Subgroups — #26234415

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	2	yes	maximal, complemented
2	#20246337	C₂	1	2	yes	maximal
3	#20244147	C₂	1	2	yes	maximal, complemented
4	#20243589	C₂	1	4	yes	maximal, complemented
5	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243413	C₂	1	2	yes	maximal, complemented
7	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	4	yes	—
10	#15689161	—	2	8	no	—
11	C₂×#13093958	C₂²	1	4	yes	—
12	C₂×#13093956	C₂²	1	2	yes	—
13	C₂×#13093956	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	1	yes	characteristic
16	#15689026	—	2	2	no	—
17	#15689016	—	2	4	no	—
18	#15688993	C₂²	1	2	yes	—
19	#15688983	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₄₇	—	2	2	no	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688886	C₂²	1	2	yes	—
23	#15688886	C₂²	1	4	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₄₇	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	#13093983	—	2	8	no	—
31	#13093983	—	4	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	4	no	—
41	#13093958	—	2	8	no	—
42	#13093956	—	2	4	no	—
43	#13093956	—	2	2	no	—
44	#13093956	—	4	2	no	—
45	#13093956	D₄	1	4	yes	complemented
46	C₄₇⋊#95	—	2	2	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₇	—	2	4	no	—
51	C₂²×C₈×C₄₇	—	2	4	no	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₄₇	Q₈	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
62	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	1	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	16	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	#41	1	2	yes	—
90	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
91	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂²⋊Dic₄₇	—	4	8	no	—
95	C₂²⋊Dic₄₇	—	8	4	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	4	4	no	—
102	C₄⋊Dic₄₇	—	8	2	no	—
103	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
104	C₄×Dic₄₇	—	4	8	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	—	2	4	no	—
110	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
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	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	#135	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₈×C₄₇	—	2	2	no	—
141	C₈×C₄₇	C₄×D₄	1	4	yes	—
142	#22657	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	—	2	4	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₇	D₄²	1	1	yes	characteristic
150	C₂²×C₄₇	#547	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#135	1	2	yes	—
152	C₂²×C₄₇	#577	1	2	yes	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	#427	1	4	yes	complemented
156	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₄₇	#520	1	1	yes	characteristic
158	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
159	C₄×C₄₇	#554	1	1	yes	characteristic
160	Dic₄₇	—	4	4	no	—
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	—	8	4	no	—
163	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
164	C₂×#427	—	47	2	no	—
165	#2354	—	47	2	no	—
166	#1947	—	47	2	no	—
167	#1854	—	47	4	no	—
168	#1844	—	47	1	no	—
169	#1807	—	47	2	no	—
170	C₄×#106	—	47	1	no	—
171	C₂×C₄₇	—	4	4	no	—
172	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
174	C₂×C₄₇	#2912	1	1	yes	characteristic
175	C₂×C₄₇	#3284	1	2	yes	—
176	C₂×C₄₇	#3340	1	2	yes	—
177	C₂³×C₈	C₂×D₄₇	1	2	yes	—
178	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
179	C₂×#117	—	47	4	no	—
180	C₈×D₄	Dic₄₇	1	8	yes	complemented
181	#427	—	94	8	no	—
182	C₂×#107	—	47	4	no	—
183	C₂×#106	—	47	1	no	—
184	C₂×#106	—	47	1	no	—
185	C₂×#106	—	47	2	no	—
186	C₂×#106	—	47	2	no	—
187	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
188	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
189	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
190	#390	—	94	2	no	—
191	#388	—	94	4	no	—
192	#384	—	47	2	no	—
193	#380	—	47	4	no	—
194	C₄×C₄⋊C₄	—	47	2	no	—
195	C₄×C₄⋊C₄	—	94	2	no	—
196	#340	—	47	4	no	—
197	#340	—	47	2	no	—
198	C₄₇	#22657	1	1	yes	characteristic, complemented, Sylow
199	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
201	C₂²×C₈	—	2	4	no	—
202	C₂²×C₈	—	2	4	no	—
203	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
204	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
