Subgroups — #26374492

Order 12032.

3630 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093983	C₂	1	2	yes	maximal
2	#20245745	C₂	1	4	yes	maximal, complemented
3	#20245744	C₂	1	8	yes	maximal, complemented
4	#20245741	C₂	1	4	yes	maximal
5	#117×Dic₄₇	C₂	1	8	yes	maximal, complemented
6	C₄×#13093983	C₂	1	4	yes	maximal, complemented
7	C₂²×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₄	1	32	yes	complemented
9	C₂×#13093983	C₂²	1	4	yes	—
10	C₂×#13093983	C₂²	1	4	yes	—
11	C₂×#13093983	C₂²	1	4	yes	—
12	C₂×#13093983	C₂²	1	4	yes	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
14	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	32	yes	complemented
17	C₄×C₄⋊Dic₄₇	C₂²	1	16	yes	complemented
18	C₂²⋊C₄×Dic₄₇	C₂²	1	32	yes	—
19	C₄×C₂²⋊Dic₄₇	C₂²	1	16	yes	—
20	#15688969	C₂²	1	8	yes	complemented
21	C₄²×Dic₄₇	C₂²	1	8	yes	complemented
22	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₄	1	16	yes	complemented
24	C₂×#117×C₄₇	C₂²	1	4	yes	—
25	C₂×#117×C₄₇	C₂²	1	4	yes	—
26	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
29	C₂³×Dic₄₇	C₂³	1	8	yes	—
30	#13093983	C₂×C₄	1	32	yes	complemented
31	#13093983	C₂×C₄	1	32	yes	complemented
32	#13093983	C₂×C₄	1	32	yes	complemented
33	#13093983	C₂×C₄	1	32	yes	complemented
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	32	yes	complemented
35	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
37	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	32	yes	—
38	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
39	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	32	yes	complemented
41	C₂×C₄×Dic₄₇	C₂³	1	32	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
44	C₄₇⋊#95	C₂³	1	32	yes	—
45	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
47	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
53	#117×C₄₇	C₂×C₄	1	32	yes	complemented
54	#117×C₄₇	C₂×C₄	1	32	yes	complemented
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	complemented
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
57	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
58	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	complemented
60	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
62	C₂⁴×C₄₇	—	4	4	no	—
63	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
66	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	—	4	8	no	—
68	C₂²×C₄×C₄₇	C₄²	1	64	yes	complemented
69	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	4	16	no	—
85	C₂²×Dic₄₇	C₂²×C₄	1	32	yes	—
86	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
88	C₂²⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
91	C₂²⋊Dic₄₇	C₂²×C₄	1	32	yes	—
92	C₂²⋊Dic₄₇	C₂²×C₄	1	32	yes	—
93	C₄⋊Dic₄₇	C₂²×C₄	1	32	yes	—
94	C₄⋊Dic₄₇	C₂²×C₄	1	32	yes	—
95	C₄×Dic₄₇	—	4	32	no	—
96	C₄×Dic₄₇	C₂²×C₄	1	32	yes	—
97	C₄×Dic₄₇	C₂²×C₄	1	32	yes	—
98	C₂³×C₄₇	—	2	8	no	—
99	C₂³×C₄₇	—	2	4	no	—
100	C₂³×C₄₇	—	4	16	no	—
101	C₂³×C₄₇	—	4	8	no	—
102	C₂³×C₄₇	C₂×C₄²	1	16	yes	—
103	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
104	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
105	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
106	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
107	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
108	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	16	no	—
110	C₂×C₄×C₄₇	—	4	16	no	—
