Subgroups — #26216790

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	2	yes	maximal
2	#20243542	C₂	1	2	yes	maximal, complemented
3	#20243505	C₂	1	2	yes	maximal, complemented
4	#20243372	C₂	1	2	yes	maximal
5	#20243368	C₂	1	2	yes	maximal
6	#20243245	C₂	1	2	yes	maximal, complemented
7	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
10	#15689184	—	2	8	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
15	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	#15688878	C₂²	1	2	yes	—
19	#15688878	C₂²	1	2	yes	—
20	#15688878	C₂²	1	4	yes	—
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
28	C₄₇×#376	C₂²	1	1	yes	characteristic
29	C₄₇×#376	C₂²	1	1	yes	characteristic
30	C₄³×C₄₇	C₂²	1	1	yes	characteristic
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
39	#13093935	—	2	4	no	—
40	C₄×C₄₇⋊C₈	—	2	4	no	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	#117×C₄₇	—	2	4	no	—
44	#117×C₄₇	D₄	1	4	yes	—
45	#117×C₄₇	Q₈	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	—	2	4	no	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#95×C₄₇	C₂×C₄	1	2	yes	—
65	#95×C₄₇	C₂×C₄	1	2	yes	—
66	#95×C₄₇	C₂×C₄	1	2	yes	—
67	#95×C₄₇	C₂×C₄	1	2	yes	—
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
89	C₂²⋊C₄×C₄₇	—	2	2	no	—
90	C₂²⋊C₄×C₄₇	—	2	2	no	—
91	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
92	C₄²×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	2	4	no	—
94	C₄²×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	4	2	no	—
98	C₄²×C₄₇	—	4	2	no	—
99	C₄²×C₄₇	C₂×D₄	1	2	yes	—
100	C₄²×C₄₇	C₂×D₄	1	2	yes	—
101	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
103	C₄²×C₄₇	#41	1	4	yes	—
104	C₄²×C₄₇	#41	1	4	yes	—
105	C₂×C₄₇⋊C₈	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	—	2	4	no	—
109	C₂×C₄₇⋊C₈	—	2	4	no	—
110	C₂×C₄₇⋊C₈	—	2	4	no	—
111	C₂³×C₄₇	—	2	1	no	—
112	C₂³×C₄₇	—	2	1	no	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	—	2	2	no	—
115	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
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	4	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	4	8	no	—
136	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
138	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₄₇	#117	1	4	yes	—
140	C₂×C₄×C₄₇	#117	1	2	yes	—
141	C₂×C₄×C₄₇	#117	1	2	yes	—
142	C₂×C₄×C₄₇	#131	1	4	yes	—
143	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
147	C₄₇⋊C₈	—	4	4	no	—
148	C₄₇⋊C₈	—	4	4	no	—
149	#20813	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	4	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	4	1	no	—
153	C₂²×C₄₇	—	4	1	no	—
154	C₂²×C₄₇	—	4	2	no	—
155	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#131	1	2	yes	—
159	C₂²×C₄₇	#568	1	2	yes	—
160	C₄×C₄₇	—	2	4	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	2	4	no	—
165	C₄×C₄₇	—	2	4	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	4	1	no	—
168	C₄×C₄₇	—	4	1	no	—
169	C₄×C₄₇	#532	1	2	yes	—
170	C₄×C₄₇	#533	1	2	yes	—
171	C₂×#433	—	47	2	no	—
172	C₄×#117	D₄₇	1	1	yes	characteristic
173	#1843	—	47	2	no	—
174	#1831	—	47	2	no	—
175	#1798	—	47	2	no	—
176	#1797	—	47	2	no	—
177	#1768	—	47	2	no	—
178	#1767	—	47	1	no	—
179	C₄×C₄⋊C₈	—	47	1	no	—
180	C₂×C₄₇	—	4	2	no	—
181	C₂×C₄₇	#2285	1	1	yes	characteristic
182	C₂×C₄₇	#2906	1	2	yes	—
183	C₂×C₄₇	#2922	1	1	yes	characteristic
184	C₂×C₄₇	#2923	1	1	yes	characteristic
185	C₂×C₄₇	#2924	1	2	yes	—
186	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
187	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
188	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
189	#433	—	94	8	no	—
190	C₂×C₄⋊C₈	—	47	4	no	—
191	C₂×C₄⋊C₈	—	47	2	no	—
192	C₂×C₄⋊C₈	—	47	2	no	—
193	C₂×C₄⋊C₈	—	47	4	no	—
