Subgroups — #26371810

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253962	C₂	1	2	yes	maximal, complemented
2	#20245662	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245636	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245610	C₂	1	2	yes	maximal, complemented
5	#20245474	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245472	C₂	1	2	yes	maximal, complemented
7	D₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245234	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245232	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2215	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689668	—	2	2	no	—
13	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
14	C₂×#13093965	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#15689527	C₂²	1	2	yes	—
17	#15689498	C₂²	1	2	yes	—
18	#15689468	—	2	2	no	—
19	#15689359	C₂²	1	2	yes	—
20	#15689357	—	2	2	no	—
21	D₈×Dic₄₇	C₂²	1	2	yes	—
22	#15689205	C₂²	1	2	yes	—
23	#15689203	—	2	2	no	—
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688949	C₂²	1	1	yes	characteristic
26	#15688948	C₂²	1	2	yes	—
27	#15688925	C₂²	1	2	yes	—
28	#15688924	C₂²	1	2	yes	—
29	#15688923	C₂²	1	2	yes	—
30	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#466	C₂²	1	2	yes	—
37	C₄₇×#366	C₂²	1	1	yes	characteristic
38	C₄₇×#357	C₂²	1	2	yes	—
39	C₂×#10625352	—	4	2	no	—
40	C₂×D₃₇₆	C₂³	1	2	yes	—
41	C₂×#10625334	—	2	2	no	—
42	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
43	#13094023	—	2	2	no	—
44	#13094020	—	2	2	no	—
45	#13094010	—	4	2	no	—
46	#13093998	—	2	2	no	—
47	#13093996	—	4	2	no	—
48	D₄×Dic₄₇	—	2	2	no	—
49	#13093986	—	2	2	no	—
50	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
51	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
52	D₇₅₂	—	2	2	no	—
53	#13093965	—	2	4	no	—
54	C₂×C₄₇⋊C₁₆	—	2	2	no	—
55	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	#13093958	C₂³	1	2	yes	—
58	#13093957	—	2	2	no	—
59	#13093946	—	2	2	no	—
60	#13093945	—	2	2	no	—
61	#13093944	—	2	2	no	—
62	C₈×Dic₄₇	D₄	1	1	yes	characteristic
63	C₈×Dic₄₇	D₄	1	1	yes	characteristic
64	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
65	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#121×C₄₇	—	2	2	no	—
69	D₁₆×C₄₇	—	2	2	no	—
70	C₂×C₁₆×C₄₇	—	2	2	no	—
71	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
73	#107×C₄₇	C₂³	1	2	yes	—
74	#102×C₄₇	—	2	2	no	—
75	C₂×D₄×C₄₇	—	2	2	no	—
76	C₂×D₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₇	—	4	2	no	—
79	C₂×C₂²⋊D₄₇	—	4	2	no	—
80	C₂×C₂²⋊D₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
83	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
84	#10625352	—	8	8	no	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×Dic₉₄	—	4	2	no	—
88	D₈×C₄₇	—	2	2	no	—
89	D₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₂²⋊C₄×C₄₇	—	4	2	no	—
96	C₂²⋊Dic₄₇	—	4	2	no	—
97	#10625334	—	4	4	no	—
98	(C₂×D₄₇)⋊C₄	—	4	2	no	—
99	(C₂×D₄₇)⋊C₄	—	4	2	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	Dic₄₇⋊C₄	—	4	2	no	—
102	C₄×Dic₄₇	D₈	1	1	yes	characteristic
103	C₄×Dic₄₇	D₈	1	1	yes	characteristic
104	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	D₃₇₆	—	2	2	no	—
