Subgroups — #26214935

Order 12032.

866 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	#20245865	C₂	1	1	yes	maximal, characteristic
4	#20243264	C₂	1	2	yes	maximal
5	#20243256	C₂	1	2	yes	maximal
6	#20243255	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	#20243248	C₂	1	2	yes	maximal
9	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
10	#20243076	C₂	1	2	yes	maximal
11	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	#15689253	C₄	1	8	yes	complemented
14	C₈×Dic₉₄	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
19	C₂×#13093934	C₂²	1	2	yes	—
20	C₂×#13093934	C₂²	1	2	yes	—
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
24	#15689029	C₄	1	2	yes	—
25	#15689021	C₄	1	2	yes	—
26	#15689007	C₄	1	4	yes	—
27	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
28	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688969	C₂²	1	2	yes	—
30	#15688878	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	#15688877	C₂²	1	2	yes	—
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄³×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
43	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
44	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
53	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
54	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
55	#13093934	—	2	4	no	—
56	#13093934	C₂×C₄	1	8	yes	—
57	C₄×C₄₇⋊C₈	—	2	4	no	—
58	C₈×Dic₄₇	—	2	4	no	—
59	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
60	C₄₇⋊#95	C₂×C₄	1	4	yes	—
61	C₄₇⋊#95	C₂×C₄	1	4	yes	—
62	C₄₇⋊#95	C₂×C₄	1	2	yes	—
63	C₄₇⋊#95	C₂×C₄	1	4	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#97×C₄₇	—	2	2	no	—
74	#97×C₄₇	—	2	2	no	—
75	#97×C₄₇	—	2	2	no	—
76	#97×C₄₇	Q₈	1	4	yes	complemented
77	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
85	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₉₄	—	2	4	no	—
88	C₂×Dic₉₄	C₄²	1	4	yes	—
89	C₂×C₈×C₄₇	—	2	1	no	—
90	C₂×C₈×C₄₇	—	2	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₄₇	#41	1	2	yes	—
100	C₂×C₈×C₄₇	#41	1	2	yes	—
101	C₄²×C₄₇	—	2	4	no	—
102	C₄²×C₄₇	—	2	1	no	—
103	C₄²×C₄₇	—	2	1	no	—
104	C₄²×C₄₇	—	4	2	no	—
105	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
107	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
108	C₄²×C₄₇	#41	1	2	yes	—
109	C₄⋊Dic₄₇	—	2	4	no	—
110	C₄⋊Dic₄₇	C₄²	1	4	yes	—
111	Dic₄₇⋊C₄	—	4	4	no	—
112	Dic₄₇⋊C₄	C₄²	1	8	yes	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₄×Dic₄₇	—	2	2	no	—
115	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
116	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
123	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
133	C₂×C₄×C₄₇	#131	1	4	yes	—
134	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	2	4	no	—
142	C₂×Dic₄₇	—	2	4	no	—
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	2	4	no	—
145	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
146	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
147	Dic₉₄	—	2	8	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	—	2	2	no	—
150	C₈×C₄₇	C₄×Q₈	1	4	yes	—
151	C₈×C₄₇	C₄×Q₈	1	4	yes	—
152	C₄₇⋊C₈	—	2	4	no	—
153	C₄₇⋊C₈	—	2	4	no	—
154	#20521	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#131	1	2	yes	—
159	C₂²×C₄₇	#568	1	2	yes	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	4	2	no	—
163	C₄×C₄₇	#405	1	4	yes	—
164	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
167	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
168	Dic₄₇	—	2	4	no	—
169	Dic₄₇	—	2	4	no	—
170	Dic₄₇	—	4	4	no	—
171	C₂×#446	—	47	1	no	—
172	C₂×C₈×Q₈	—	47	1	no	—
173	#2295	—	47	1	no	—
174	#1771	—	47	2	no	—
175	#1770	—	47	2	no	—
176	#1770	—	47	1	no	—
177	C₈×C₄⋊C₄	—	47	1	no	—
178	#1768	—	47	2	no	—
179	C₄×C₄⋊C₈	—	47	1	no	—
180	#1726	—	47	2	no	—
181	C₄×#97	D₄₇	1	1	yes	characteristic
182	C₂×C₄₇	C₄²×Q₈	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
184	C₂×C₄₇	#2875	1	1	yes	characteristic
185	C₂×C₄₇	#2962	1	2	yes	—
186	C₂×C₄₇	#2969	1	2	yes	—
187	C₂×C₄×Q₈	—	47	1	no	—
188	#446	—	47	8	no	—
189	C₈×Q₈	—	47	8	no	—
190	C₂×C₄⋊C₈	—	47	2	no	—
191	C₂×C₄⋊C₈	—	47	2	no	—
192	C₂×C₄⋊C₈	—	47	1	no	—
193	C₂×C₄⋊C₈	—	47	1	no	—
194	C₂×#97	—	47	2	no	—
195	C₂×#97	—	47	2	no	—
196	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
197	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
198	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₄×C₈	—	47	2	no	—
201	C₂×C₄×C₈	—	47	2	no	—
202	C₂×C₄×C₈	—	47	2	no	—
