Subgroups — #26115337

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251746	C₂	1	1	yes	maximal, characteristic
2	#20243539	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688869	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688866	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688866	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094021	C₂²	1	1	yes	characteristic
9	C₂×#13094013	—	2	2	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₈	—	2	2	no	—
13	C₂×C₈×Dic₄₇	—	2	1	no	—
14	#15689024	—	2	1	no	—
15	#15689024	—	2	1	no	—
16	#15689024	C₄	1	4	yes	—
17	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
18	#15688877	—	2	1	no	—
19	#15688869	—	2	2	no	—
20	#15688866	—	2	2	no	—
21	#15688866	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#332	—	2	2	no	—
26	C₂²×D₁₈₈	—	2	1	no	—
27	C₂²×D₁₈₈	—	2	1	no	—
28	C₂²×D₁₈₈	C₂×C₄	1	2	yes	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	#13094021	—	2	1	no	—
32	#13094021	—	2	1	no	—
33	#13094021	—	2	1	no	—
34	#13094021	—	2	1	no	—
35	#13094013	—	4	4	no	—
36	#13094013	—	4	4	no	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
38	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
47	Dic₄₇⋊C₈	—	4	4	no	—
48	C₈×Dic₄₇	—	4	2	no	—
49	C₂²×Q₈×C₄₇	—	2	1	no	—
50	C₂²×C₈×C₄₇	—	2	1	no	—
51	C₂²×C₈×C₄₇	—	2	1	no	—
52	C₄⋊Q₈×C₄₇	—	2	1	no	—
53	C₄⋊Q₈×C₄₇	—	2	1	no	—
54	C₄⋊Q₈×C₄₇	—	2	1	no	—
55	C₄⋊Q₈×C₄₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	—	2	1	no	—
60	C₄⋊C₈×C₄₇	—	2	1	no	—
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
63	C₂³×D₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	—	4	1	no	—
65	C₂×Q₈×C₄₇	—	4	1	no	—
66	C₂×Q₈×C₄₇	—	4	2	no	—
67	C₂×Q₈×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×D₁₈₈	—	4	4	no	—
77	C₂×D₁₈₈	—	4	1	no	—
78	C₂×D₁₈₈	—	4	1	no	—
79	C₂×D₁₈₈	—	4	1	no	—
80	C₂×D₁₈₈	—	4	1	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	4	2	no	—
83	C₂×D₁₈₈	—	4	1	no	—
84	C₂×D₁₈₈	—	4	1	no	—
85	C₂×D₁₈₈	—	4	1	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	4	1	no	—
89	C₂×C₈×C₄₇	—	4	1	no	—
90	C₂×C₈×C₄₇	—	4	1	no	—
91	C₂×C₈×C₄₇	—	4	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	—	4	1	no	—
95	C₄⋊C₄×C₄₇	—	4	1	no	—
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₇	#41	1	1	yes	characteristic
99	C₄²×C₄₇	#41	1	1	yes	characteristic
100	(C₂×D₄₇)⋊C₄	—	4	8	no	—
101	(C₂×D₄₇)⋊C₄	—	4	8	no	—
102	C₄×Dic₄₇	—	2	1	no	—
103	C₄×Dic₄₇	—	2	1	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	—	2	1	no	—
106	C₄×Dic₄₇	—	4	2	no	—
107	C₄×Dic₄₇	D₈	1	4	yes	—
108	C₄×Dic₄₇	SD₁₆	1	4	yes	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	4	1	no	—
111	C₂×C₄₇⋊C₈	—	4	1	no	—
112	C₂×C₄₇⋊C₈	—	4	1	no	—
113	C₂×C₄₇⋊C₈	—	4	2	no	—
114	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×D₄₇	—	4	2	no	—
116	C₂²×D₄₇	—	4	4	no	—
117	C₂²×D₄₇	—	4	4	no	—
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	—	4	2	no	—
120	Q₈×C₄₇	—	8	2	no	—
121	Q₈×C₄₇	—	8	1	no	—
122	Q₈×C₄₇	—	8	1	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
136	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	—	2	2	no	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	#102	1	4	yes	—
