Subgroups — #26213939

Order 12032.

1606 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689097	C₂	1	2	yes	maximal
2	#20245745	C₂	1	2	yes	maximal, complemented
3	C₄×#13093939	C₂	1	8	yes	maximal, complemented
4	#20243121	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#520	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₂²	1	4	yes	—
7	#15689097	C₄	1	16	yes	complemented
8	C₂×#13093939	C₂²	1	8	yes	—
9	C₄⋊C₄×Dic₄₇	—	2	8	no	—
10	C₄×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
11	#15688890	C₂²	1	8	yes	—
12	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#520	—	2	8	no	—
14	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	complemented
15	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	complemented
16	C₂×C₄×D₄×C₄₇	C₂²	1	4	yes	—
17	C₂×#117×C₄₇	C₄	1	2	yes	complemented
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	complemented
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
21	#13093983	—	2	8	no	—
22	#13093983	C₂×C₄	1	16	yes	complemented
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
25	C₂×C₄×Dic₄₇	—	2	4	no	—
26	C₂×C₄×Dic₄₇	—	2	8	no	—
27	#13093939	C₂×C₄	1	32	yes	complemented
28	C₄₇⋊#95	—	2	8	no	—
29	C₂×#41×C₄₇	—	2	4	no	—
30	C₂×#41×C₄₇	—	2	4	no	—
31	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
32	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₄×D₄×C₄₇	—	2	16	no	—
41	C₄×D₄×C₄₇	—	2	8	no	—
42	#117×C₄₇	—	2	4	no	—
43	#117×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
57	#41×C₄₇	—	2	16	no	—
58	#41×C₄₇	—	2	16	no	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	C₄²	1	4	yes	complemented
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	2	2	no	—
66	C₂×D₄×C₄₇	C₄²	1	4	yes	complemented
67	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	—	8	2	no	—
72	C₂²×C₄×C₄₇	—	8	4	no	—
73	C₂²×C₄×C₄₇	C₄²	1	8	yes	complemented
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	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	4	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₇	#41	1	4	yes	—
84	C₂²×Dic₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	—	2	8	no	—
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
89	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
90	C₂²⋊C₄×C₄₇	—	2	4	no	—
91	C₂²⋊C₄×C₄₇	—	2	8	no	—
92	C₄²×C₄₇	—	2	8	no	—
93	C₄²×C₄₇	—	2	4	no	—
94	C₂²⋊Dic₄₇	—	2	8	no	—
95	C₂²⋊Dic₄₇	C₂²×C₄	1	16	yes	—
96	C₄⋊Dic₄₇	—	2	16	no	—
97	C₄×Dic₄₇	—	2	16	no	—
98	C₄×Dic₄₇	—	8	8	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	—	4	1	no	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	—	4	2	no	—
105	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
106	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
107	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
109	C₂³×C₄₇	C₂×#41	1	2	yes	—
110	Q₈×C₄₇	—	2	8	no	—
111	D₄×C₄₇	—	2	16	no	—
112	D₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	16	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	4	8	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	8	8	no	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	#117	1	2	yes	—
130	C₂×C₄×C₄₇	#117	1	2	yes	—
131	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
133	C₂×Dic₄₇	—	2	16	no	—
134	C₂×Dic₄₇	—	4	8	no	—
135	C₂×Dic₄₇	—	4	8	no	—
136	#20393	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	—	4	2	no	—
143	C₂²×C₄₇	—	4	2	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	—	4	4	no	—
146	C₂²×C₄₇	—	4	2	no	—
147	C₂²×C₄₇	—	4	2	no	—
148	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×C₄×D₄	1	4	yes	—
152	C₄×C₄₇	—	2	8	no	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	—	4	2	no	—
156	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
157	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
158	Dic₄₇	—	4	16	no	—
159	C₂×#520	D₄₇	1	1	yes	characteristic
160	C₂×#410	—	47	2	no	—
161	#2274	—	47	2	no	—
162	C₄×#99	—	47	8	no	—
163	#1740	—	47	2	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	4	4	no	—
168	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄₇	#2883	1	1	yes	characteristic
170	C₂×C₄₇	#2884	1	1	yes	characteristic
171	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
172	#520	—	2	8	no	—
173	C₂×C₄×Q₈	Dic₄₇	1	2	yes	complemented
174	C₂×C₄×D₄	Dic₄₇	1	2	yes	complemented
175	C₂×C₄×D₄	C₂×D₄₇	1	4	yes	—
176	C₂×#117	—	47	4	no	—
177	C₂×#117	Dic₄₇	1	2	yes	complemented
178	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	complemented
180	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
181	#410	—	47	16	no	—
182	C₂×#99	—	47	8	no	—
183	C₄×C₄⋊C₄	—	94	8	no	—
184	C₄×C₂²⋊C₄	—	47	8	no	—
185	#342	—	47	8	no	—
186	C₄₇	#20393	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	2	4	no	—
188	C₂×#41	—	2	4	no	—
189	C₂×#41	C₄×D₄₇	1	8	yes	—
190	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
191	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
193	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₄×Q₈	—	2	8	no	—
198	C₄×D₄	—	2	16	no	—
199	C₄×D₄	—	2	8	no	—
200	#117	—	2	4	no	—
201	#117	—	2	8	no	—
202	#117	—	47	16	no	—
203	#117	—	94	8	no	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	94	4	no	—
206	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	8	yes	—
212	C₂×C₂²⋊C₄	—	47	8	no	—
213	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
214	C₂×C₂²⋊C₄	C₂²×D₄₇	1	2	yes	—
215	C₂×C₄²	—	94	4	no	—
216	C₂×C₄²	—	94	8	no	—
217	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
218	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
219	#99	—	47	32	no	—
220	#95	—	94	8	no	—
221	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
222	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
223	#41	—	2	16	no	—
224	#41	—	2	16	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₄×Dic₄₇	1	4	yes	complemented
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₄×Dic₄₇	1	4	yes	complemented
233	C₂×D₄	C₂×C₄×D₄₇	1	8	yes	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	8	2	no	—
238	C₂²×C₄	—	8	4	no	—
239	C₂²×C₄	—	94	8	no	—
240	C₂²×C₄	C₄×Dic₄₇	1	8	yes	complemented
241	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
242	C₂²×C₄	#10625350	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
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	8	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	94	16	no	—
255	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
256	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₂²⋊C₄	—	47	16	no	—
260	C₂²⋊C₄	—	94	8	no	—
261	C₄²	—	2	8	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	94	16	no	—
264	C₄²	—	376	8	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
272	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
273	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
274	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
275	C₂³	C₂×#10625350	1	2	yes	—
276	Q₈	—	2	8	no	—
277	D₄	—	2	16	no	—
278	D₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	16	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	8	no	—
289	C₂×C₄	—	94	16	no	—
290	C₂×C₄	—	188	8	no	—
291	C₂×C₄	—	188	8	no	—
292	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
293	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
294	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
296	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
298	C₂×C₄	#13093983	1	2	yes	—
299	C₂×C₄	#13093983	1	2	yes	—
300	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
301	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
314	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
315	C₂²	C₂×#13093983	1	1	yes	characteristic
316	C₂²	C₂×C₄×C₂²⋊D₄₇	1	4	yes	—
317	C₄	—	2	8	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	188	16	no	—
322	C₄	C₄×C₂²⋊Dic₄₇	1	4	yes	—
323	C₄	C₄×C₂²⋊Dic₄₇	1	4	yes	—
324	C₂	—	2	4	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	4	4	no	—
328	C₂	C₂×C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
329	C₂	#20251849	1	1	yes	characteristic
330	C₂	#20251852	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₂×C₄₇×#520	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#520	D₄₇
hypocenter	C₄₇	#20393
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ C₂×#520

