4	no	—
115	C₄₇⋊C₈	—	8	2	no	—
116	#9707	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	8	4	no	—
118	C₂²×C₄₇	—	8	4	no	—
119	C₂²×C₄₇	—	8	2	no	—
120	C₂²×C₄₇	—	8	2	no	—
121	C₂²×C₄₇	—	8	2	no	—
122	C₂²×C₄₇	#390	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#100	1	2	yes	—
124	C₂²×C₄₇	#416	1	1	yes	characteristic
125	C₂²×C₄₇	#417	1	2	yes	—
126	C₂²×C₄₇	#417	1	1	yes	characteristic
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	4	2	no	—
130	C₄×C₄₇	—	8	2	no	—
131	C₄×C₄₇	#463	1	2	yes	—
132	C₄×C₄₇	#464	1	2	yes	—
133	C₄×C₄₇	#489	1	2	yes	—
134	C₄×C₄₇	#496	1	1	yes	characteristic
135	C₄×C₄₇	#497	1	1	yes	characteristic
136	#2858	D₄₇	1	1	yes	characteristic
137	#1842	—	47	1	no	—
138	#1767	—	47	1	no	—
139	C₂×#333	—	47	2	no	—
140	C₂×#331	—	47	2	no	—
141	C₂×C₄₇	—	16	2	no	—
142	C₂×C₄₇	#1551	1	2	yes	—
143	C₂×C₄₇	#1560	1	2	yes	—
144	C₂×C₄₇	#1959	1	1	yes	characteristic
145	C₂×C₄₇	#1965	1	1	yes	characteristic
146	C₂×C₄₇	#1983	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	Dic₄₇	1	4	yes	—
150	C₂×C₄⋊C₈	—	47	2	no	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×C₄⋊C₈	—	94	2	no	—
153	C₂×#97	—	94	1	no	—
154	#390	—	2	1	no	—
155	#390	—	2	1	no	—
156	#390	—	2	2	no	—
157	C₄³	C₂×D₄₇	1	1	yes	characteristic
158	#336	—	94	1	no	—
159	#333	—	94	4	no	—
160	#331	—	94	4	no	—
161	C₄₇	#9707	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
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	8	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₂²⋊C₄	—	2	4	no	—
177	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₄⋊C₈	—	47	4	no	—
183	C₄⋊C₈	—	94	2	no	—
184	C₄⋊C₈	—	94	2	no	—
185	C₄⋊C₈	—	94	2	no	—
186	C₄⋊C₈	—	188	4	no	—
187	#97	—	188	2	no	—
188	C₂⁴	—	2	2	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	4	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
203	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	188	2	no	—
209	C₂×C₈	—	188	4	no	—
210	C₂×C₈	—	188	2	no	—
211	C₂×C₈	—	188	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	D₄×D₄₇	1	2	yes	—
224	C₄²	#10625352	1	2	yes	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
231	Q₈	—	8	4	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	2	no	—
234	D₄	—	8	1	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	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	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	#13093941	1	4	yes	—
247	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
248	C₂×C₄	#13093995	1	2	yes	—
249	C₂×C₄	#13094011	1	1	yes	characteristic
250	C₂×C₄	#13094020	1	1	yes	characteristic
251	C₂×C₄	#13094044	1	2	yes	—
252	C₂×C₄	#13094049	1	2	yes	—
253	C₂×C₄	#13094049	1	1	yes	characteristic
254	C₂×C₄	#13094054	1	2	yes	—
255	C₂×C₄	#13094054	1	1	yes	characteristic
256	C₈	—	376	2	no	—
257	C₈	—	188	4	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	4	no	—
261	C₂²	—	8	4	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#15689026	1	1	yes	characteristic
264	C₂²	C₂×#13093941	1	2	yes	—
265	C₂²	#15689119	1	1	yes	characteristic
266	C₂²	#15689124	1	1	yes	characteristic
267	C₂²	#15689124	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	2	no	—
272	C₄	#15689335	1	2	yes	—
273	C₄	#15689342	1	2	yes	—
274	C₄	#15689482	1	2	yes	—
275	C₄	#15689513	1	1	yes	characteristic
276	C₄	#15689518	1	1	yes	characteristic
277	C₂	—	16	2	no	—
278	C₂	#20242083	1	2	yes	—
279	C₂	#20242123	1	2	yes	—
280	C₂	#20244205	1	1	yes	characteristic
281	C₂	#20244229	1	1	yes	characteristic
282	C₂	#20244321	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	#9707
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. #20243540	C₂	semidirect
2. #20243241	C₂	semidirect
3. C₂×#15688872	C₂	semidirect
161. C₄₇	#9707	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20243241	2
C₂×#15688872	2
C₂×#15688867	2
C₄₇×#2858	2
#9707	47