Subgroups — #26207591

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255737	C₂	1	1	yes	maximal, characteristic
2	#20255034	C₂	1	2	yes	maximal, complemented
3	#20243005	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242875	C₂	1	2	yes	maximal, complemented
5	#20242782	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242774	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242637	C₂	1	2	yes	maximal, complemented
8	#20242622	C₂	1	2	yes	maximal, complemented
9	#20242273	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625336	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094055	—	2	2	no	—
13	#15689778	C₂²	1	2	yes	—
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689526	C₂²	1	2	yes	—
16	#15689502	C₂²	1	2	yes	—
17	#15689435	—	2	2	no	—
18	#15689403	C₂²	1	2	yes	—
19	#15689379	—	2	2	no	—
20	#15689356	—	2	1	no	—
21	#15689333	C₂²	1	2	yes	—
22	#15689325	C₂²	1	2	yes	—
23	#15689283	—	2	2	no	—
24	#15689279	—	2	2	no	—
25	C₄×#10625336	C₂²	1	1	yes	characteristic
26	#15689201	C₂²	1	2	yes	—
27	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
28	#107×D₄₇	—	2	1	no	—
29	#15688876	C₂²	1	1	yes	characteristic
30	#15688873	C₂²	1	1	yes	characteristic
31	#15688869	C₂²	1	2	yes	—
32	#15688868	C₂²	1	2	yes	—
33	Q₈⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
34	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
35	#15688850	C₂²	1	1	yes	characteristic
36	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
37	D₄×Q₈×C₄₇	C₂²	1	2	yes	—
38	C₄₇×#500	—	2	1	no	—
39	C₄₇×#500	C₂²	1	1	yes	characteristic
40	C₄₇×#474	—	2	2	no	—
41	C₄₇×#474	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₄₇	—	2	1	no	—
44	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×#10625350	—	2	2	no	—
46	#13094055	—	4	8	no	—
47	C₂×#10625337	—	2	2	no	—
48	C₂×#10625337	—	2	2	no	—
49	C₂×#10625337	—	2	2	no	—
50	C₂×#10625336	—	2	2	no	—
51	C₂×#10625336	D₄	1	1	yes	characteristic
52	C₂×#10625329	—	2	2	no	—
53	C₂×C₈×D₄₇	—	2	1	no	—
54	#13094024	C₂³	1	2	yes	—
55	#13094006	—	4	2	no	—
56	#13094003	—	2	2	no	—
57	#13093995	—	2	2	no	—
58	C₄×D₁₈₈	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×D₄₇	—	2	2	no	—
60	#13093958	—	2	1	no	—
61	#13093958	—	2	1	no	—
62	#13093957	—	2	2	no	—
63	#13093957	—	4	2	no	—
64	#13093955	D₄	1	1	yes	characteristic
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
67	#13093948	—	2	2	no	—
68	#13093948	D₄	1	1	yes	characteristic
69	#13093947	—	2	2	no	—
70	#13093945	—	2	2	no	—
71	#13093944	D₄	1	1	yes	characteristic
72	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
73	(C₂×D₄₇)⋊C₈	—	2	1	no	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₄₇	—	2	2	no	—
77	C₂×D₈×C₄₇	D₄	1	1	yes	characteristic
78	C₄⋊Q₈×C₄₇	—	2	2	no	—
79	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
80	C₂²⋊Q₈×C₄₇	—	2	2	no	—
81	C₂²⋊Q₈×C₄₇	—	4	2	no	—
82	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₄₇	—	2	2	no	—
84	C₄×D₄×C₄₇	C₂³	1	2	yes	—
85	#107×C₄₇	—	2	1	no	—
86	#107×C₄₇	—	2	1	no	—
87	#107×C₄₇	—	2	2	no	—