203	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
205	#391	—	47	2	no	—
206	#389	—	47	2	no	—
207	#387	—	47	4	no	—
208	C₄×C₄⋊C₄	—	47	1	no	—
209	C₄×C₄⋊C₄	—	47	2	no	—
210	#376	—	47	2	no	—
211	C₄³	C₂×D₄₇	1	1	yes	characteristic
212	#336	—	47	2	no	—
213	#336	—	47	2	no	—
214	#336	—	47	2	no	—
215	C₄₇	#20521	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	47	1	no	—
217	C₂²×C₈	—	47	2	no	—
218	C₂²×C₈	—	47	2	no	—
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
223	C₄×Q₈	—	47	4	no	—
224	C₄×Q₈	—	47	4	no	—
225	C₂×C₄⋊C₄	—	47	2	no	—
226	C₂×C₄⋊C₄	—	47	1	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	—	47	4	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₂×C₄²	—	47	2	no	—
231	C₂×C₄²	C₄×D₄₇	1	2	yes	—
232	C₂×C₄²	C₄×D₄₇	1	2	yes	—
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	C₄⋊C₈	—	47	8	no	—
237	C₄⋊C₈	—	47	8	no	—
238	C₄⋊C₈	—	47	4	no	—
239	C₄⋊C₈	—	47	4	no	—
240	#97	—	2	2	no	—
241	#97	—	2	2	no	—
242	#97	—	2	2	no	—
243	#97	—	47	8	no	—
244	#97	—	94	4	no	—
245	#97	Dic₉₄	1	4	yes	complemented
246	C₄×C₈	—	47	8	no	—
247	C₄×C₈	—	94	4	no	—
248	C₄×C₈	—	94	4	no	—
249	C₄×C₈	C₄×D₄₇	1	4	yes	—
250	C₄×C₈	C₄×D₄₇	1	4	yes	—
251	#95	—	47	4	no	—
252	#95	—	47	4	no	—
253	#95	—	47	2	no	—
254	#95	—	47	4	no	—
255	C₂×Q₈	—	47	4	no	—
256	C₂×Q₈	—	94	4	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	2	no	—
259	C₂²×C₄	—	47	4	no	—
260	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
261	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
263	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	47	4	no	—
270	C₂×C₈	—	47	4	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	—	94	2	no	—
274	C₂×C₈	—	94	2	no	—
275	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
276	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
277	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
278	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
279	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
280	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
281	C₂×C₈	#10625350	1	2	yes	—
282	C₂×C₈	#10625350	1	2	yes	—
283	C₄⋊C₄	—	47	8	no	—
284	C₄⋊C₄	—	47	4	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	188	4	no	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	4	2	no	—
291	C₄²	—	47	4	no	—
292	C₄²	—	47	4	no	—
293	C₄²	—	94	2	no	—
294	C₄²	—	94	2	no	—
295	C₄²	C₂×Dic₉₄	1	2	yes	—
296	C₄²	C₂×C₄×D₄₇	1	2	yes	—
297	C₄²	C₂×C₄×D₄₇	1	2	yes	—
298	C₄²	#10625350	1	2	yes	—
299	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	Q₈	—	94	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	47	4	no	—
308	C₂×C₄	—	47	4	no	—
309	C₂×C₄	—	94	4	no	—
310	C₂×C₄	—	94	4	no	—
311	C₂×C₄	—	94	4	no	—
312	C₂×C₄	—	94	4	no	—
313	C₂×C₄	—	94	4	no	—
314	C₂×C₄	C₄²×D₄₇	1	2	yes	—
315	C₂×C₄	C₄²×D₄₇	1	2	yes	—
316	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
317	C₂×C₄	#13094032	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
321	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
322	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
323	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	94	4	no	—
327	C₈	—	94	4	no	—
328	C₈	C₄×Dic₉₄	1	4	yes	—
329	C₈	C₄×Dic₉₄	1	4	yes	—
330	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
331	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
332	C₂²	C₄×#10625350	1	1	yes	characteristic
333	C₂²	C₂×#13094032	1	2	yes	—
334	C₂²	#15689889	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	94	4	no	—
339	C₄	—	94	4	no	—
340	C₄	—	188	4	no	—
341	C₄	#15689080	1	4	yes	—
342	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
343	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
344	C₄	C₄×#10625350	1	1	yes	characteristic
345	C₄	C₄×#10625350	1	1	yes	characteristic
346	C₂	C₄²×Dic₉₄	1	1	yes	characteristic
347	C₂	C₂×#15689080	1	1	yes	characteristic
348	C₂	#20251813	1	1	yes	characteristic
349	C₂	#20252335	1	2	yes	—
350	C₂	#20252392	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂³×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#20521
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
13. #15689253	C₄	semidirect
14. C₈×Dic₉₄	C₄	semidirect
76. #97×C₄₇	Q₈	semidirect
215. C₄₇	#20521	semidirect
245. #97	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
C₂×C₈×Dic₉₄	2
#20245865	2
#20243264	2
#20243256	2
#20243255	2
C₈×C₄⋊Dic₄₇	2
#20243248	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243076	2
C₄×#97×C₄₇	2
#20521	47