Subgroups — #26214239

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689150	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
5	#20243263	C₂	1	4	yes	maximal, complemented
6	#20243257	C₂	1	4	yes	maximal, complemented
7	#20243085	C₂	1	1	yes	maximal, characteristic
8	#20243082	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂²×#10625329	C₂²	1	1	yes	characteristic
12	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15689150	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
16	#15689079	—	2	4	no	—
17	#15689079	C₄	1	8	yes	complemented
18	C₂×#13093935	—	2	4	no	—
19	C₂×#13093935	C₂²	1	2	yes	—
20	C₂×#13093935	C₂²	1	2	yes	—
21	C₂×#13093935	C₂²	1	2	yes	—
22	C₂×#13093935	C₂²	1	2	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	#15688878	C₂²	1	4	yes	—
26	#15688878	C₂²	1	4	yes	—
27	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#389	C₄	1	4	yes	—
31	C₄₇×#385	C₄	1	4	yes	—
32	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
33	C₂×#10625329	—	2	2	no	—
34	C₂×#10625329	—	2	2	no	—
35	C₂×#10625329	—	2	2	no	—
36	C₂×#10625329	—	2	2	no	—
37	C₂×#10625329	C₂×C₄	1	4	yes	—
38	C₂×#10625329	C₂×C₄	1	4	yes	—
39	C₂²×C₄₇⋊C₈	—	2	2	no	—
40	C₂²×C₄₇⋊C₈	—	2	2	no	—
41	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
43	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
44	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
45	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
46	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
48	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
51	#13093935	—	2	4	no	—
52	#13093935	—	2	4	no	—
53	#13093935	—	2	4	no	—
54	#13093935	C₂×C₄	1	8	yes	complemented
55	C₄×C₄₇⋊C₈	—	2	4	no	—
56	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
57	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
58	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	#117×C₄₇	—	2	2	no	—
60	#117×C₄₇	—	2	2	no	—
61	#117×C₄₇	C₂×C₄	1	4	yes	—
62	#117×C₄₇	C₂×C₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₄₇	—	2	2	no	—
69	C₂×C₄²×C₄₇	—	2	2	no	—
70	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
74	#95×C₄₇	C₂×C₄	1	8	yes	—
75	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	4	4	no	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₄₇	#41	1	2	yes	—
87	C₂²×C₄×C₄₇	#41	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	4	8	no	—
91	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
92	C₂²⋊C₄×C₄₇	—	2	4	no	—
93	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
94	C₄²×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
101	#10625329	—	2	8	no	—
102	#10625329	—	2	8	no	—
103	C₂×C₄₇⋊C₈	—	2	8	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	complemented
111	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
117	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
118	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂³×C₄₇	—	2	4	no	—
121	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂³×C₄₇	C₄×D₄	1	2	yes	—
123	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
124	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	2	8	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	4	8	no	—
132	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
133	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₄₇	#117	1	4	yes	—
136	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
137	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
139	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₄₇⋊C₈	—	2	4	no	—
147	C₄₇⋊C₈	—	2	4	no	—
148	C₄₇⋊C₈	—	2	4	no	—
149	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
150	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
151	#20436	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	4	no	—
153	C₂²×C₄₇	—	2	4	no	—
154	C₂²×C₄₇	—	2	4	no	—
155	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
156	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₂²×C₄₇	#568	1	4	yes	—
160	C₄×C₄₇	—	2	8	no	—
161	C₄×C₄₇	—	4	4	no	—
162	C₄×C₄₇	—	4	4	no	—
163	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
164	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
166	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂×#424	—	47	1	no	—
169	C₂×#405	—	47	1	no	—
170	C₂×#405	—	47	1	no	—
171	C₂²×#97	—	47	1	no	—
172	#2296	D₄₇	1	1	yes	characteristic
173	#1771	—	47	4	no	—
174	#1770	—	47	4	no	—
175	#1729	—	47	1	no	—
176	#1728	—	47	1	no	—
177	C₂×C₄₇	—	2	4	no	—
178	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄₇	#2875	1	2	yes	—
180	C₂×C₄₇	#2930	1	2	yes	—
181	C₂×C₄₇	#2961	1	2	yes	—
182	C₂²×M₁₆	—	47	1	no	—
183	C₂²×M₁₆	—	47	1	no	—
184	C₂³×C₈	—	47	1	no	—
185	C₂³×C₈	—	47	1	no	—
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	#424	—	47	8	no	—
