Subgroups — #26373216

Order 12032.

558 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255811	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245615	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245593	C₂	1	2	yes	maximal, complemented
4	#20245535	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245486	C₂	1	2	yes	maximal, complemented
6	#20245430	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245311	C₂	1	2	yes	maximal, complemented
8	C₄×#13093965	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245095	C₂	1	2	yes	maximal, complemented
11	C₄₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689838	—	2	1	no	—
13	C₂×#13093967	C₂²	1	2	yes	—
14	C₂×#13093965	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689448	—	2	2	no	—
18	#15689414	—	2	1	no	—
19	#15689359	C₂²	1	2	yes	—
20	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
21	#15689161	C₂²	1	2	yes	—
22	#15688950	C₂²	1	2	yes	—
23	#15688948	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#15688927	C₂²	1	2	yes	—
26	#15688927	C₂²	1	2	yes	—
27	#15688927	C₂²	1	2	yes	—
28	#15688924	C₂²	1	2	yes	—
29	#15688924	C₂²	1	1	yes	characteristic
30	#15688923	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂×#135×C₄₇	C₂²	1	2	yes	—
34	C₄₇×#547	—	2	1	no	—
35	C₄₇×#494	C₂²	1	1	yes	characteristic
36	C₄₇×#467	C₂²	1	2	yes	—
37	C₄₇×#461	—	2	1	no	—
38	C₄₇×#452	—	2	2	no	—
39	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
42	#13094024	—	2	1	no	—
43	#13094010	—	4	2	no	—
44	#13093998	—	2	2	no	—
45	#13093995	—	4	1	no	—
46	C₄×D₁₈₈	—	2	1	no	—
47	#13093983	—	2	2	no	—
48	#13093982	—	4	1	no	—
49	#13093967	—	4	2	no	—
50	#13093965	—	2	2	no	—
51	C₂×C₄₇⋊C₁₆	—	4	2	no	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	#13093958	—	2	1	no	—
56	#13093958	C₂³	1	2	yes	—
57	#13093958	C₂³	1	1	yes	characteristic
58	#13093956	—	2	2	no	—
59	#13093946	—	2	2	no	—
60	#13093946	—	2	1	no	—
61	C₂×#41×C₄₇	—	2	2	no	—
62	#135×C₄₇	—	2	4	no	—
63	#135×C₄₇	—	4	2	no	—
64	C₂×Q₁₆×C₄₇	—	2	1	no	—
65	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
66	C₂×SD₁₆×C₄₇	—	2	2	no	—
67	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	#123×C₄₇	—	4	2	no	—
71	C₂²⋊Q₈×C₄₇	—	2	2	no	—
72	#121×C₄₇	—	4	1	no	—
73	C₄×D₄×C₄₇	—	2	1	no	—
74	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
76	#107×C₄₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
78	#103×C₄₇	—	2	2	no	—
79	#102×C₄₇	—	2	1	no	—
80	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
81	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	#41×C₄₇	—	4	4	no	—
83	#41×C₄₇	—	8	2	no	—
84	C₂×Q₈×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	1	no	—
86	C₂×D₄×C₄₇	—	4	2	no	—
87	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	—	4	1	no	—
89	C₂²×C₄×C₄₇	—	4	2	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₂²⋊D₄₇	—	4	2	no	—
92	C₂×D₁₈₈	—	2	1	no	—
93	C₂×C₄×D₄₇	—	4	1	no	—
94	C₂×Dic₉₄	—	4	1	no	—
95	Q₁₆×C₄₇	—	2	2	no	—
96	Q₁₆×C₄₇	—	2	2	no	—
97	Q₁₆×C₄₇	—	8	1	no	—
98	SD₁₆×C₄₇	—	4	4	no	—
99	D₈×C₄₇	—	2	1	no	—
100	D₈×C₄₇	#41	1	2	yes	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	—	4	1	no	—
108	C₄⋊C₄×C₄₇	—	4	2	no	—
109	C₄⋊C₄×C₄₇	—	4	1	no	—
110	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₂²⋊C₄×C₄₇	—	4	1	no	—
113	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₂²⋊Dic₄₇	—	4	2	no	—
116	(C₂×D₄₇)⋊C₄	—	4	2	no	—
117	(C₂×D₄₇)⋊C₄	—	4	1	no	—