111	C₂×C₄×C₄₇	C₂×C₄²	1	32	yes	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	32	yes	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	16	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	16	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
123	C₂×Dic₄₇	—	2	32	no	—
124	C₂×Dic₄₇	—	4	32	no	—
125	C₂×Dic₄₇	C₂³×C₄	1	32	yes	—
126	#33996	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	8	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	8	no	—
130	C₂²×C₄₇	—	2	8	no	—
131	C₂²×C₄₇	—	2	8	no	—
132	C₂²×C₄₇	—	4	16	no	—
133	C₂²×C₄₇	C₂²×C₄²	1	4	yes	—
134	C₂²×C₄₇	C₂²×C₄²	1	8	yes	—
135	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
136	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
137	C₂²×C₄₇	#585	1	4	yes	—
138	C₄×C₄₇	—	2	16	no	—
139	C₄×C₄₇	C₂²×C₄²	1	4	yes	—
140	C₄×C₄₇	C₂²×C₄²	1	8	yes	—
141	C₄×C₄₇	C₂²×C₄²	1	4	yes	—
142	Dic₄₇	—	2	32	no	—
143	C₂²×#117	—	47	2	no	—
144	C₂²×#117	D₄₇	1	1	yes	characteristic
145	#2274	—	47	8	no	—
146	#2274	—	47	4	no	—
147	#2273	—	47	4	no	—
148	C₄×#117	—	47	4	no	—
149	C₄×#117	—	47	8	no	—
150	C₂×C₄₇	—	2	8	no	—
151	C₂×C₄₇	—	2	4	no	—
152	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
153	C₂×C₄₇	#3428	1	4	yes	—
154	C₂×C₄₇	#3428	1	2	yes	—
155	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#117	—	47	32	no	—
157	C₂×#117	—	47	4	no	—
158	C₂×#117	—	47	4	no	—
159	C₂×#117	—	47	4	no	—
160	C₂×#117	—	47	4	no	—
161	C₂×#117	Dic₄₇	1	16	yes	complemented
162	C₂×#117	C₂×D₄₇	1	4	yes	—
163	C₂×#117	C₂×D₄₇	1	4	yes	—
164	C₂²×C₄⋊C₄	—	47	4	no	—
165	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
166	C₂²×C₂²⋊C₄	—	47	4	no	—
167	C₂²×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
168	C₂²×C₄²	—	47	4	no	—
169	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
170	C₄×C₄⋊C₄	—	47	32	no	—
171	C₄×C₄⋊C₄	—	47	16	no	—
172	C₄×C₂²⋊C₄	—	47	16	no	—
173	C₄×C₂²⋊C₄	—	47	32	no	—
174	#376	—	47	8	no	—
175	C₄³	—	47	8	no	—
176	C₄₇	#33996	1	1	yes	characteristic, complemented, Sylow
177	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
178	C₂³×C₄	—	47	8	no	—
179	C₂³×C₄	C₄×D₄₇	1	16	yes	—
180	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
181	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
186	#117	—	47	32	no	—
187	#117	—	47	32	no	—
188	#117	—	47	32	no	—
189	#117	—	47	32	no	—
190	#117	C₂×Dic₄₇	1	32	yes	complemented
191	#117	C₂×Dic₄₇	1	32	yes	complemented
192	C₂×C₄⋊C₄	—	47	32	no	—
193	C₂×C₄⋊C₄	—	47	8	no	—
194	C₂×C₄⋊C₄	—	47	8	no	—
195	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	complemented
196	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
197	C₂×C₂²⋊C₄	—	47	32	no	—
198	C₂×C₂²⋊C₄	—	47	8	no	—
199	C₂×C₂²⋊C₄	—	47	8	no	—
200	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	16	yes	—
201	C₂×C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
202	C₂×C₄²	—	47	32	no	—
203	C₂×C₄²	—	47	8	no	—
204	C₂×C₄²	—	47	8	no	—
205	C₂×C₄²	—	47	32	no	—
206	C₂×C₄²	C₂×Dic₄₇	1	16	yes	complemented
207	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
209	#95	—	47	32	no	—
210	C₂⁴	—	4	4	no	—
211	C₂⁴	C₂×C₄×D₄₇	1	8	yes	—
212	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
213	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
214	C₂⁴	C₂³×D₄₇	1	2	yes	—
215	C₂²×C₄	—	4	8	no	—
216	C₂²×C₄	—	47	16	no	—
217	C₂²×C₄	—	47	32	no	—
218	C₂²×C₄	—	188	16	no	—
219	C₂²×C₄	C₄×Dic₄₇	1	64	yes	complemented
220	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
221	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
