Subgroups — #26369670

Order 12032.

674 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	C₂×#15689552	C₂	1	1	yes	maximal, characteristic
2	#20252258	C₂	1	2	yes	maximal, complemented
3	#20245264	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245260	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245258	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245243	C₂	1	2	yes	maximal, complemented
7	C₄×#13093968	C₂	1	2	yes	maximal, complemented
8	#367×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245097	C₂	1	2	yes	maximal, complemented
10	#20244986	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094043	C₂²	1	2	yes	—
13	#15689602	—	2	2	no	—
14	#15689552	C₄	1	8	yes	complemented
15	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
16	C₂×#13093968	C₂²	1	2	yes	—
17	C₂×#13093963	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
19	#15689223	—	2	2	no	—
20	#15689220	—	2	2	no	—
21	C₄×Dic₁₈₈	C₂²	1	2	yes	—
22	#15689217	C₂²	1	2	yes	—
23	D₈×Dic₄₇	C₂²	1	2	yes	—
24	#107×D₄₇	C₂²	1	2	yes	—
25	#15689121	—	2	2	no	—
26	#15689077	C₂²	1	1	yes	characteristic
27	#15688951	C₂²	1	1	yes	characteristic
28	#15688950	C₂²	1	2	yes	—
29	#15688928	C₂²	1	2	yes	—
30	#15688926	C₂²	1	2	yes	—
31	#15688924	C₂²	1	2	yes	—
32	#15688898	C₂²	1	1	yes	characteristic
33	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#367	C₂²	1	1	yes	characteristic
38	C₄₇×#357	C₂²	1	2	yes	—
39	C₄₇×#346	C₂²	1	1	yes	characteristic
40	C₂×#10625352	—	2	2	no	—
41	#13094043	—	2	4	no	—
42	#13094043	C₂×C₄	1	8	yes	complemented
43	C₂×Dic₁₈₈	C₂³	1	2	yes	—
44	C₂×#10625335	—	2	2	no	—
45	C₂×#10625324	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
47	C₄×Dic₉₄	—	2	2	no	—
48	#13093989	—	2	2	no	—
49	D₄×Dic₄₇	—	2	2	no	—
50	C₄×C₂²⋊D₄₇	—	4	2	no	—
51	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
52	#13093981	D₄	1	1	yes	characteristic
53	#13093980	—	4	2	no	—
54	C₂×Dic₄₇⋊C₄	—	4	2	no	—
55	C₄⋊C₄×D₄₇	—	2	2	no	—
56	Dic₃₇₆	C₂×C₄	1	4	yes	complemented
57	#13093968	C₂×C₄	1	8	yes	complemented
58	#13093963	—	2	2	no	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₄₇	C₂×C₄	1	4	yes	complemented
62	#13093958	C₂³	1	2	yes	—
63	#13093957	—	2	2	no	—
64	#13093949	—	2	2	no	—
65	#13093947	—	2	2	no	—
66	#13093945	—	2	2	no	—
67	#13093935	D₄	1	1	yes	characteristic
68	#13093934	D₄	1	1	yes	characteristic
69	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₄₇	C₂³	1	2	yes	—
71	C₄×D₄×C₄₇	—	2	2	no	—
72	D₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
75	#107×C₄₇	C₂³	1	2	yes	—
76	#102×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	4	2	no	—
80	C₂×C₂²⋊D₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	—	4	2	no	—
83	#10625352	—	2	8	no	—
84	#10625352	—	4	4	no	—
85	C₂×C₄×D₄₇	—	4	2	no	—
86	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₉₄	—	2	2	no	—
88	D₈×C₄₇	—	2	2	no	—
89	D₈×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₂²⋊C₄×C₄₇	—	4	2	no	—
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²⋊Dic₄₇	—	4	2	no	—
96	#10625335	—	2	8	no	—
97	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	Dic₄₇⋊C₄	—	4	2	no	—
100	Dic₄₇⋊C₄	—	4	2	no	—
101	Dic₄₇⋊C₄	—	4	2	no	—
102	Dic₄₇⋊C₄	D₈	1	2	yes	—
103	C₄×Dic₄₇	—	4	2	no	—