110	D₃₇₆	—	2	2	no	—
111	C₁₆×C₄₇	—	2	1	no	—
112	C₁₆×C₄₇	#41	1	2	yes	complemented
113	C₄₇⋊C₁₆	—	2	1	no	—
114	C₄₇⋊C₁₆	#41	1	2	yes	complemented
115	C₂³×C₄₇	—	4	2	no	—
116	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₄₇	—	4	2	no	—
118	D₄×C₄₇	—	4	2	no	—
119	D₄×C₄₇	—	4	2	no	—
120	D₄×C₄₇	—	8	4	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₄₇	—	8	4	no	—
127	C₂²⋊D₄₇	—	8	4	no	—
128	C₂²⋊D₄₇	—	8	4	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	8	4	no	—
132	C₂×Dic₄₇	D₁₆	1	4	yes	complemented
133	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
137	D₁₈₈	—	4	2	no	—
138	D₁₈₈	—	4	2	no	—
139	C₄×D₄₇	—	8	4	no	—
140	Dic₉₄	—	8	4	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₄₇⋊C₈	—	2	1	no	—
145	C₄₇⋊C₈	#135	1	2	yes	—
146	#33733	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	8	2	no	—
150	C₂²×C₄₇	—	8	2	no	—
151	C₂²×C₄₇	—	8	2	no	—
152	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
153	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₄₇	—	8	2	no	—
155	C₂×D₄₇	—	8	2	no	—
156	C₂×D₄₇	—	8	2	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	8	2	no	—
159	C₄×C₄₇	#532	1	1	yes	characteristic
160	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
161	Dic₄₇	—	2	1	no	—
162	Dic₄₇	—	8	2	no	—
163	Dic₄₇	C₂×D₁₆	1	2	yes	—
164	Dic₄₇	C₂×D₁₆	1	2	yes	—
165	Dic₄₇	#508	1	2	yes	—
166	#3169	—	47	2	no	—
167	#2252	—	47	1	no	—
168	#2246	—	47	1	no	—
169	#2240	—	47	2	no	—
170	#2215	—	47	1	no	—
171	#2215	—	47	2	no	—
172	#2215	D₄₇	1	1	yes	characteristic, complemented
173	C₄×D₁₆	—	47	1	no	—
174	C₄×D₁₆	—	47	1	no	—
175	C₄×D₁₆	—	47	2	no	—
176	C₂×C₄×C₁₆	—	47	1	no	—
177	C₂×C₄₇	—	2	2	no	—
178	C₂×C₄₇	—	16	2	no	—
179	C₂×C₄₇	#3169	1	1	yes	characteristic
180	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
182	D₄₇	—	16	2	no	—
183	#532	—	94	2	no	—
184	C₂×D₁₆	—	47	2	no	—
185	C₂×D₁₆	—	47	1	no	—
186	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
187	C₂²×C₁₆	—	47	1	no	—
188	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
189	#500	—	47	2	no	—
190	#493	—	47	2	no	—
191	#486	—	94	2	no	—
192	#466	—	47	2	no	—
193	#466	—	94	2	no	—
194	#466	C₂×D₄₇	1	2	yes	—
195	C₄×D₈	—	47	2	no	—
196	C₄×D₈	—	47	2	no	—
197	C₄×D₈	—	94	2	no	—
198	C₂×C₄×C₈	—	47	1	no	—
199	#366	—	47	1	no	—
200	#366	—	47	2	no	—
201	#366	C₂×D₄₇	1	1	yes	characteristic
202	#357	—	47	2	no	—
203	#357	—	47	2	no	—
204	#357	—	47	2	no	—
205	#357	C₂×D₄₇	1	2	yes	—
206	C₄×C₁₆	—	47	1	no	—
207	C₄×C₁₆	—	47	1	no	—
208	C₄×C₁₆	—	47	1	no	—
209	C₄×C₁₆	—	47	1	no	—
210	C₄₇	#33733	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	188	2	no	—
212	C₂×D₈	—	47	2	no	—
213	C₂×D₈	—	94	2	no	—
214	C₂×D₈	C₂²×D₄₇	1	2	yes	—
215	C₂²×C₈	—	47	1	no	—
216	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₄⋊Q₈	—	94	2	no	—
218	#127	—	94	2	no	—
219	#124	—	188	2	no	—
220	#121	—	2	2	no	—
221	#121	—	94	2	no	—
222	#121	—	188	2	no	—
223	C₄×D₄	—	94	2	no	—
224	C₄×D₄	—	94	2	no	—
225	C₄×D₄	—	188	2	no	—
226	C₂×C₄²	—	47	1	no	—
227	D₁₆	—	2	2	no	—
228	D₁₆	—	94	4	no	—
229	D₁₆	—	94	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	47	1	no	—
232	C₂×C₁₆	—	47	1	no	—
233	C₂×C₁₆	—	94	2	no	—
