Subgroups — #26121351

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244194	C₂	1	1	yes	maximal, characteristic
2	#20244193	C₂	1	1	yes	maximal, characteristic
3	#20244193	C₂	1	1	yes	maximal, characteristic
4	#20243515	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243515	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094021	C₂²	1	1	yes	characteristic
9	C₂×#13093946	—	2	2	no	—
10	C₂×#13093944	—	2	2	no	—
11	C₂×#13093944	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15688993	C₂²	1	1	yes	characteristic
15	#15688885	—	2	2	no	—
16	#15688885	—	2	2	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#384	C₂²	1	1	yes	characteristic
19	C₄₇×#384	C₂²	1	1	yes	characteristic
20	C₄₇×#340	—	2	2	no	—
21	C₂²×D₁₈₈	—	2	1	no	—
22	C₂²×D₁₈₈	—	2	1	no	—
23	C₂²×D₁₈₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
26	#13094021	—	2	1	no	—
27	#13094021	—	2	1	no	—
28	#13094021	—	2	1	no	—
29	#13094021	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	#13093946	—	4	4	no	—
34	#13093944	—	4	4	no	—
35	#13093944	—	4	4	no	—
36	C₄×C₄₇⋊C₈	—	2	1	no	—
37	C₄×C₄₇⋊C₈	—	2	1	no	—
38	C₄×C₄₇⋊C₈	—	2	1	no	—
39	C₄×C₄₇⋊C₈	—	2	1	no	—
40	C₂²×C₈×C₄₇	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₄₇	—	2	1	no	—
49	C₄×C₈×C₄₇	—	2	1	no	—
50	C₂³×D₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	—	4	1	no	—
52	C₂²×C₄×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	4	1	no	—
57	C₂×D₁₈₈	—	2	2	no	—
58	C₂×D₁₈₈	—	2	2	no	—
59	C₂×D₁₈₈	—	2	2	no	—
60	C₂×D₁₈₈	—	4	2	no	—
61	C₂×D₁₈₈	—	4	2	no	—
62	C₂×D₁₈₈	—	4	2	no	—
63	C₂×D₁₈₈	—	4	2	no	—
64	C₂×D₁₈₈	—	4	2	no	—
65	C₂×D₁₈₈	—	4	2	no	—
66	C₂×D₁₈₈	—	4	1	no	—
67	C₂×D₁₈₈	—	4	1	no	—
68	C₂×D₁₈₈	—	4	1	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	—	4	2	no	—
73	C₄⋊C₄×C₄₇	—	4	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₄⋊C₄×C₄₇	—	4	2	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₄₇	#41	1	1	yes	characteristic
80	C₄²×C₄₇	#41	1	1	yes	characteristic
81	C₄²×C₄₇	#41	1	1	yes	characteristic
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄⋊Dic₄₇	—	4	2	no	—
84	C₄⋊Dic₄₇	—	4	2	no	—
85	C₂×C₄₇⋊C₈	—	2	2	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂³×C₄₇	#127	1	1	yes	characteristic
92	C₂²×D₄₇	—	4	2	no	—
93	C₂²×D₄₇	—	8	1	no	—
94	C₂²×D₄₇	—	8	1	no	—
95	C₂²×D₄₇	—	8	1	no	—
96	C₂²×D₄₇	—	8	1	no	—
97	C₂²×D₄₇	—	8	1	no	—
98	C₂²×D₄₇	—	8	1	no	—
99	C₂×C₄×C₄₇	—	8	1	no	—
100	C₂×C₄×C₄₇	—	8	1	no	—
101	C₂×C₄×C₄₇	—	8	1	no	—
102	C₂×C₄×C₄₇	—	8	1	no	—
103	C₂×C₄×C₄₇	—	8	1	no	—
104	C₂×C₄×C₄₇	—	8	1	no	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#135	1	2	yes	—
112	C₂×C₄×C₄₇	#135	1	2	yes	—
113	C₂×C₄×C₄₇	#135	1	2	yes	—
114	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×Dic₄₇	—	8	1	no	—
118	C₂×Dic₄₇	—	8	1	no	—
119	C₂×Dic₄₇	—	8	1	no	—