194	C₂×C₂²⋊C₈	—	47	4	no	—
195	C₂×#97	—	47	2	no	—
196	C₂×C₄×C₈	—	47	2	no	—
197	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
198	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
199	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
200	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
201	#376	C₂×D₄₇	1	1	yes	characteristic
202	#376	C₂×D₄₇	1	1	yes	characteristic
203	C₄³	C₂×D₄₇	1	1	yes	characteristic
204	C₄³	C₂×D₄₇	1	1	yes	characteristic
205	#336	—	47	2	no	—
206	#336	—	47	2	no	—
207	#336	—	47	4	no	—
208	C₄₇	#20813	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
210	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
211	C₂²×C₈	—	47	4	no	—
212	C₂²×C₈	—	47	4	no	—
213	#117	—	2	4	no	—
214	#117	Dic₉₄	1	4	yes	—
215	#117	D₁₈₈	1	4	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
219	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
224	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
225	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
226	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
227	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
228	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₄⋊C₈	—	94	8	no	—
235	C₄⋊C₈	—	94	8	no	—
236	C₄⋊C₈	—	94	4	no	—
237	C₄⋊C₈	—	94	4	no	—
238	C₂²⋊C₈	—	94	8	no	—
239	#97	—	94	4	no	—
240	C₄×C₈	—	94	4	no	—
241	#95	C₂×Dic₄₇	1	2	yes	—
242	#95	C₂×Dic₄₇	1	2	yes	—
243	#95	C₂×Dic₄₇	1	2	yes	—
244	#95	C₂×Dic₄₇	1	2	yes	—
245	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
252	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
262	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
263	C₂×C₈	—	94	4	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	94	4	no	—
266	C₂×C₈	—	94	4	no	—
267	C₂×C₈	—	94	4	no	—
268	C₂×C₈	—	94	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	2	no	—
282	C₄²	C₂×Dic₉₄	1	2	yes	—
283	C₄²	C₂×Dic₉₄	1	2	yes	—
284	C₄²	C₂×D₁₈₈	1	2	yes	—
285	C₄²	C₂×D₁₈₈	1	2	yes	—
286	C₄²	#10625350	1	4	yes	—
287	C₄²	#10625350	1	4	yes	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
293	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	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	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	8	no	—
313	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
316	C₂×C₄	#13093983	1	4	yes	—
317	C₂×C₄	#13093983	1	2	yes	—
318	C₂×C₄	#13093983	1	2	yes	—
319	C₂×C₄	#13094031	1	4	yes	—
320	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
321	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
322	C₂×C₄	C₂×#10625350	1	2	yes	—
323	C₂×C₄	C₂×#10625350	1	2	yes	—
324	C₈	—	188	4	no	—
325	C₈	—	188	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
332	C₂²	C₂×#13093983	1	1	yes	characteristic
333	C₂²	C₂×#13093983	1	1	yes	characteristic
334	C₂²	C₂×#13094031	1	2	yes	—
335	C₂²	#15689890	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	4	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	#15689673	1	2	yes	—
346	C₄	#15689680	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	#20245795	1	1	yes	characteristic
349	C₂	#20251956	1	2	yes	—
350	C₂	#20252032	1	1	yes	characteristic
351	C₂	#20252039	1	1	yes	characteristic
352	C₂	#20252046	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20813
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #20243542	C₂	semidirect
3. #20243505	C₂	semidirect
6. #20243245	C₂	semidirect
7. #20243241	C₂	semidirect
8. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
208. C₄₇	#20813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
#20243542	2
#20243505	2
#20243372	2
#20243368	2
#20243245	2
#20243241	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄×#117×C₄₇	2
#20813	47