Subgroups — #26275161

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253235	C₂	1	2	yes	maximal
2	#20251980	C₂	1	1	yes	maximal, characteristic
3	#20244896	C₂	1	2	yes	maximal, complemented
4	#20244735	C₂	1	2	yes	maximal, complemented
5	#20244168	C₂	1	1	yes	maximal, characteristic, complemented
6	#20244137	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243618	C₂	1	2	yes	maximal
8	#20243549	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243448	C₂	1	1	yes	maximal, characteristic
10	#20243227	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094030	C₂²	1	1	yes	characteristic
13	#15689603	C₂²	1	2	yes	—
14	#15689427	—	2	2	no	—
15	#15689420	—	2	2	no	—
16	#15689415	—	2	2	no	—
17	#15689400	—	2	2	no	—
18	C₂×#13093960	C₂²	1	1	yes	characteristic
19	#15689166	C₂²	1	2	yes	—
20	#15689166	C₂²	1	2	yes	—
21	#15689164	—	2	2	no	—
22	#15689158	—	2	1	no	—
23	C₂×#13093957	C₂²	1	2	yes	—
24	C₂×#13093956	C₂²	1	1	yes	characteristic
25	#106×D₄₇	—	2	1	no	—
26	#15689150	C₂²	1	2	yes	—
27	C₂×#13093950	C₂²	1	1	yes	characteristic
28	#15689122	C₂²	1	2	yes	—
29	#15689117	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	#15688988	C₂²	1	2	yes	—
32	#15688892	C₂²	1	2	yes	—
33	#15688891	C₂²	1	2	yes	—
34	#15688887	C₂²	1	1	yes	characteristic
35	#15688882	C₂²	1	1	yes	characteristic
36	#15688882	C₂²	1	2	yes	—
37	C₄₇×#544	C₂²	1	2	yes	—
38	C₄₇×#501	—	2	2	no	—
39	C₄₇×#499	—	2	1	no	—
40	C₄₇×#498	—	2	1	no	—
41	C₄₇×#428	C₂²	1	2	yes	—
42	C₄₇×#427	C₂²	1	1	yes	characteristic
43	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
44	C₄₇×#405	C₂²	1	1	yes	characteristic
45	C₂×#10625350	C₂³	1	1	yes	characteristic
46	#13094030	—	2	2	no	—
47	#13094030	Q₈	1	4	yes	complemented
48	C₂×#10625329	C₂³	1	2	yes	—
49	C₂×#10625329	C₂³	1	1	yes	characteristic
50	C₂×#10625324	—	2	1	no	—
51	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₄₇	—	2	1	no	—
53	C₄×Dic₉₄	—	2	2	no	—
54	C₄×D₁₈₈	—	2	2	no	—
55	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
56	#13093982	—	2	2	no	—
57	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
58	C₄⋊C₄×D₄₇	—	2	2	no	—
59	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
60	#13093960	—	2	2	no	—
61	#13093957	—	2	2	no	—
62	#13093957	—	2	2	no	—
63	#13093957	—	2	2	no	—
64	#13093957	—	2	2	no	—
65	#13093956	—	2	1	no	—
66	#13093956	—	2	1	no	—
67	#13093955	—	2	2	no	—
68	#13093955	—	2	2	no	—
69	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
70	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
71	#13093950	Q₈	1	4	yes	complemented
72	#13093947	—	2	2	no	—
73	#13093947	D₄	1	2	yes	—
74	#13093944	—	2	2	no	—
75	#13093944	D₄	1	2	yes	—
76	C₈×Dic₄₇	—	2	2	no	—
77	C₄₇⋊#95	—	4	2	no	—
78	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₄₇	—	2	2	no	—
81	C₄⋊Q₈×C₄₇	—	2	2	no	—
82	#125×C₄₇	—	2	2	no	—
83	#125×C₄₇	—	2	2	no	—
84	C₂²⋊Q₈×C₄₇	—	4	2	no	—
85	#117×C₄₇	C₂³	1	1	yes	characteristic
86	#117×C₄₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
88	#107×C₄₇	—	2	2	no	—
89	#107×C₄₇	D₄	1	2	yes	—
90	#106×C₄₇	—	2	1	no	—