120	D₁₈₈	—	4	4	no	—
121	D₁₈₈	—	4	4	no	—
122	D₁₈₈	—	4	4	no	—
123	D₁₈₈	—	8	2	no	—
124	D₁₈₈	—	8	2	no	—
125	D₁₈₈	—	8	2	no	—
126	C₈×C₄₇	—	4	2	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	C₄₇⋊C₈	—	4	2	no	—
129	#10678	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	#399	1	1	yes	characteristic
131	C₂²×C₄₇	#495	1	1	yes	characteristic
132	C₂²×C₄₇	#495	1	1	yes	characteristic
133	C₂²×C₄₇	#495	1	1	yes	characteristic
134	C₂²×C₄₇	#496	1	1	yes	characteristic
135	C₂²×C₄₇	#496	1	1	yes	characteristic
136	C₂²×C₄₇	#496	1	1	yes	characteristic
137	C₂×D₄₇	—	8	2	no	—
138	C₂×D₄₇	—	8	2	no	—
139	C₂×D₄₇	—	8	2	no	—
140	C₂×D₄₇	—	16	1	no	—
141	C₂×D₄₇	—	16	1	no	—
142	C₂×D₄₇	—	16	1	no	—
143	C₂×D₄₇	—	16	1	no	—
144	C₄×C₄₇	—	16	1	no	—
145	C₄×C₄₇	—	16	1	no	—
146	C₄×C₄₇	#478	1	2	yes	—
147	C₄×C₄₇	#478	1	2	yes	—
148	C₄×C₄₇	#478	1	2	yes	—
149	C₄×C₄₇	#482	1	2	yes	—
150	C₄×C₄₇	#482	1	2	yes	—
151	C₄×C₄₇	#482	1	2	yes	—
152	Dic₄₇	—	16	1	no	—
153	#1957	—	47	1	no	—
154	#1957	—	47	1	no	—
155	#1957	—	47	1	no	—
156	#1834	—	47	1	no	—
157	#1834	—	47	1	no	—
158	#1834	D₄₇	1	1	yes	characteristic, complemented
159	#1273	—	47	1	no	—
160	C₂×C₄₇	#1711	1	1	yes	characteristic
161	C₂×C₄₇	#1717	1	1	yes	characteristic
162	C₂×C₄₇	#1717	1	1	yes	characteristic
163	C₂×C₄₇	#1717	1	1	yes	characteristic
164	C₂×C₄₇	#2092	1	1	yes	characteristic
165	C₂×C₄₇	#2092	1	1	yes	characteristic
166	C₂×C₄₇	#2092	1	1	yes	characteristic
167	D₄₇	—	16	2	no	—
168	C₂×#127	—	47	1	no	—
169	C₂×#102	—	94	2	no	—
170	C₂×#102	—	94	2	no	—
171	C₂×#102	—	94	2	no	—
172	C₂×C₄×C₈	—	47	1	no	—
173	C₂×C₄×C₈	—	47	1	no	—
174	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
175	#384	—	47	1	no	—
176	#384	C₂×D₄₇	1	1	yes	characteristic
177	#384	C₂×D₄₇	1	1	yes	characteristic
178	#340	—	2	2	no	—
179	#340	—	94	2	no	—
180	#340	—	94	2	no	—
181	C₄₇	#10678	1	1	yes	characteristic, complemented, Sylow
182	C₂²×D₄	—	94	1	no	—
183	C₂²×D₄	—	94	1	no	—
184	C₂²×D₄	—	94	1	no	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
188	#127	—	47	2	no	—
189	#127	—	94	1	no	—
190	#127	—	94	1	no	—
191	#127	—	94	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	94	1	no	—
199	C₂×C₄⋊C₄	—	94	1	no	—
200	C₂×C₄⋊C₄	—	94	1	no	—
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	#102	—	188	4	no	—
203	#102	—	188	4	no	—
204	#102	—	188	4	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	94	1	no	—
208	C₄×C₈	—	94	1	no	—
209	C₄×C₈	—	94	1	no	—
210	C₄×C₈	—	94	1	no	—
211	C₂⁴	—	188	1	no	—
212	C₂×D₄	—	94	2	no	—
213	C₂×D₄	—	94	2	no	—
214	C₂×D₄	—	94	2	no	—
215	C₂×D₄	—	188	1	no	—
216	C₂×D₄	—	188	1	no	—
217	C₂×D₄	—	188	1	no	—
218	C₂×D₄	—	188	2	no	—
219	C₂×D₄	—	188	2	no	—
220	C₂×D₄	—	188	2	no	—
221	C₂×D₄	—	188	2	no	—
222	C₂×D₄	—	188	2	no	—