118	C₄⋊Dic₄₇	—	2	1	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	4	1	no	—
121	C₄×Dic₄₇	—	4	2	no	—
122	D₃₇₆	—	2	1	no	—
123	D₃₇₆	—	2	1	no	—
124	C₄₇⋊C₁₆	—	2	1	no	—
125	C₄₇⋊C₁₆	—	2	1	no	—
126	C₄₇⋊C₁₆	—	2	2	no	—
127	C₂³×C₄₇	—	4	1	no	—
128	C₂³×C₄₇	C₂×D₈	1	2	yes	—
129	C₂²×D₄₇	—	4	1	no	—
130	Q₈×C₄₇	—	4	2	no	—
131	Q₈×C₄₇	—	4	2	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	2	2	no	—
134	D₄×C₄₇	—	4	1	no	—
135	D₄×C₄₇	—	8	4	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	4	1	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	8	4	no	—
141	C₂×C₄×C₄₇	—	8	2	no	—
142	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₄₇	—	16	2	no	—
147	C₂×Dic₄₇	—	4	1	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	D₁₈₈	—	4	1	no	—
150	D₁₈₈	—	4	1	no	—
151	C₄×D₄₇	—	8	2	no	—
152	Dic₉₄	—	16	1	no	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	2	1	no	—
155	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
157	#33870	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	4	2	no	—
162	C₂²×C₄₇	—	8	1	no	—
163	C₂²×C₄₇	—	8	1	no	—
164	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₄₇	—	8	1	no	—
166	C₂×D₄₇	—	8	1	no	—
167	C₂×D₄₇	—	8	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	8	1	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	#508	1	2	yes	—
173	C₄×C₄₇	#546	1	1	yes	characteristic
174	C₄×C₄₇	#577	1	1	yes	characteristic
175	Dic₄₇	—	8	1	no	—
176	Dic₄₇	—	8	2	no	—
177	#3341	—	47	1	no	—
178	#3283	D₄₇	1	1	yes	characteristic, complemented
179	#2241	—	47	1	no	—
180	#2237	—	47	2	no	—
181	#2226	—	47	1	no	—
182	#2217	—	47	2	no	—
183	#2208	—	47	1	no	—
184	#2186	—	47	2	no	—
185	C₄×D₁₆	—	47	1	no	—
186	D₄×C₁₆	—	47	1	no	—
187	#2138	—	47	2	no	—
188	C₂×C₄₇	—	4	2	no	—
189	C₂×C₄₇	—	8	2	no	—
190	C₂×C₄₇	#3294	1	1	yes	characteristic
191	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
192	C₂×C₄₇	#3416	1	1	yes	characteristic
193	D₄₇	—	16	1	no	—
194	C₂×#135	C₂×D₄₇	1	2	yes	—
195	#555	—	94	1	no	—
196	#547	—	2	1	no	—
197	C₂×SD₃₂	—	47	2	no	—
198	C₂×D₁₆	—	47	1	no	—
199	C₂²×C₁₆	—	47	2	no	—
200	#500	—	47	1	no	—
201	#494	C₂×D₄₇	1	1	yes	characteristic
202	#482	—	94	2	no	—
203	#476	—	94	1	no	—
204	#467	C₂×D₄₇	1	2	yes	—
205	#466	—	47	2	no	—
206	#461	—	2	1	no	—
207	#452	—	2	2	no	—
208	C₄×D₈	—	47	1	no	—
209	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
210	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
211	#427	—	47	2	no	—
212	#367	—	47	2	no	—
213	#366	—	47	1	no	—
214	C₄⋊C₁₆	—	47	1	no	—
215	#358	—	47	2	no	—
216	#358	—	47	2	no	—
217	#358	—	47	2	no	—
218	#357	—	47	1	no	—
219	#357	—	47	2	no	—
220	#357	—	47	1	no	—
221	C₂²⋊C₁₆	—	47	2	no	—
222	C₄×C₁₆	—	47	1	no	—
223	C₄₇	#33870	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	2	2	no	—
225	#135	—	2	4	no	—
226	#135	—	4	2	no	—
227	C₂×Q₁₆	—	2	1	no	—
228	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
229	C₂×SD₁₆	—	2	2	no	—
230	C₂×D₈	—	47	1	no	—
231	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	94	1	no	—
235	#124	—	188	2	no	—
236	#123	—	4	2	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	#121	—	4	1	no	—
239	#121	—	94	2	no	—
240	#121	—	188	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	94	1	no	—
243	C₄×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
244	#117	—	94	2	no	—
245	#117	—	188	1	no	—
246	C₂×C₄⋊C₄	—	4	1	no	—
247	SD₃₂	—	188	2	no	—
248	D₁₆	—	94	2	no	—
249	C₂×C₁₆	—	47	1	no	—
250	C₂×C₁₆	—	47	1	no	—
251	C₂×C₁₆	—	47	2	no	—
252	C₂×C₁₆	—	188	2	no	—