143	C₂×Dic₄₇	#102	1	4	yes	—
144	D₁₈₈	—	8	2	no	—
145	D₁₈₈	—	8	2	no	—
146	D₁₈₈	—	8	1	no	—
147	D₁₈₈	—	8	1	no	—
148	D₁₈₈	—	8	1	no	—
149	D₁₈₈	—	8	1	no	—
150	D₁₈₈	—	8	2	no	—
151	D₁₈₈	—	8	1	no	—
152	D₁₈₈	—	8	1	no	—
153	C₈×C₄₇	—	4	2	no	—
154	C₈×C₄₇	—	8	1	no	—
155	C₄₇⋊C₈	—	4	2	no	—
156	C₄₇⋊C₈	—	8	1	no	—
157	#9698	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	#390	1	1	yes	characteristic
159	C₂²×C₄₇	#413	1	2	yes	—
160	C₂²×C₄₇	C₂×#102	1	2	yes	—
161	C₂²×C₄₇	#416	1	1	yes	characteristic
162	C₂²×C₄₇	#417	1	1	yes	characteristic
163	C₂×D₄₇	—	8	4	no	—
164	C₂×D₄₇	—	8	4	no	—
165	C₂×D₄₇	—	8	2	no	—
166	C₂×D₄₇	—	8	2	no	—
167	C₂×D₄₇	—	8	2	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	#463	1	1	yes	characteristic
172	C₄×C₄₇	#463	1	1	yes	characteristic
173	C₄×C₄₇	#464	1	1	yes	characteristic
174	C₄×C₄₇	#464	1	1	yes	characteristic
175	C₄×C₄₇	#486	1	2	yes	—
176	C₄×C₄₇	#492	1	1	yes	characteristic
177	C₄×C₄₇	#493	1	1	yes	characteristic
178	Dic₄₇	—	2	4	no	—
179	Dic₄₇	—	2	4	no	—
180	#2858	—	47	1	no	—
181	#1842	—	47	1	no	—
182	C₄×C₄⋊C₈	—	47	1	no	—
183	C₂×#332	—	47	1	no	—
184	C₂×#332	D₄₇	1	1	yes	characteristic, complemented
185	C₂×#331	—	47	1	no	—
186	C₂×#331	—	47	1	no	—
187	C₂×C₄₇	#1540	1	2	yes	—
188	C₂×C₄₇	#1547	1	2	yes	—
189	C₂×C₄₇	#1957	1	1	yes	characteristic
190	C₂×C₄₇	#1967	1	1	yes	characteristic
191	C₂×C₄₇	#1983	1	1	yes	characteristic
192	D₄₇	—	16	2	no	—
193	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
194	C₂×#127	—	47	1	no	—
195	C₂×#124	—	94	2	no	—
196	C₂×C₄⋊C₈	—	47	1	no	—
197	C₂×C₄⋊C₈	—	47	1	no	—
198	C₂×C₄⋊C₈	—	94	2	no	—
199	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₄×C₈	—	94	1	no	—
202	#390	—	47	4	no	—
203	#390	—	94	1	no	—
204	#390	—	94	1	no	—
205	C₄³	—	47	1	no	—
206	#336	—	94	1	no	—
207	#332	—	2	2	no	—
208	#332	—	94	2	no	—
209	#331	—	94	2	no	—
210	#331	—	94	2	no	—
211	C₄₇	#9698	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	2	1	no	—
213	C₂²×D₄	—	47	2	no	—
214	C₂²×D₄	—	94	1	no	—
215	C₂²×D₄	—	94	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	94	1	no	—
219	C₂²×C₈	—	94	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	#127	—	94	1	no	—
225	#127	—	94	1	no	—
226	#127	—	94	1	no	—
227	#127	—	94	1	no	—
228	#124	—	188	4	no	—
229	#124	—	188	4	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
232	C₂×C₂²⋊C₄	—	94	4	no	—
233	C₂×C₂²⋊C₄	—	94	4	no	—
234	C₂×C₄²	—	47	2	no	—
235	C₂×C₄²	—	47	1	no	—
236	C₂×C₄²	—	47	1	no	—
237	C₂×C₄²	—	47	2	no	—
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	47	2	no	—
243	C₄⋊C₈	—	94	1	no	—
244	C₄⋊C₈	—	94	1	no	—
245	C₄⋊C₈	—	94	1	no	—
246	C₄⋊C₈	—	188	4	no	—
247	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
248	C₄×C₈	—	188	2	no	—
249	C₂⁴	—	94	2	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×D₄	—	94	2	no	—
255	C₂×D₄	—	94	2	no	—
256	C₂×D₄	—	188	4	no	—
257	C₂×D₄	—	188	1	no	—
258	C₂×D₄	—	188	1	no	—
259	C₂×D₄	—	188	1	no	—
260	C₂×D₄	—	188	1	no	—
261	C₂×D₄	—	188	2	no	—
262	C₂×D₄	—	188	2	no	—
263	C₂×D₄	—	188	1	no	—
264	C₂×D₄	—	188	1	no	—
265	C₂×D₄	—	188	1	no	—