223	C₂×D₄	—	188	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	188	1	no	—
227	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	2	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	2	no	—
247	C₄⋊C₄	—	188	2	no	—
248	C₄²	#10625350	1	1	yes	characteristic
249	C₄²	#10625350	1	1	yes	characteristic
250	C₄²	Q₈×D₄₇	1	1	yes	characteristic
251	C₄²	#10625354	1	1	yes	characteristic
252	C₂³	—	376	1	no	—
253	C₂³	—	376	1	no	—
254	C₂³	—	376	1	no	—
255	C₂³	—	376	1	no	—
256	C₂³	—	376	1	no	—
257	C₂³	—	376	1	no	—
258	C₂³	—	188	2	no	—
259	C₂³	#13094020	1	1	yes	characteristic
260	D₄	—	376	2	no	—
261	D₄	—	376	2	no	—
262	D₄	—	376	2	no	—
263	D₄	—	188	4	no	—
264	D₄	—	188	4	no	—
265	D₄	—	188	4	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	376	1	no	—
273	C₂×C₄	—	376	1	no	—
274	C₂×C₄	—	376	1	no	—
275	C₂×C₄	#13094000	1	1	yes	characteristic
276	C₂×C₄	#13094002	1	1	yes	characteristic
277	C₂×C₄	#13094002	1	1	yes	characteristic
278	C₂×C₄	#13094007	1	1	yes	characteristic
279	C₂×C₄	#13094009	1	1	yes	characteristic
280	C₂×C₄	#13094009	1	1	yes	characteristic
281	C₂×C₄	#13094044	1	2	yes	—
282	C₂×C₄	#13094045	1	2	yes	—
283	C₂×C₄	#13094047	1	2	yes	—
284	C₂×C₄	#13094048	1	2	yes	—
285	C₂×C₄	#13094049	1	2	yes	—
286	C₂×C₄	#13094051	1	2	yes	—
287	C₈	—	4	2	no	—
288	C₈	—	188	2	no	—
289	C₈	—	188	2	no	—
290	C₂²	—	376	2	no	—
291	C₂²	—	376	2	no	—
292	C₂²	—	376	2	no	—
293	C₂²	—	752	1	no	—
294	C₂²	—	752	1	no	—
295	C₂²	—	752	1	no	—
296	C₂²	—	752	1	no	—
297	C₂²	#15689058	1	1	yes	characteristic
298	C₂²	#15689504	1	1	yes	characteristic
299	C₂²	#15689505	1	1	yes	characteristic
300	C₂²	#15689508	1	1	yes	characteristic
301	C₂²	#15689509	1	1	yes	characteristic
302	C₂²	#15689511	1	1	yes	characteristic
303	C₂²	#15689513	1	1	yes	characteristic
304	C₄	—	16	1	no	—
305	C₄	—	16	1	no	—
306	C₄	—	752	1	no	—
307	C₄	#15689420	1	2	yes	—
308	C₄	#15689422	1	2	yes	—
309	C₄	#15689426	1	2	yes	—
310	C₄	#15689447	1	2	yes	—
311	C₄	#15689449	1	2	yes	—
312	C₄	#15689450	1	2	yes	—
313	C₂	—	752	2	no	—
314	C₂	#20243001	1	1	yes	characteristic
315	C₂	#20243028	1	1	yes	characteristic
316	C₂	#20243029	1	1	yes	characteristic
317	C₂	#20243030	1	1	yes	characteristic
318	C₂	#20244906	1	1	yes	characteristic
319	C₂	#20244910	1	1	yes	characteristic
320	C₂	#20244912	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1834	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	#1834	D₄₇
hypocenter	C₄₇	#10678
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #20243515	C₂	semidirect
5. #20243515	C₂	semidirect
6. C₄₇⋊#1273	C₂	semidirect
7. C₄₇×#1834	C₂	semidirect
158. #1834	D₄₇	semidirect
181. C₄₇	#10678	semidirect

Maximal subgroups

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

subgroup	index
#20244194	2
#20244193	2
#20243515	2
C₄₇⋊#1273	2
C₄₇×#1834	2
#10678	47