190	C₂×C₄⋊C₈	—	47	4	no	—
191	#405	—	47	8	no	—
192	#405	—	94	4	no	—
193	C₂×#97	—	47	2	no	—
194	C₂×#97	—	47	2	no	—
195	C₂×#97	—	47	2	no	—
196	C₂×#97	—	47	2	no	—
197	C₂×#97	—	94	4	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₂×C₄×C₈	—	47	2	no	—
200	#389	Dic₄₇	1	4	yes	—
201	#385	Dic₄₇	1	4	yes	—
202	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
203	#336	—	47	4	no	—
204	#336	—	47	4	no	—
205	C₄₇	#20436	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×M₁₆	—	47	4	no	—
209	C₂×M₁₆	—	47	4	no	—
210	C₂×M₁₆	—	94	2	no	—
211	C₂×M₁₆	—	94	2	no	—
212	C₂×M₁₆	—	94	2	no	—
213	C₂×M₁₆	—	94	2	no	—
214	C₂²×C₈	—	47	4	no	—
215	C₂²×C₈	—	47	4	no	—
216	C₂²×C₈	—	47	4	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	—	47	1	no	—
219	C₂²×C₈	—	47	4	no	—
220	C₂²×C₈	—	47	2	no	—
221	C₂²×C₈	—	47	2	no	—
222	C₂²×C₈	—	47	2	no	—
223	C₂²×C₈	—	94	2	no	—
224	C₂²×C₈	—	94	2	no	—
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	C₄×D₄₇	1	4	yes	—
228	#117	C₄×D₄₇	1	4	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
232	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
233	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
237	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
238	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
239	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
240	C₄⋊C₈	—	47	16	no	—
241	#97	—	47	8	no	—
242	#97	—	94	4	no	—
243	#97	—	94	4	no	—
244	#97	—	94	4	no	—
245	C₄×C₈	—	47	8	no	—
246	C₄×C₈	—	94	4	no	—
247	#95	C₂×Dic₄₇	1	8	yes	—
248	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
252	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
253	C₂²×C₄	D₄×D₄₇	1	2	yes	—
254	C₂²×C₄	#10625352	1	2	yes	—
255	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
256	C₂²×C₄	#10625354	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
258	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
259	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
260	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	M₁₆	—	94	8	no	—
263	M₁₆	—	94	8	no	—
264	C₂×C₈	—	47	16	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	47	2	no	—
267	C₂×C₈	—	47	2	no	—
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	47	2	no	—
270	C₂×C₈	—	47	2	no	—
271	C₂×C₈	—	47	4	no	—
272	C₂×C₈	—	47	4	no	—
273	C₂×C₈	—	47	4	no	—
274	C₂×C₈	—	94	8	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	94	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	4	8	no	—
283	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	C₂×C₄×D₄₇	1	4	yes	—
292	C₄²	C₂×C₄×D₄₇	1	4	yes	—
293	C₂³	—	2	4	no	—
294	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
295	C₂³	D₄×Dic₄₇	1	2	yes	—
296	C₂³	Q₈×Dic₄₇	1	2	yes	—
297	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
306	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
307	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
308	C₂×C₄	#13093982	1	4	yes	—
309	C₂×C₄	#13093986	1	4	yes	—
310	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
311	C₂×C₄	#13093989	1	4	yes	—
312	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
313	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
318	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
319	C₈	—	47	4	no	—
320	C₈	—	47	4	no	—
321	C₈	—	94	4	no	—
322	C₈	—	94	4	no	—
323	C₈	—	94	4	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	4	no	—
327	C₂²	C₄⋊C₄×Dic₄₇	1	4	yes	—
328	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
329	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
330	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
331	C₂²	#15689890	1	4	yes	—
332	C₄	—	2	8	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	4	no	—
335	C₄	C₄⋊C₄×Dic₄₇	1	4	yes	—
336	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
337	C₄	C₂×#13093982	1	1	yes	characteristic
338	C₄	C₂×#13093986	1	1	yes	characteristic
339	C₄	C₂×#13093989	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	C₂×C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
342	C₂	#20251816	1	2	yes	—
343	C₂	#20252092	1	2	yes	—
344	C₂	#20252329	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2296	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#20436
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
5. #20243263	C₂	semidirect
6. #20243257	C₂	semidirect
14. #15689150	C₄	semidirect
17. #15689079	C₄	semidirect
50. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
54. #13093935	C₂×C₄	semidirect
56. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
110. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
205. C₄₇	#20436	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689150	2
C₂×#15689079	2
C₂²×#13093935	2
#20243263	2
#20243257	2
#20243085	2
#20243082	2
C₄₇×#2296	2
#20436	47