266	C₂×D₄	—	188	1	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	47	2	no	—
269	C₂²×C₄	—	47	2	no	—
270	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	188	1	no	—
280	C₂×C₈	—	188	2	no	—
281	C₂×C₈	—	188	1	no	—
282	C₂×C₈	—	188	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	188	8	no	—
288	C₂²⋊C₄	—	188	8	no	—
289	C₄²	—	47	4	no	—
290	C₄²	—	47	4	no	—
291	C₄²	—	94	1	no	—
292	C₄²	—	94	1	no	—
293	C₄²	—	94	1	no	—
294	C₄²	—	94	1	no	—
295	C₄²	—	188	2	no	—
296	C₄²	#10625350	1	1	yes	characteristic
297	C₄²	D₄×D₄₇	1	1	yes	characteristic
298	C₄²	#10625352	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂³	—	188	2	no	—
301	C₂³	—	188	4	no	—
302	C₂³	—	188	4	no	—
303	C₂³	—	188	2	no	—
304	C₂³	—	188	2	no	—
305	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
306	Q₈	—	8	1	no	—
307	Q₈	—	8	1	no	—
308	Q₈	—	8	2	no	—
309	D₄	—	376	2	no	—
310	D₄	—	376	2	no	—
311	D₄	—	376	1	no	—
312	D₄	—	376	1	no	—
313	D₄	—	376	1	no	—
314	D₄	—	376	1	no	—
315	D₄	—	376	1	no	—
316	D₄	—	376	1	no	—
317	D₄	—	376	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	47	4	no	—
322	C₂×C₄	—	47	4	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
329	C₂×C₄	#13093996	1	1	yes	characteristic
330	C₂×C₄	#13093997	1	1	yes	characteristic
331	C₂×C₄	#13094010	1	1	yes	characteristic
332	C₂×C₄	#13094020	1	1	yes	characteristic
333	C₂×C₄	D₈×D₄₇	1	2	yes	—
334	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
335	C₂×C₄	#13094045	1	1	yes	characteristic
336	C₂×C₄	#13094047	1	1	yes	characteristic
337	C₂×C₄	#13094051	1	1	yes	characteristic
338	C₂×C₄	#13094056	1	1	yes	characteristic
339	C₈	—	4	2	no	—
340	C₈	—	8	1	no	—
341	C₈	—	376	1	no	—
342	C₈	—	188	2	no	—
343	C₂²	—	376	2	no	—
344	C₂²	—	376	2	no	—
345	C₂²	—	376	4	no	—
346	C₂²	—	376	4	no	—
347	C₂²	—	376	2	no	—
348	C₂²	#15689023	1	1	yes	characteristic
349	C₂²	#15689106	1	2	yes	—
350	C₂²	#102×D₄₇	1	2	yes	—
351	C₂²	#15689118	1	1	yes	characteristic
352	C₂²	#15689123	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	94	4	no	—
357	C₄	—	94	4	no	—
358	C₄	#15689340	1	1	yes	characteristic
359	C₄	#15689341	1	1	yes	characteristic
360	C₄	#15689347	1	1	yes	characteristic
361	C₄	#15689348	1	1	yes	characteristic
362	C₄	#15689468	1	2	yes	—
363	C₄	#15689494	1	1	yes	characteristic
364	C₄	#15689498	1	1	yes	characteristic
365	C₂	—	752	2	no	—
366	C₂	#20242036	1	2	yes	—
367	C₂	#20242064	1	2	yes	—
368	C₂	#20244192	1	1	yes	characteristic
369	C₂	#20244236	1	1	yes	characteristic
370	C₂	#20244317	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#332	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#13094021	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#332	D₄₇
hypocenter	C₄₇	#9698
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #20243539	C₂	semidirect
3. C₄⋊C₈×Dic₄₇	C₂	semidirect
4. C₂×#15688869	C₂	semidirect
7. C₂×C₄₇×#332	C₂	semidirect
184. C₂×#332	D₄₇	semidirect
211. C₄₇	#9698	semidirect

Maximal subgroups

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

subgroup	index
#20251746	2
#20243539	2
C₄⋊C₈×Dic₄₇	2
C₂×#15688869	2
C₂×#15688866	2
C₂×C₄₇×#332	2
#9698	47