88	#107×C₄₇	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₄₇	—	2	1	no	—
90	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
91	#102×C₄₇	—	2	2	no	—
92	#102×C₄₇	D₄	1	2	yes	—
93	C₄×C₈×C₄₇	—	2	1	no	—
94	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₄₇	—	2	2	no	—
96	C₂×Q₈×C₄₇	—	4	2	no	—
97	C₂×Q₈×C₄₇	—	4	4	no	—
98	C₂×Q₈×C₄₇	—	8	2	no	—
99	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₄₇	—	2	2	no	—
102	C₂²×C₄×C₄₇	—	4	2	no	—
103	C₂×C₂²⋊D₄₇	—	4	2	no	—
104	#10625350	—	4	4	no	—
105	#10625350	—	8	2	no	—
106	C₂×D₁₈₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₇	—	2	1	no	—
108	C₂×C₄×D₄₇	—	4	2	no	—
109	C₂×Dic₉₄	—	2	2	no	—
110	D₈×C₄₇	—	2	2	no	—
111	C₂×C₈×C₄₇	—	2	1	no	—
112	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₄⋊C₄×C₄₇	—	2	1	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	2	2	no	—
117	C₄⋊C₄×C₄₇	—	4	2	no	—
118	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
120	C₂²⋊C₄×C₄₇	—	2	2	no	—
121	C₂²⋊C₄×C₄₇	—	4	2	no	—
122	C₄²×C₄₇	—	2	1	no	—
123	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₄₇	—	4	2	no	—
125	#10625337	—	4	4	no	—
126	#10625337	—	4	4	no	—
127	#10625337	—	8	2	no	—
128	#10625336	—	4	4	no	—
129	#10625336	D₈	1	4	yes	complemented
130	(C₂×D₄₇)⋊C₄	—	2	1	no	—
131	C₄⋊Dic₄₇	—	2	2	no	—
132	C₄⋊Dic₄₇	—	2	2	no	—
133	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
134	Dic₄₇⋊C₄	—	4	2	no	—
135	#10625329	—	4	4	no	—
136	C₂×C₄₇⋊C₈	—	2	1	no	—
137	C₂×C₄₇⋊C₈	—	2	1	no	—
138	C₂×C₄₇⋊C₈	—	2	2	no	—
139	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
141	C₈×D₄₇	—	4	2	no	—
142	C₂³×C₄₇	—	2	2	no	—
143	C₂²×D₄₇	—	2	1	no	—
144	Q₈×C₄₇	—	2	1	no	—
145	Q₈×C₄₇	—	4	2	no	—
146	Q₈×C₄₇	—	4	2	no	—
147	Q₈×C₄₇	—	8	4	no	—
148	Q₈×C₄₇	C₂×D₈	1	2	yes	—
149	D₄×C₄₇	—	2	2	no	—
150	D₄×C₄₇	—	2	2	no	—
151	C₂×C₄×C₄₇	—	2	1	no	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	2	1	no	—
154	C₂×C₄×C₄₇	—	4	2	no	—
155	C₂×C₄×C₄₇	—	4	2	no	—
156	C₂×C₄×C₄₇	—	4	2	no	—
157	C₂×C₄×C₄₇	—	8	4	no	—
158	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²⋊D₄₇	—	8	4	no	—
162	C₂×Dic₄₇	—	2	1	no	—
163	C₂×Dic₄₇	—	4	2	no	—
164	D₁₈₈	—	2	1	no	—
165	D₁₈₈	C₂×D₈	1	2	yes	—
166	C₄×D₄₇	—	4	1	no	—
167	C₄×D₄₇	—	4	1	no	—
168	C₄×D₄₇	—	8	4	no	—
169	Dic₉₄	—	4	2	no	—
170	Dic₉₄	—	4	2	no	—
171	C₈×C₄₇	—	2	1	no	—
172	C₈×C₄₇	—	4	1	no	—
173	C₈×C₄₇	#137	1	2	yes	complemented
174	C₄₇⋊C₈	—	2	2	no	—
175	C₄₇⋊C₈	—	2	1	no	—
176	C₄₇⋊C₈	—	4	2	no	—
177	C₄₇⋊C₈	—	4	1	no	—
178	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
179	#19887	—	47	1	no	maximal, Sylow
180	C₂²×C₄₇	—	4	2	no	—
181	C₂²×C₄₇	—	4	2	no	—
182	C₂²×C₄₇	—	4	2	no	—
183	C₂²×C₄₇	D₄²	1	1	yes	characteristic
184	C₂×D₄₇	—	2	2	no	—
185	C₂×D₄₇	—	4	1	no	—
186	C₂×D₄₇	—	4	1	no	—
187	C₄×C₄₇	—	2	2	no	—
188	C₄×C₄₇	—	2	1	no	—
189	C₄×C₄₇	—	4	2	no	—
190	C₄×C₄₇	—	4	1	no	—
191	C₄×C₄₇	—	8	2	no	—
192	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
193	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
194	C₄×C₄₇	#576	1	1	yes	characteristic
195	C₄×C₄₇	#578	1	1	yes	characteristic
196	Dic₄₇	—	4	1	no	—
