Subgroups — #26104422

Order 12032.

676 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243456	C₂	1	1	yes	maximal, characteristic
2	#20243082	C₂	1	1	yes	maximal, characteristic
3	#20241214	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#391	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#391	—	2	1	no	—
15	C₄₇×#391	—	2	1	no	—
16	C₄₇×#391	C₄	1	2	yes	—
17	C₄₇×#391	C₂²	1	2	yes	—
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	—	2	1	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	4	2	no	—
22	C₂×#10625329	—	4	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
26	C₂²×Q₈×C₄₇	—	2	4	no	—
27	C₂²×Q₈×C₄₇	—	2	1	no	—
28	C₂²×Q₈×C₄₇	—	4	2	no	—
29	C₂²×Q₈×C₄₇	—	4	2	no	—
30	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	—	2	1	no	—
32	C₂³×C₄×C₄₇	—	2	1	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	—	4	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#95×C₄₇	—	2	2	no	—
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	2	no	—
48	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₄₇	—	2	4	no	—
50	C₂×Q₈×C₄₇	—	2	2	no	—
51	C₂×Q₈×C₄₇	—	4	16	no	—
52	C₂×Q₈×C₄₇	—	4	2	no	—
53	C₂×Q₈×C₄₇	—	4	2	no	—
54	C₂×Q₈×C₄₇	—	4	2	no	—
55	C₂×Q₈×C₄₇	—	4	2	no	—
56	C₂×Q₈×C₄₇	M₁₆	1	4	yes	complemented
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	4	2	no	—
83	C₄²×C₄₇	—	4	1	no	—
84	C₄²×C₄₇	—	4	1	no	—
85	C₄²×C₄₇	—	4	1	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	#10625329	—	4	8	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	#99	1	2	yes	—
99	C₂³×C₄₇	#103	1	2	yes	—
100	C₂³×C₄₇	#104	1	2	yes	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₄₇	—	4	4	no	—
103	Q₈×C₄₇	—	4	8	no	—
104	Q₈×C₄₇	—	4	4	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	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	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	8	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	8	4	no	—
130	C₂×C₄×C₄₇	#100	1	2	yes	—
131	C₂×C₄×C₄₇	#103	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#137	1	2	yes	—
136	C₄₇⋊C₈	—	4	4	no	—
137	C₄₇⋊C₈	—	8	2	no	—
138	#7609	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	#328	1	4	yes	—
146	C₂²×C₄₇	#407	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
150	C₂²×C₄₇	#419	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
152	C₂²×C₄₇	#421	1	1	yes	characteristic
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	4	4	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	—	8	2	no	—
162	C₂×#391	D₄₇	1	1	yes	characteristic
163	#1817	—	47	1	no	—
164	#1728	—	47	1	no	—
165	#1272	—	47	4	no	—
166	C₂×C₄₇	—	2	4	no	—
167	C₂×C₄₇	#1462	1	1	yes	characteristic
168	C₂×C₄₇	#1487	1	1	yes	characteristic
169	C₂×C₄₇	#1491	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
171	C₂×C₄₇	#1511	1	1	yes	characteristic
172	C₂×C₄₇	#1527	1	1	yes	characteristic
173	C₂×C₄₇	#1548	1	1	yes	characteristic
174	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
175	C₂²×M₁₆	—	94	1	no	—
176	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	—	94	1	no	—
179	#391	—	2	1	no	—
180	#391	—	2	1	no	—
181	#391	Dic₄₇	1	2	yes	—
182	#391	C₂×D₄₇	1	2	yes	—
183	C₂×#95	—	2	1	no	—
184	C₂×#95	—	2	1	no	—
185	#336	—	47	2	no	—
186	#336	—	47	2	no	—
187	#336	—	94	1	no	—
188	#336	—	94	1	no	—
189	C₄₇	#7609	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×Q₈	—	4	2	no	—
193	C₂²×Q₈	—	4	2	no	—
194	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
195	C₂³×C₄	—	2	1	no	—
196	C₂³×C₄	—	2	1	no	—
197	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂×M₁₆	—	94	4	no	—
201	C₂×M₁₆	—	188	2	no	—
202	C₂×M₁₆	—	188	2	no	—
203	C₂²×C₈	—	94	2	no	—
204	C₂²×C₈	—	94	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²⋊C₈	—	188	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	16	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	#10625329	1	4	yes	complemented
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	#10625336	1	2	yes	—
244	C₂²×C₄	#10625337	1	2	yes	—
245	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	M₁₆	—	188	8	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	188	2	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#13093939	1	2	yes	—
269	C₂³	#13093948	1	2	yes	—
270	C₂³	#13093951	1	2	yes	—
271	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
272	Q₈	—	4	4	no	—
273	Q₈	—	4	8	no	—
274	Q₈	—	4	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	8	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	#13093941	1	2	yes	—
301	C₂×C₄	#13093948	1	2	yes	—
302	C₂×C₄	C₂×#10625329	1	2	yes	—
303	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
304	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
305	C₂×C₄	#13094055	1	2	yes	—
306	C₈	—	376	2	no	—
307	C₈	—	188	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	#15688860	1	4	yes	—
315	C₂²	#15689087	1	1	yes	characteristic
316	C₂²	C₂×#13093939	1	1	yes	characteristic
317	C₂²	C₂×#13093941	1	1	yes	characteristic
318	C₂²	C₂×#13093948	1	1	yes	characteristic
319	C₂²	#15689132	1	1	yes	characteristic
320	C₂²	C₂×#13093951	1	1	yes	characteristic
321	C₂²	#15689140	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	4	no	—
330	C₄	—	8	2	no	—
331	C₂	—	2	4	no	—
332	C₂	#20241715	1	1	yes	characteristic
333	C₂	#20241817	1	1	yes	characteristic
334	C₂	#20241835	1	1	yes	characteristic
335	C₂	C₂×#15688860	1	1	yes	characteristic
336	C₂	#20241921	1	1	yes	characteristic
337	C₂	#20241986	1	1	yes	characteristic
338	C₂	#20242069	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#391	D₄₇
hypocenter	C₄₇	#7609
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#391

Complements

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

N	H ≅ G/N	type
3. #20241214	C₂	semidirect
56. C₂×Q₈×C₄₇	M₁₆	semidirect
189. C₄₇	#7609	semidirect
226. C₂×Q₈	#10625329	semidirect

Maximal subgroups

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

subgroup	index
#20243456	2
#20243082	2
#20241214	2
C₂×C₄₇×#391	2
#7609	47