104	C₄×Dic₄₇	—	4	2	no	—
105	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	Dic₁₈₈	—	2	2	no	—
109	Dic₁₈₈	C₂²×C₄	1	4	yes	—
110	#10625324	—	2	2	no	—
111	C₈×D₄₇	C₂²×C₄	1	2	yes	—
112	C₈×D₄₇	#41	1	2	yes	—
113	C₁₆×C₄₇	—	2	1	no	—
114	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
115	C₄₇⋊C₁₆	—	2	1	no	—
116	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
119	D₄×C₄₇	—	2	4	no	—
120	D₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	16	2	no	—
123	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₄₇	—	4	8	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	8	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	16	2	no	—
130	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
131	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
132	C₄×D₄₇	—	16	2	no	—
133	C₄×D₄₇	C₄×D₄	1	2	yes	—
134	C₄×D₄₇	#135	1	2	yes	—
135	Dic₉₄	—	2	4	no	—
136	Dic₉₄	—	4	2	no	—
137	C₈×C₄₇	—	2	1	no	—
138	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₄₇⋊C₈	—	2	1	no	—
141	C₄₇⋊C₈	C₄×D₄	1	2	yes	—
142	#33460	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	—	8	2	no	—
145	C₂²×C₄₇	—	8	2	no	—
146	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₄₇	—	2	2	no	—
148	C₂×D₄₇	C₄×D₈	1	2	yes	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	8	2	no	—
151	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	4	4	no	—
155	Dic₄₇	—	8	2	no	—
156	Dic₄₇	C₄×D₈	1	2	yes	—
157	C₂×#508	—	47	1	no	—
158	#2951	—	47	2	no	—
159	#2177	—	47	1	no	—
160	#2177	D₄₇	1	1	yes	characteristic, complemented
161	#2176	—	47	1	no	—
162	#2176	—	47	1	no	—
163	C₄×SD₃₂	—	47	2	no	—
164	C₄×SD₃₂	—	47	2	no	—
165	C₂×#367	—	47	1	no	—
166	#2139	—	47	2	no	—
167	#2108	—	47	1	no	—
168	C₂×C₄₇	—	8	4	no	—
169	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
170	C₂×C₄₇	#3416	1	2	yes	—
171	D₄₇	—	2	2	no	—
172	C₂×#135	—	47	2	no	—
173	#520	—	94	2	no	—
174	#508	—	47	8	no	—
175	C₂×Q₃₂	—	47	1	no	—
176	C₂×SD₃₂	—	47	2	no	—
177	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
178	C₂×M₃₂	—	47	1	no	—
179	C₂²×C₁₆	—	47	1	no	—
180	#440	—	94	2	no	—
181	#440	—	94	2	no	—
182	C₄×Q₁₆	—	47	2	no	—
183	C₄×Q₁₆	—	47	2	no	—
184	C₄×D₈	—	47	2	no	—
185	C₄×D₈	C₂×D₄₇	1	2	yes	—
186	C₂×#107	—	47	2	no	—
187	#417	—	94	2	no	—
188	#405	—	47	1	no	—
189	#367	—	47	1	no	—
190	#367	—	47	2	no	—
191	#367	C₂×D₄₇	1	1	yes	characteristic
192	#358	—	47	2	no	—
193	#358	—	47	2	no	—
194	#357	—	47	2	no	—
195	#357	C₂×D₄₇	1	2	yes	—
196	#346	—	47	1	no	—
197	#346	C₂×D₄₇	1	1	yes	characteristic
198	C₄×C₁₆	—	47	1	no	—
199	C₄×C₁₆	—	47	1	no	—
200	C₄₇	#33460	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	94	2	no	—
202	#135	—	47	8	no	—
203	#135	—	94	4	no	—
204	C₂×Q₁₆	—	47	2	no	—
205	C₂×SD₁₆	—	94	2	no	—
206	C₂×D₈	C₂²×D₄₇	1	2	yes	—
207	C₂×M₁₆	—	47	1	no	—
208	C₂²×C₈	—	47	1	no	—
209	C₄×Q₈	—	94	2	no	—
210	C₄×Q₈	—	94	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	94	2	no	—
213	C₄×D₄	—	188	2	no	—
214	C₄×D₄	—	188	2	no	—
215	#117	—	47	1	no	—
216	#117	—	188	2	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄⋊C₄	—	188	2	no	—
219	Q₃₂	—	47	4	no	—
220	SD₃₂	—	47	8	no	—
221	D₁₆	C₄×D₄₇	1	4	yes	complemented
222	M₃₂	—	94	2	no	—
223	C₂×C₁₆	—	47	1	no	—
224	C₂×C₁₆	—	47	4	no	—
225	C₂×C₁₆	—	47	1	no	—