253	#107	—	47	1	no	—
254	#107	—	47	2	no	—
255	#107	—	94	1	no	—
256	#107	C₂²×D₄₇	1	1	yes	characteristic
257	#106	—	94	2	no	—
258	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
259	#103	—	2	2	no	—
260	#102	—	2	1	no	—
261	#102	—	94	1	no	—
262	#102	—	94	2	no	—
263	C₂²⋊C₈	C₂²⋊D₄₇	1	2	yes	—
264	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
265	#41	—	4	4	no	—
266	#41	—	8	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	188	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	94	1	no	—
272	C₂×D₄	—	188	2	no	—
273	C₂×D₄	#10625334	1	1	yes	characteristic
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	188	1	no	—
277	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	8	1	no	—
281	SD₁₆	—	4	4	no	—
282	D₈	—	2	1	no	—
283	D₈	—	94	1	no	—
284	D₈	—	94	1	no	—
285	D₈	#10625352	1	2	yes	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
290	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	94	1	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	188	1	no	—
298	C₄⋊C₄	#10625334	1	1	yes	characteristic
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	188	2	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₂²⋊C₄	—	188	1	no	—
304	C₂²⋊C₄	#10625334	1	2	yes	—
305	C₄²	—	188	2	no	—
306	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
307	C₁₆	—	94	1	no	—
308	C₁₆	—	94	1	no	—
309	C₁₆	—	94	2	no	—
310	C₂³	—	4	1	no	—
311	C₂³	—	188	1	no	—
312	C₂³	C₂×#10625334	1	2	yes	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	752	1	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	1	no	—
319	D₄	—	8	4	no	—
320	D₄	—	188	1	no	—
321	D₄	—	188	1	no	—
322	D₄	—	752	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	376	2	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	C₂×#10625334	1	2	yes	—
333	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
334	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	1	no	—
338	C₈	C₂×#10625352	1	1	yes	characteristic
339	C₈	#13094069	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	8	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	376	1	no	—
347	C₂²	—	376	1	no	—
348	C₂²	—	376	1	no	—
349	C₂²	C₂²×#10625334	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	1	no	—
354	C₄	—	376	2	no	—
355	C₄	—	376	1	no	—
356	C₄	#15689553	1	2	yes	—
357	C₄	#15689767	1	1	yes	characteristic
358	C₄	#15689934	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	—	8	2	no	—
361	C₂	—	752	1	no	—
362	C₂	#20255201	1	1	yes	characteristic
363	C₂	C₂×#15689553	1	1	yes	characteristic
364	C₂	#20256532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3283	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3283	D₄₇
hypocenter	C₄₇	#33870
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3283

Complements

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

N	H ≅ G/N	type
1. #20255811	C₂	semidirect
2. #20245615	C₂	semidirect
3. #20245593	C₂	semidirect
4. #20245535	C₂	semidirect
5. #20245486	C₂	semidirect
6. #20245430	C₂	semidirect
7. #20245311	C₂	semidirect
8. C₄×#13093965	C₂	semidirect
9. D₄×C₄₇⋊C₁₆	C₂	semidirect
10. #20245095	C₂	semidirect
11. C₄₇×#3283	C₂	semidirect
178. #3283	D₄₇	semidirect
223. C₄₇	#33870	semidirect

Maximal subgroups

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

subgroup	index
#20255811	2
#20245615	2
#20245593	2
#20245535	2
#20245486	2
#20245430	2
#20245311	2
C₄×#13093965	2
D₄×C₄₇⋊C₁₆	2
#20245095	2
C₄₇×#3283	2
#33870	47