91	#106×C₄₇	—	2	1	no	—
92	#106×C₄₇	—	2	2	no	—
93	#106×C₄₇	D₄	1	1	yes	characteristic
94	#106×C₄₇	D₄	1	1	yes	characteristic
95	#97×C₄₇	—	2	1	no	—
96	C₄×C₈×C₄₇	—	2	1	no	—
97	C₂×Q₈×C₄₇	—	4	2	no	—
98	C₂²×C₄×C₄₇	—	2	2	no	—
99	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₂²⋊D₄₇	—	2	1	no	—
101	C₂²×Dic₄₇	—	2	1	no	—
102	C₂²×Dic₄₇	—	2	2	no	—
103	#10625350	—	2	1	no	—
104	#10625350	—	2	1	no	—
105	#10625350	C₂×Q₈	1	2	yes	—
106	#10625350	C₂×Q₈	1	2	yes	—
107	C₂×D₁₈₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₄₇	—	2	1	no	—
109	C₂×C₄×D₄₇	—	4	2	no	—
110	C₂×Dic₉₄	C₂×D₄	1	1	yes	characteristic
111	M₁₆×C₄₇	—	2	1	no	—
112	M₁₆×C₄₇	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₄₇	—	2	1	no	—
114	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₄₇	#41	1	2	yes	—
118	C₄⋊C₄×C₄₇	—	2	2	no	—
119	C₄⋊C₄×C₄₇	—	2	2	no	—
120	C₄⋊C₄×C₄₇	—	4	2	no	—
121	C₄⋊C₄×C₄₇	—	4	2	no	—
122	C₄⋊C₄×C₄₇	—	4	2	no	—
123	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
125	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
126	C₂²⋊C₄×C₄₇	—	2	1	no	—
127	C₂²⋊C₄×C₄₇	—	2	2	no	—
128	C₄²×C₄₇	—	2	1	no	—
129	C₄²×C₄₇	—	2	2	no	—
130	(C₂×D₄₇)⋊C₄	—	4	2	no	—
131	C₄⋊Dic₄₇	—	2	2	no	—
132	C₄⋊Dic₄₇	—	2	2	no	—
133	Dic₄₇⋊C₄	—	4	2	no	—
134	C₄×Dic₄₇	—	2	1	no	—
135	C₄×Dic₄₇	—	4	2	no	—
136	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
137	#10625329	—	2	1	no	—
138	#10625329	—	2	2	no	—
139	#10625329	—	2	2	no	—
140	#10625329	C₂×Q₈	1	2	yes	—
141	C₂×C₄₇⋊C₈	—	2	1	no	—
142	C₂×C₄₇⋊C₈	—	2	1	no	—
143	C₂×C₄₇⋊C₈	—	2	1	no	—
144	C₂×C₄₇⋊C₈	—	2	2	no	—
145	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
146	#10625324	—	2	4	no	—
147	C₈×D₄₇	—	2	4	no	—
148	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×D₄₇	—	2	1	no	—
150	Q₈×C₄₇	—	16	2	no	—
151	C₂×C₄×C₄₇	—	2	2	no	—
152	C₂×C₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	2	2	no	—
154	C₂×C₄×C₄₇	—	4	2	no	—
155	C₂×C₄×C₄₇	—	8	2	no	—
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂²⋊D₄₇	—	2	2	no	—
163	C₂²⋊D₄₇	—	2	2	no	—
164	C₂×Dic₄₇	—	2	1	no	—
165	C₂×Dic₄₇	—	2	2	no	—
166	C₂×Dic₄₇	—	4	2	no	—
167	C₂×Dic₄₇	—	4	2	no	—
168	C₂×Dic₄₇	—	8	2	no	—
169	D₁₈₈	—	2	1	no	—
170	D₁₈₈	C₄⋊Q₈	1	2	yes	—
171	C₄×D₄₇	—	2	2	no	—
172	C₄×D₄₇	—	2	2	no	—
173	C₄×D₄₇	—	16	2	no	—
174	Dic₉₄	—	2	1	no	—
175	Dic₉₄	C₄⋊Q₈	1	2	yes	—
176	C₈×C₄₇	—	2	2	no	—
177	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
178	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
179	C₄₇⋊C₈	—	2	2	no	—
180	C₄₇⋊C₈	—	2	2	no	—
181	C₄₇⋊C₈	—	4	2	no	—
182	#26080	—	47	1	no	maximal, Sylow
183	C₂²×C₄₇	—	2	2	no	—
184	C₂²×C₄₇	#544	1	1	yes	characteristic
185	C₂²×C₄₇	D₄²	1	1	yes	characteristic
186	C₂²×C₄₇	#554	1	1	yes	characteristic
187	C₂×D₄₇	—	2	2	no	—
188	C₂×D₄₇	—	4	2	no	—
189	C₄×C₄₇	—	4	2	no	—
190	C₄×C₄₇	—	4	2	no	—
191	C₄×C₄₇	—	4	2	no	—
192	C₄×C₄₇	—	8	2	no	—
193	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₄×C₄₇	#547	1	1	yes	characteristic
196	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
197	Dic₄₇	—	2	2	no	—
198	Dic₄₇	—	4	2	no	—