197	Dic₄₇	—	8	2	no	—
198	Q₈×D₈	D₄₇	1	1	yes	characteristic, complemented
199	#3334	—	47	1	no	—
200	#3276	—	47	2	no	—
201	#1713	—	47	1	no	—
202	#1689	—	47	2	no	—
203	#1674	—	47	1	no	—
204	#1673	—	47	1	no	—
205	#1649	—	47	2	no	—
206	#1647	—	47	2	no	—
207	#1589	—	47	1	no	—
208	C₈×SD₁₆	—	47	1	no	—
209	C₂×C₄₇	—	8	2	no	—
210	C₂×C₄₇	#3276	1	1	yes	characteristic
211	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
212	C₂×C₄₇	#3293	1	1	yes	characteristic
213	D₄₇	—	4	2	no	—
214	C₂×#137	—	94	2	no	—
215	D₄×Q₈	C₂×D₄₇	1	2	yes	—
216	#547	—	47	2	no	—
217	#500	—	2	1	no	—
218	#500	—	47	2	no	—
219	#500	—	47	1	no	—
220	#500	C₂×D₄₇	1	1	yes	characteristic
221	#494	—	47	2	no	—
222	#480	—	94	2	no	—
223	#474	—	2	2	no	—
224	#474	—	47	2	no	—
225	#474	C₂×D₄₇	1	2	yes	—
226	#470	—	94	2	no	—
227	#466	—	94	1	no	—
228	#462	—	47	2	no	—
229	#461	—	47	2	no	—
230	#452	—	94	2	no	—
231	#451	—	94	2	no	—
232	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
233	C₄×SD₁₆	—	47	1	no	—
234	C₄×D₈	—	2	1	no	—
235	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
236	#436	—	47	2	no	—
237	C₈×D₄	—	47	1	no	—
238	C₂×#107	—	94	1	no	—
239	#335	—	47	1	no	—
240	#334	—	47	1	no	—
241	#332	—	47	2	no	—
242	#332	—	47	2	no	—
243	Q₈⋊C₈	—	47	1	no	—
244	D₄⋊C₈	—	47	2	no	—
245	D₄⋊C₈	—	47	1	no	—
246	C₈²	—	47	1	no	—
247	C₄₇	#19887	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	94	2	no	—
249	C₂²×Q₈	—	2	2	no	—
250	#137	—	188	8	no	—
251	C₂×Q₁₆	—	94	2	no	—
252	C₂×Q₁₆	—	94	2	no	—
253	C₂×Q₁₆	—	94	2	no	—
254	C₂×SD₁₆	—	47	1	no	—
255	C₂×SD₁₆	—	94	2	no	—
256	C₂×D₈	C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×M₁₆	—	94	2	no	—
258	C₂²×C₈	—	94	1	no	—
259	C₄⋊Q₈	—	2	2	no	—
260	C₄⋊Q₈	—	47	2	no	—
261	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
262	#123	—	188	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₂²⋊Q₈	—	94	2	no	—
266	#121	—	94	2	no	—
267	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	—	47	1	no	—
270	C₄×D₄	C₂²×D₄₇	1	2	yes	—
271	C₂×C₄⋊C₄	—	94	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	2	2	no	—
275	#107	—	94	1	no	—
276	#107	—	94	1	no	—
277	#107	—	94	2	no	—
278	#107	—	188	2	no	—
279	#107	C₂²⋊D₄₇	1	1	yes	characteristic
280	#106	—	47	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	47	2	no	—
283	C₄⋊C₈	—	47	1	no	—
284	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
285	#103	—	47	1	no	—
286	#103	—	94	2	no	—
287	#103	—	94	2	no	—
288	#102	—	2	2	no	—
289	#102	—	47	1	no	—
290	#102	—	94	2	no	—
291	#102	C₂²⋊D₄₇	1	2	yes	—
292	C₂²⋊C₈	—	94	2	no	—
293	C₂²⋊C₈	—	94	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	47	1	no	—
296	C₄×C₈	—	47	1	no	—
297	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
298	#41	—	376	2	no	—
299	#41	—	188	4	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	2	no	—
302	C₂×Q₈	—	4	4	no	—
303	C₂×Q₈	—	8	2	no	—
304	C₂×Q₈	—	94	2	no	—
305	C₂×Q₈	D₄×D₄₇	1	1	yes	characteristic
306	C₂×D₄	—	47	1	no	—
307	C₂×D₄	—	188	2	no	—
308	C₂×D₄	C₂×C₂²⋊D₄₇	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	4	2	no	—
