Subgroups — #26116451

Order 12032.

684 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	#20241267	C₂	1	4	yes	maximal
4	C₄×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15689029	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	C₄	1	2	yes	complemented
12	#15688993	C₄	1	1	yes	characteristic, complemented
13	#15688983	—	2	1	no	—
14	#15688983	C₂²	1	2	yes	—
15	#15688878	—	2	1	no	—
16	C₂²×C₄²×C₄₇	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	4	2	no	—
23	#13094003	—	4	2	no	—
24	#13094003	—	4	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₇	—	2	2	no	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	—	2	4	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	—	4	2	no	—
43	C₂×C₄²×C₄₇	—	4	2	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	#95×C₄₇	—	2	4	no	—
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	Q₈	1	4	yes	complemented
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	1	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	D₈	1	2	yes	—
71	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂×Dic₉₄	—	4	2	no	—
79	C₂×Dic₉₄	—	4	2	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	4	1	no	—
84	C₄²×C₄₇	—	4	1	no	—
85	C₄²×C₄₇	—	4	8	no	—
86	C₂²⋊Dic₄₇	—	4	2	no	—
87	C₂²⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	Dic₄₇⋊C₄	—	4	4	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂³×C₄₇	#137	1	2	yes	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	#99	1	2	yes	—
130	C₂×C₄×C₄₇	#102	1	2	yes	—
131	C₂×C₄×C₄₇	#102	1	2	yes	—
132	C₂×C₄×C₄₇	#104	1	2	yes	—
133	C₂×C₄×C₄₇	#104	1	2	yes	—
134	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	Dic₉₄	—	8	4	no	—
144	C₄₇⋊C₈	—	8	2	no	—
145	#9909	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	#391	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
156	C₂²×C₄₇	#410	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
158	C₂²×C₄₇	#419	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
161	C₂²×C₄₇	#474	1	2	yes	—
162	C₂²×C₄₇	#475	1	2	yes	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	4	8	no	—
169	C₄×C₄₇	—	4	1	no	—
170	C₄×C₄₇	#327	1	4	yes	—
171	Dic₄₇	—	8	2	no	—
172	Dic₄₇	—	8	2	no	—
173	#2857	—	47	1	no	—
174	#1797	—	47	1	no	—
175	C₄×#95	D₄₇	1	1	yes	characteristic
176	#1294	—	47	4	no	—
177	C₂×C₄₇	—	2	2	no	—
178	C₂×C₄₇	—	2	2	no	—
179	C₂×C₄₇	C₂×#327	1	1	yes	characteristic
180	C₂×C₄₇	#1502	1	1	yes	characteristic
181	C₂×C₄₇	#1506	1	1	yes	characteristic
182	C₂×C₄₇	#1508	1	1	yes	characteristic
183	C₂×C₄₇	#1988	1	1	yes	characteristic
184	C₂×C₄₇	#1994	1	1	yes	characteristic
185	C₂×C₄₇	#1996	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	—	47	1	no	—
187	C₂×C₂²⋊Q₈	—	94	1	no	—
188	C₂²×C₄²	—	2	1	no	—
189	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	94	1	no	—
191	C₂×C₂²⋊C₈	—	47	2	no	—
192	#391	—	94	1	no	—
193	#389	—	47	2	no	—
194	#389	—	94	1	no	—
195	#384	—	47	1	no	—
196	#380	—	47	2	no	—
197	#380	—	94	1	no	—
198	C₂×#95	—	2	1	no	—
199	C₂×#95	C₂×D₄₇	1	2	yes	—
200	#336	—	94	1	no	—
201	C₄₇	#9909	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	94	1	no	—
203	C₂³×C₄	—	2	2	no	—
204	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	—	94	2	no	—
208	C₄⋊Q₈	—	188	2	no	—
209	C₂²⋊Q₈	—	188	2	no	—
210	C₂²⋊Q₈	—	188	2	no	—
211	C₂×C₄⋊C₄	—	47	2	no	—
212	C₂×C₄⋊C₄	—	47	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	94	1	no	—
215	C₂×C₄⋊C₄	—	94	1	no	—
216	C₂×C₄⋊C₄	—	94	1	no	—
217	C₂×C₂²⋊C₄	—	94	2	no	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	—	4	2	no	—
220	C₂×C₄²	—	4	2	no	—
221	C₂×C₄²	—	4	2	no	—
222	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
225	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
226	C₄⋊C₈	—	188	2	no	—
227	C₂²⋊C₈	—	94	4	no	—
228	#95	—	2	4	no	—
229	#95	—	2	2	no	—
230	#95	—	94	2	no	—
231	#95	—	94	2	no	—
232	#95	Dic₉₄	1	4	yes	complemented
233	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×Q₈	—	188	2	no	—
235	C₂×Q₈	—	188	2	no	—
236	C₂×Q₈	—	188	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	#10625334	1	2	yes	—
258	C₂²×C₄	#10625336	1	2	yes	—
259	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
260	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
262	C₂²×C₄	#10625350	1	2	yes	—
263	C₂×C₈	—	188	2	no	—
264	C₂×C₈	—	188	2	no	—
265	C₂×C₈	—	188	2	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₄⋊C₄	—	188	2	no	—
268	C₄⋊C₄	—	188	2	no	—
269	C₄⋊C₄	—	188	4	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	—	188	2	no	—
272	C₂²⋊C₄	—	188	2	no	—
273	C₂²⋊C₄	—	188	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	8	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂³	#13094003	1	2	yes	—
285	C₂³	C₂×#10625334	1	1	yes	characteristic
286	C₂³	C₂×#10625336	1	1	yes	characteristic
287	C₂³	#13094054	1	2	yes	—
288	Q₈	—	376	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	#13093938	1	2	yes	—
319	C₂×C₄	#13093944	1	2	yes	—
320	C₂×C₄	#13093944	1	2	yes	—
321	C₂×C₄	#13093952	1	2	yes	—
322	C₂×C₄	#13093952	1	2	yes	—
323	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
324	C₂×C₄	#13094012	1	1	yes	characteristic
325	C₂×C₄	#13094024	1	1	yes	characteristic
326	C₈	—	376	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	#15689029	1	1	yes	characteristic
336	C₂²	C₂×#13093938	1	1	yes	characteristic
337	C₂²	#15689096	1	1	yes	characteristic
338	C₂²	C₂×#13093944	1	1	yes	characteristic
339	C₂²	#15689131	1	1	yes	characteristic
340	C₂²	C₂×#13093952	1	1	yes	characteristic
341	C₂²	C₂×#13093952	1	1	yes	characteristic
342	C₂²	#15689403	1	2	yes	—
343	C₂²	#15689408	1	2	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	8	no	—
350	C₄	—	4	1	no	—
351	C₄	—	376	2	no	—
352	C₄	—	376	2	no	—
353	C₄	#15688856	1	4	yes	—
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	C₂×#15688856	1	1	yes	characteristic
357	C₂	#20241882	1	1	yes	characteristic
358	C₂	#20241899	1	1	yes	characteristic
359	C₂	#20241907	1	1	yes	characteristic
360	C₂	#20244340	1	1	yes	characteristic
361	C₂	#20244365	1	1	yes	characteristic
362	C₂	#20244373	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄×#95×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×#95	D₄₇
hypocenter	C₄₇	#9909
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
11. #15689021	C₄	semidirect
12. #15688993	C₄	semidirect
50. #95×C₄₇	Q₈	semidirect
201. C₄₇	#9909	semidirect
232. #95	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
#20243368	2
#20241267	2
C₄×#95×C₄₇	2
#9909	47