205	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₄×D₄	Dic₉₄	1	4	yes	—
208	C₄×D₄	C₂×Dic₄₇	1	4	yes	—
209	#117	—	94	8	no	—
210	#117	—	188	4	no	—
211	C₂×C₄⋊C₄	—	47	2	no	—
212	C₂×C₄⋊C₄	—	47	4	no	—
213	C₂×C₄⋊C₄	—	47	2	no	—
214	C₂×C₄⋊C₄	—	94	2	no	—
215	C₂×C₄⋊C₄	—	94	2	no	—
216	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	—	94	4	no	—
218	C₂×C₂²⋊C₄	—	94	4	no	—
219	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
220	C₂×C₄²	—	94	4	no	—
221	C₂×C₄²	—	94	2	no	—
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	#107	—	94	8	no	—
224	#106	—	47	4	no	—
225	#106	—	94	4	no	—
226	#106	—	94	2	no	—
227	#106	—	188	2	no	—
228	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
229	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
230	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
231	#95	—	94	2	no	—
232	#95	—	94	4	no	—
233	C₂⁴	C₂×D₁₈₈	1	2	yes	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	C₄⋊Dic₄₇	1	4	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	94	2	no	—
241	C₂²×C₄	—	94	2	no	—
242	C₂²×C₄	—	94	4	no	—
243	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
244	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
245	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
248	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	16	no	—
254	C₂×C₈	D₄×D₄₇	1	2	yes	—
255	C₂×C₈	#10625352	1	2	yes	—
256	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
257	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
258	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
259	C₄⋊C₄	—	47	4	no	—
260	C₄⋊C₄	—	94	2	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	94	2	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	—	94	2	no	—
265	C₄⋊C₄	—	376	2	no	—
266	C₄⋊C₄	—	188	4	no	—
267	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
268	C₂²⋊C₄	—	376	4	no	—
269	C₂²⋊C₄	—	188	8	no	—
270	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
271	C₄²	—	188	8	no	—
272	C₄²	—	188	4	no	—
273	C₄²	C₂×Dic₉₄	1	2	yes	—
274	C₄²	C₂²×Dic₄₇	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	4	no	—
279	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
280	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
281	D₄	—	2	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	188	4	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	4	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	—	188	2	no	—
297	C₂×C₄	—	188	2	no	—
298	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
300	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
301	C₂×C₄	#13094046	1	4	yes	—
302	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
305	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
306	C₂×C₄	#13094070	1	1	yes	characteristic
307	C₂×C₄	#13094071	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	2	2	no	—
310	C₈	D₄×Dic₄₇	1	4	yes	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
317	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
318	C₂²	#15689777	1	1	yes	characteristic
319	C₂²	C₂×#13094046	1	2	yes	—
320	C₂²	#15689935	1	2	yes	—
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	376	2	no	—
324	C₄	—	376	4	no	—
325	C₄	—	188	4	no	—
326	C₄	#15689161	1	4	yes	complemented
327	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
328	C₄	#15689606	1	1	yes	characteristic
329	C₄	D₄×Dic₉₄	1	1	yes	characteristic
330	C₄	#15689830	1	1	yes	characteristic
331	C₂	—	4	4	no	—
332	C₂	D₄×C₄⋊Dic₄₇	1	1	yes	characteristic
333	C₂	C₂×#15689161	1	1	yes	characteristic
334	C₂	#20251983	1	1	yes	characteristic
335	C₂	#20255120	1	2	yes	—
336	C₂	#20255802	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₄×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#22657
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15689161	C₂	semidirect
3. #20244147	C₂	semidirect
4. #20243589	C₂	semidirect
5. #20243545	C₂	semidirect
6. #20243413	C₂	semidirect
7. C₄×#13093956	C₂	semidirect
26. C₈×D₄×C₄₇	C₄	semidirect
45. #13093956	D₄	semidirect
155. C₄×C₄₇	#427	semidirect
180. C₈×D₄	Dic₄₇	semidirect
198. C₄₇	#22657	semidirect
326. C₄	#15689161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20246337	2
#20244147	2
#20243589	2
#20243545	2
#20243413	2
C₄×#13093956	2
C₂×C₈×D₄×C₄₇	2
#22657	47