222	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
225	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
228	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
229	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
231	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
232	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	C₄⋊C₄	—	47	32	no	—
236	C₄⋊C₄	—	47	32	no	—
237	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
238	C₂²⋊C₄	—	47	32	no	—
239	C₂²⋊C₄	—	47	32	no	—
240	C₂²⋊C₄	C₂²×Dic₄₇	1	32	yes	—
241	C₄²	—	47	32	no	—
242	C₄²	—	47	32	no	—
243	C₄²	—	188	32	no	—
244	C₄²	C₂²×Dic₄₇	1	16	yes	—
245	C₄²	C₂²×Dic₄₇	1	16	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	8	no	—
248	C₂³	—	4	16	no	—
249	C₂³	—	4	8	no	—
250	C₂³	C₂×C₄×Dic₄₇	1	16	yes	—
251	C₂³	C₂²×C₄×D₄₇	1	4	yes	—
252	C₂³	C₂²×C₄×D₄₇	1	4	yes	—
253	C₂³	C₂²×C₄×D₄₇	1	4	yes	—
254	C₂³	C₂³×Dic₄₇	1	4	yes	—
255	C₂³	C₂³×Dic₄₇	1	2	yes	—
256	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	47	32	no	—
260	C₂×C₄	—	94	32	no	—
261	C₂×C₄	—	188	32	no	—
262	C₂×C₄	C₂×C₄×Dic₄₇	1	32	yes	—
263	C₂×C₄	C₂×C₄×Dic₄₇	1	32	yes	—
264	C₂×C₄	C₂×C₄×Dic₄₇	1	16	yes	—
265	C₂×C₄	C₂×C₄×Dic₄₇	1	16	yes	—
266	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
267	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
268	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
269	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
270	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
271	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
272	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
273	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	4	16	no	—
280	C₂²	C₂²×C₄×Dic₄₇	1	4	yes	—
281	C₂²	C₂²×C₄×Dic₄₇	1	8	yes	—
282	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
283	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
284	C₂²	#15689959	1	4	yes	—
285	C₄	—	2	16	no	—
286	C₄	—	94	32	no	—
287	C₄	C₂²×C₄×Dic₄₇	1	8	yes	—
288	C₄	C₂²×C₄×Dic₄₇	1	4	yes	—
289	C₄	C₂²×C₄×Dic₄₇	1	4	yes	—
290	C₂	—	2	8	no	—
291	C₂	—	2	4	no	—
292	C₂	C₂³×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂	#20256589	1	4	yes	—
294	C₂	#20256591	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×#117×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂²×#117	D₄₇
hypocenter	C₄₇	#33996
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. #20245745	C₂	semidirect
3. #20245744	C₂	semidirect
5. #117×Dic₄₇	C₂	semidirect
6. C₄×#13093983	C₂	semidirect
8. C₂×#13093983	C₄	semidirect
16. C₄⋊C₄×Dic₄₇	C₂²	semidirect
17. C₄×C₄⋊Dic₄₇	C₂²	semidirect
20. #15688969	C₂²	semidirect
21. C₄²×Dic₄₇	C₂²	semidirect
23. C₂×#117×C₄₇	C₄	semidirect
30. #13093983	C₂×C₄	semidirect
31. #13093983	C₂×C₄	semidirect
32. #13093983	C₂×C₄	semidirect
33. #13093983	C₂×C₄	semidirect
34. C₂×C₄⋊Dic₄₇	C₂×C₄	semidirect
40. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
53. #117×C₄₇	C₂×C₄	semidirect
54. #117×C₄₇	C₂×C₄	semidirect
55. C₂×C₄⋊C₄×C₄₇	C₂×C₄	semidirect
59. C₂×C₄²×C₄₇	C₂×C₄	semidirect
68. C₂²×C₄×C₄₇	C₄²	semidirect
161. C₂×#117	Dic₄₇	semidirect
176. C₄₇	#33996	semidirect
190. #117	C₂×Dic₄₇	semidirect
191. #117	C₂×Dic₄₇	semidirect
195. C₂×C₄⋊C₄	C₂×Dic₄₇	semidirect
206. C₂×C₄²	C₂×Dic₄₇	semidirect
219. 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
C₂²×#13093983	2
#20245745	2
#20245744	2
#20245741	2
#117×Dic₄₇	2
C₄×#13093983	2
C₂²×#117×C₄₇	2
#33996	47