199	Dic₄₇	—	8	2	no	—
200	#3163	D₄₇	1	1	yes	characteristic, complemented
201	#3083	—	47	2	no	—
202	#2912	—	47	1	no	—
203	#2089	—	47	2	no	—
204	#2061	—	47	2	no	—
205	#1951	—	47	1	no	—
206	#1945	—	47	1	no	—
207	#1860	—	47	2	no	—
208	#1846	—	47	1	no	—
209	#1815	—	47	1	no	—
210	#1764	—	47	1	no	—
211	C₂×C₄₇	—	2	1	no	—
212	C₂×C₄₇	—	2	1	no	—
213	C₂×C₄₇	#2440	1	1	yes	characteristic
214	C₂×C₄₇	#3290	1	2	yes	—
215	D₄₇	—	4	2	no	—
216	C₂×#131	—	47	1	no	—
217	#544	C₂×D₄₇	1	2	yes	—
218	#520	—	47	2	no	—
219	#501	—	2	2	no	—
220	#499	—	2	1	no	—
221	#498	—	2	1	no	—
222	#479	—	94	2	no	—
223	#478	—	94	2	no	—
224	#477	—	94	2	no	—
225	#474	—	94	2	no	—
226	C₂×#108	—	47	1	no	—
227	#428	—	47	2	no	—
228	#428	—	47	2	no	—
229	#428	—	94	2	no	—
230	#428	C₂×D₄₇	1	2	yes	—
231	#427	—	94	1	no	—
232	#427	C₂×D₄₇	1	1	yes	characteristic
233	C₂×#107	—	47	2	no	—
234	C₂×#106	—	47	1	no	—
235	C₂×#106	—	94	1	no	—
236	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
237	#424	—	47	2	no	—
238	C₂×#104	—	47	1	no	—
239	#417	—	47	2	no	—
240	#416	—	47	1	no	—
241	#405	C₂×D₄₇	1	1	yes	characteristic
242	C₂×C₄×C₈	—	47	1	no	—
243	#382	—	47	2	no	—
244	#343	—	47	2	no	—
245	#343	—	47	2	no	—
246	#341	—	47	1	no	—
247	#339	—	47	1	no	—
248	#339	—	47	2	no	—
249	C₄₇	#26080	1	1	yes	characteristic, complemented, Sylow
250	C₂×#41	—	47	1	no	—
251	#131	—	47	4	no	—
252	#131	—	94	2	no	—
253	C₂×M₁₆	—	47	1	no	—
254	C₂×M₁₆	—	47	2	no	—
255	C₂×M₁₆	—	94	1	no	—
256	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
257	C₂²×C₈	—	47	1	no	—
258	C₂²×C₈	—	94	1	no	—
259	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	#125	—	2	2	no	—
263	#125	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₄×Q₈	—	94	2	no	—
266	C₄×D₄	—	94	2	no	—
267	C₄×D₄	—	188	2	no	—
268	#117	—	94	2	no	—
269	#117	C₂²×D₄₇	1	1	yes	characteristic
270	#117	C₂²×D₄₇	1	2	yes	—
271	C₂×C₄⋊C₄	—	47	2	no	—
272	C₂×C₄⋊C₄	—	94	2	no	—
273	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
274	C₂×C₄²	—	47	1	no	—
275	#108	—	94	2	no	—
276	#107	—	2	2	no	—
277	#107	—	94	2	no	—
278	#107	—	94	2	no	—
279	#107	—	94	2	no	—
280	#107	—	94	2	no	—
281	#107	C₂²⋊D₄₇	1	2	yes	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	2	2	no	—
285	#106	—	94	1	no	—
286	#106	—	94	1	no	—
287	#106	—	94	2	no	—
288	#106	—	94	2	no	—
289	#106	C₂²⋊D₄₇	1	1	yes	characteristic
290	#106	C₂²⋊D₄₇	1	1	yes	characteristic
291	C₄⋊C₈	—	94	2	no	—
292	C₄⋊C₈	—	94	2	no	—
293	#104	—	47	4	no	—
294	#103	—	47	2	no	—
295	#103	—	94	2	no	—
296	#102	—	47	2	no	—
297	#102	—	94	2	no	—
298	#97	—	2	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	94	2	no	—
301	#95	—	188	2	no	—
302	#41	—	47	2	no	—
303	#41	—	47	2	no	—
304	#41	—	94	1	no	—
305	#41	—	94	1	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂×Q₈	—	47	1	no	—
308	C₂×D₄	—	47	1	no	—
309	C₂×D₄	—	94	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	94	1	no	—
312	C₂²×C₄	—	94	1	no	—
313	C₂²×C₄	—	94	2	no	—
314	C₂²×C₄	—	188	2	no	—
315	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
316	M₁₆	—	2	1	no	—