226	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
228	#107	—	47	2	no	—
229	#107	—	94	2	no	—
230	#107	C₂²×D₄₇	1	2	yes	—
231	#103	—	94	2	no	—
232	#103	—	94	2	no	—
233	#102	—	2	2	no	—
234	#102	—	94	2	no	—
235	#97	—	47	1	no	—
236	#97	—	47	1	no	—
237	C₄×C₈	—	47	1	no	—
238	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
239	#41	—	94	8	no	—
240	#41	—	188	4	no	—
241	C₂×Q₈	—	94	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	188	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	47	1	no	—
246	C₂²×C₄	—	188	2	no	—
247	C₂²×C₄	—	188	2	no	—
248	C₂²×C₄	—	188	2	no	—
249	Q₁₆	—	47	4	no	—
250	Q₁₆	—	94	2	no	—
251	SD₁₆	—	94	8	no	—
252	D₈	—	2	2	no	—
253	D₈	C₂×C₄×D₄₇	1	4	yes	—
254	M₁₆	—	94	2	no	—
255	C₂×C₈	—	47	1	no	—
256	C₂×C₈	—	47	2	no	—
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	47	1	no	—
259	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	47	2	no	—
263	C₄⋊C₄	—	94	2	no	—
264	C₄⋊C₄	—	188	2	no	—
265	C₄⋊C₄	—	188	2	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	47	2	no	—
269	C₂²⋊C₄	—	188	2	no	—
270	C₄²	—	47	1	no	—
271	C₄²	—	188	2	no	—
272	C₄²	—	188	2	no	—
273	C₄²	D₄×D₄₇	1	1	yes	characteristic
274	C₁₆	—	2	1	no	—
275	C₁₆	—	47	2	no	—
276	C₁₆	—	94	1	no	—
277	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	—	47	1	no	—
280	Q₈	—	94	4	no	—
281	Q₈	—	188	2	no	—
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	188	8	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	—	47	1	no	—
288	C₂×C₄	—	47	2	no	—
289	C₂×C₄	—	47	2	no	—
290	C₂×C₄	—	47	2	no	—
291	C₂×C₄	—	188	8	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	752	2	no	—
295	C₂×C₄	—	752	2	no	—
296	C₂×C₄	D₈×D₄₇	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	47	2	no	—
300	C₈	—	94	1	no	—
301	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
302	C₈	#41×D₄₇	1	1	yes	characteristic
303	C₂²	—	4	4	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	47	2	no	—
307	C₂²	—	94	2	no	—
308	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	47	2	no	—
312	C₄	—	94	2	no	—
313	C₄	—	376	2	no	—
314	C₄	—	188	4	no	—
315	C₄	C₄×D₄×D₄₇	1	1	yes	characteristic
316	C₄	#135×D₄₇	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	—	94	2	no	—
319	C₂	C₄×D₈×D₄₇	1	1	yes	characteristic
320	C₂	#20256534	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2177	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689552	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2177	D₄₇
hypocenter	C₄₇	#33460
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #20252258	C₂	semidirect
3. #20245264	C₂	semidirect
4. #20245260	C₂	semidirect
5. #20245258	C₂	semidirect
6. #20245243	C₂	semidirect
7. C₄×#13093968	C₂	semidirect
8. #367×D₄₇	C₂	semidirect
9. #20245097	C₂	semidirect
10. #20244986	C₂	semidirect
11. C₄₇×#2177	C₂	semidirect
14. #15689552	C₄	semidirect
42. #13094043	C₂×C₄	semidirect
56. Dic₃₇₆	C₂×C₄	semidirect
57. #13093968	C₂×C₄	semidirect
61. C₁₆×D₄₇	C₂×C₄	semidirect
72. D₁₆×C₄₇	C₂×C₄	semidirect
160. #2177	D₄₇	semidirect
200. C₄₇	#33460	semidirect
221. D₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689552	2
#20252258	2
#20245264	2
#20245260	2
#20245258	2
#20245243	2
C₄×#13093968	2
#367×D₄₇	2
#20245097	2
#20244986	2
C₄₇×#2177	2
#33460	47