234	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
236	#107	—	47	2	no	—
237	#107	—	94	2	no	—
238	#107	C₂²×D₄₇	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	94	2	no	—
241	#102	—	94	2	no	—
242	#102	—	94	2	no	—
243	C₄×C₈	—	47	1	no	—
244	C₄×C₈	—	47	1	no	—
245	C₄×C₈	—	47	1	no	—
246	C₄×C₈	—	47	1	no	—
247	#41	—	376	8	no	—
248	C₂×Q₈	—	188	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	94	2	no	—
252	C₂×D₄	—	188	2	no	—
253	C₂×D₄	—	188	2	no	—
254	C₂×D₄	—	188	2	no	—
255	C₂²×C₄	—	47	1	no	—
256	C₂²×C₄	—	47	1	no	—
257	C₂²×C₄	—	188	2	no	—
258	C₂²×C₄	—	188	2	no	—
259	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	94	2	no	—
263	D₈	—	94	2	no	—
264	D₈	—	188	4	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	47	1	no	—
268	C₂×C₈	—	47	1	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	188	2	no	—
277	C₂²⋊C₄	—	188	2	no	—
278	C₂²⋊C₄	—	188	2	no	—
279	C₄²	—	47	1	no	—
280	C₄²	—	47	1	no	—
281	C₄²	—	47	1	no	—
282	C₄²	—	47	1	no	—
283	C₁₆	—	2	1	no	—
284	C₁₆	—	47	2	no	—
285	C₁₆	—	94	1	no	—
286	C₁₆	#10625352	1	2	yes	complemented
287	C₂³	—	4	2	no	—
288	C₂³	—	188	2	no	—
289	C₂³	D₈×D₄₇	1	1	yes	characteristic
290	Q₈	—	376	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	376	4	no	—
295	D₄	—	376	4	no	—
296	D₄	—	376	4	no	—
297	D₄	—	188	2	no	—
298	D₄	—	188	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	47	4	no	—
303	C₂×C₄	—	47	1	no	—
304	C₂×C₄	—	47	1	no	—
305	C₂×C₄	—	47	1	no	—
306	C₂×C₄	—	47	1	no	—
307	C₂×C₄	—	94	2	no	—
308	C₂×C₄	—	376	4	no	—
309	C₂×C₄	—	376	4	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	47	2	no	—
315	C₈	—	94	1	no	—
316	C₈	#41×D₄₇	1	1	yes	characteristic
317	C₈	C₂×#10625352	1	1	yes	characteristic
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	376	2	no	—
324	C₂²	—	376	2	no	—
325	C₂²	—	376	2	no	—
326	C₂²	D₁₆×D₄₇	1	2	yes	—
327	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	47	2	no	—
331	C₄	—	47	2	no	—
332	C₄	—	47	2	no	—
333	C₄	—	94	1	no	—
334	C₄	—	376	2	no	—
335	C₄	#15689668	1	1	yes	characteristic
336	C₄	#135×D₄₇	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	16	2	no	—
339	C₂	—	752	2	no	—
340	C₂	#20253962	1	1	yes	characteristic
341	C₂	C₂×D₁₆×D₄₇	1	1	yes	characteristic
342	C₂	#508×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2215	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#2215	D₄₇
hypocenter	C₄₇	#33733
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #20253962	C₂	semidirect
2. #20245662	C₂	semidirect
3. #20245636	C₂	semidirect
4. #20245610	C₂	semidirect
5. #20245474	C₂	semidirect
6. #20245472	C₂	semidirect
7. D₁₆×Dic₄₇	C₂	semidirect
8. #20245234	C₂	semidirect
9. #20245232	C₂	semidirect
10. C₂×C₁₆×Dic₄₇	C₂	semidirect
11. C₄₇×#2215	C₂	semidirect
112. C₁₆×C₄₇	#41	semidirect
114. C₄₇⋊C₁₆	#41	semidirect
132. C₂×Dic₄₇	D₁₆	semidirect
172. #2215	D₄₇	semidirect
210. C₄₇	#33733	semidirect
286. C₁₆	#10625352	semidirect

Maximal subgroups

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

subgroup	index
#20253962	2
#20245662	2
#20245636	2
#20245610	2
#20245474	2
#20245472	2
D₁₆×Dic₄₇	2
#20245234	2
#20245232	2
C₂×C₁₆×Dic₄₇	2
C₄₇×#2215	2
#33733	47