311	C₂²×C₄	—	94	1	no	—
312	C₂²×C₄	—	188	2	no	—
313	Q₁₆	—	376	2	no	—
314	Q₁₆	—	188	4	no	—
315	Q₁₆	—	188	4	no	—
316	SD₁₆	—	47	4	no	—
317	SD₁₆	—	188	4	no	—
318	D₈	—	2	2	no	—
319	M₁₆	—	188	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	47	1	no	—
322	C₂×C₈	—	47	1	no	—
323	C₂×C₈	—	94	1	no	—
324	C₂×C₈	—	94	2	no	—
325	C₂×C₈	—	94	1	no	—
326	C₂×C₈	—	188	2	no	—
327	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
328	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	47	1	no	—
334	C₄⋊C₄	—	94	2	no	—
335	C₄⋊C₄	—	94	2	no	—
336	C₄⋊C₄	—	188	2	no	—
337	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
338	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	94	1	no	—
342	C₂²⋊C₄	—	188	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₄₇	1	1	yes	characteristic
345	C₂³	—	2	2	no	—
346	C₂³	—	94	1	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	8	4	no	—
351	Q₈	—	188	2	no	—
352	Q₈	—	188	2	no	—
353	Q₈	D₈×D₄₇	1	2	yes	—
354	D₄	—	2	2	no	—
355	D₄	—	2	2	no	—
356	D₄	—	47	2	no	—
357	D₄	—	94	1	no	—
358	D₄	—	376	4	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	4	no	—
366	C₂×C₄	—	94	1	no	—
367	C₂×C₄	—	376	4	no	—
368	C₂×C₄	—	188	2	no	—
369	C₂×C₄	—	188	1	no	—
370	C₂×C₄	—	188	1	no	—
371	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
373	C₂×C₄	#13094069	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	47	2	no	—
377	C₈	—	94	2	no	—
378	C₈	—	94	1	no	—
379	C₈	—	188	2	no	—
380	C₈	—	188	1	no	—
381	C₈	#13094055	1	2	yes	complemented
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	94	2	no	—
386	C₂²	—	188	1	no	—
387	C₂²	—	188	1	no	—
388	C₂²	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	376	2	no	—
395	C₄	—	188	1	no	—
396	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
397	C₄	C₂×#13094055	1	1	yes	characteristic
398	C₄	#15689924	1	1	yes	characteristic
399	C₄	#15689938	1	1	yes	characteristic
400	C₂	—	8	2	no	—
401	C₂	—	188	2	no	—
402	C₂	#20255034	1	1	yes	characteristic
403	C₂	D₈×C₂²⋊D₄₇	1	1	yes	characteristic
404	C₂	#20255190	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	Q₈×D₈×C₄₇	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#20255737	C₂
hypercenter	Q₈×D₈	D₄₇
hypocenter	C₄₇	#19887
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈×D₈

Complements

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

N	H ≅ G/N	type
2. #20255034	C₂	semidirect
3. #20243005	C₂	semidirect
4. #20242875	C₂	semidirect
5. #20242782	C₂	semidirect
6. #20242774	C₂	semidirect
7. #20242637	C₂	semidirect
8. #20242622	C₂	semidirect
9. #20242273	C₂	semidirect
10. C₈×#10625336	C₂	semidirect
11. Q₈×D₈×C₄₇	C₂	semidirect
129. #10625336	D₈	semidirect
173. C₈×C₄₇	#137	semidirect
198. Q₈×D₈	D₄₇	semidirect
247. C₄₇	#19887	semidirect
381. C₈	#13094055	semidirect

Maximal subgroups

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

subgroup	index
#20255737	2
#20255034	2
#20243005	2
#20242875	2
#20242782	2
#20242774	2
#20242637	2
#20242622	2
#20242273	2
C₈×#10625336	2
Q₈×D₈×C₄₇	2
#19887	47