Complements

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

N	H ≅ G/N	type
2. #20245745	C₂	semidirect
3. C₄×#13093939	C₂	semidirect
4. #20243121	C₂	semidirect
7. #15689097	C₄	semidirect
14. C₂×C₄×Q₈×C₄₇	C₄	semidirect
15. C₂×C₄×D₄×C₄₇	C₄	semidirect
17. C₂×#117×C₄₇	C₄	semidirect
19. C₂²×C₄⋊C₄×C₄₇	C₄	semidirect
22. #13093983	C₂×C₄	semidirect
27. #13093939	C₂×C₄	semidirect
61. C₂×Q₈×C₄₇	C₄²	semidirect
66. C₂×D₄×C₄₇	C₄²	semidirect
73. C₂²×C₄×C₄₇	C₄²	semidirect
173. C₂×C₄×Q₈	Dic₄₇	semidirect
174. C₂×C₄×D₄	Dic₄₇	semidirect
177. C₂×#117	Dic₄₇	semidirect
179. C₂²×C₄⋊C₄	Dic₄₇	semidirect
186. C₄₇	#20393	semidirect
227. C₂×Q₈	C₄×Dic₄₇	semidirect
232. C₂×D₄	C₄×Dic₄₇	semidirect
240. C₂²×C₄	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689097	2
#20245745	2
C₄×#13093939	2
#20243121	2
C₂×C₄₇×#520	2
#20393	47