317	M₁₆	—	47	2	no	—
318	M₁₆	—	94	1	no	—
319	M₁₆	—	94	4	no	—
320	M₁₆	—	94	2	no	—
321	M₁₆	—	94	2	no	—
322	M₁₆	Q₈×D₄₇	1	2	yes	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	47	2	no	—
325	C₂×C₈	—	94	1	no	—
326	C₂×C₈	—	94	1	no	—
327	C₂×C₈	—	94	1	no	—
328	C₂×C₈	—	94	4	no	—
329	C₂×C₈	—	94	2	no	—
330	C₂×C₈	Q₈×D₄₇	1	2	yes	—
331	C₂×C₈	#10625354	1	2	yes	—
332	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	94	2	no	—
340	C₄⋊C₄	—	94	2	no	—
341	C₄⋊C₄	—	188	2	no	—
342	C₄⋊C₄	D₄×D₄₇	1	2	yes	—
343	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
344	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	188	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	47	2	no	—
351	C₄²	—	94	1	no	—
352	C₄²	—	188	2	no	—
353	C₂³	—	94	1	no	—
354	C₂³	#13094069	1	1	yes	characteristic
355	Q₈	—	16	2	no	—
356	Q₈	—	47	2	no	—
357	Q₈	—	94	1	no	—
358	D₄	—	47	2	no	—
359	D₄	—	94	2	no	—
360	D₄	—	94	2	no	—
361	D₄	—	94	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	94	1	no	—
368	C₂×C₄	—	94	2	no	—
369	C₂×C₄	—	94	2	no	—
370	C₂×C₄	—	94	2	no	—
371	C₂×C₄	—	376	2	no	—
372	C₂×C₄	—	188	2	no	—
373	C₂×C₄	—	188	2	no	—
374	C₂×C₄	—	752	2	no	—
375	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
376	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
377	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
378	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
380	C₂×C₄	#13094072	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	94	2	no	—
383	C₈	—	94	2	no	—
384	C₈	—	188	2	no	—
385	C₈	#13094000	1	2	yes	—
386	C₈	#13094000	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	94	2	no	—
389	C₂²	—	188	2	no	—
390	C₂²	#15689750	1	1	yes	characteristic
391	C₂²	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
392	C₂²	#15689834	1	1	yes	characteristic
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	—	8	2	no	—
397	C₄	—	94	2	no	—
398	C₄	—	376	2	no	—
399	C₄	—	188	2	no	—
400	C₄	C₂×#13094000	1	1	yes	characteristic
401	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
402	C₄	#15689776	1	1	yes	characteristic
403	C₄	Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	188	2	no	—
407	C₂	#20247086	1	1	yes	characteristic
408	C₂	#20255168	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3163	C₂
socle	C₂²×C₄₇	D₄²
prime radical	C₂×#10625350	C₂³
hypercenter	#3163	D₄₇
hypocenter	C₄₇	#26080
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3163

Complements

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

N	H ≅ G/N	type
3. #20244896	C₂	semidirect
4. #20244735	C₂	semidirect
5. #20244168	C₂	semidirect
6. #20244137	C₂	semidirect
8. #20243549	C₂	semidirect
11. C₄₇×#3163	C₂	semidirect
47. #13094030	Q₈	semidirect
71. #13093950	Q₈	semidirect
200. #3163	D₄₇	semidirect
249. C₄₇	#26080	semidirect

Maximal subgroups

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

subgroup	index
#20253235	2
#20251980	2
#20244896	2
#20244735	2
#20244168	2
#20244137	2
#20243618	2
#20243549	2
#20243448	2
#20243227	2
C₄₇×#3163	2
#26080	47