Subgroups — #26216722

Order 12032.

1070 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689166	C₂	1	2	yes	maximal
2	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
4	#20246008	C₂	1	2	yes	maximal
5	#20243555	C₂	1	4	yes	maximal, complemented
6	#20243549	C₂	1	2	yes	maximal, complemented
7	#20243543	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	#15689166	—	2	8	no	—
13	C₂×#13093957	C₂²	1	4	yes	—
14	C₂×#13093956	—	2	4	no	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093955	C₂²	1	4	yes	—
18	#15689078	C₄	1	8	yes	complemented
19	C₂×#13093934	C₂²	1	2	yes	—
20	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
21	#15689021	—	2	4	no	—
22	#15688995	—	2	4	no	—
23	#15688992	C₂²	1	4	yes	—
24	#15688988	C₂²	1	4	yes	—
25	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#427	—	2	4	no	—
28	C₂×#107×C₄₇	C₂²	1	2	yes	—
29	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
31	C₂³×Dic₄₇	—	2	2	no	—
32	C₂×#10625329	C₂×C₄	1	4	yes	—
33	C₂×#10625329	D₄	1	4	yes	—
34	C₂×#10625329	Q₈	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	#13093983	C₂×C₄	1	4	yes	—
37	#13093983	Q₈	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	—	2	4	no	—
39	C₂×C₄⋊Dic₄₇	—	2	4	no	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
43	C₂×Dic₄₇⋊C₄	—	2	8	no	—
44	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
46	#13093957	—	2	8	no	—
47	#13093956	—	2	4	no	—
48	#13093956	—	2	4	no	—
49	#13093955	—	2	8	no	—
50	#13093934	C₂×C₄	1	8	yes	complemented
51	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
52	C₄₇⋊#95	—	2	8	no	—
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₄₇	D₄	1	4	yes	—
56	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	#117×C₄₇	—	2	4	no	—
60	#117×C₄₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
64	C₂×C₄²×C₄₇	—	2	2	no	—
65	#107×C₄₇	—	2	4	no	—
66	#106×C₄₇	—	2	2	no	—
67	#106×C₄₇	—	2	2	no	—
68	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	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	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	—	8	4	no	—
82	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₂²⋊C₄×C₄₇	—	4	4	no	—
95	C₄²×C₄₇	—	4	4	no	—
96	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	4	4	no	—
100	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
101	Dic₄₇⋊C₄	—	8	8	no	—
102	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
103	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
104	#10625329	C₄⋊C₄	1	16	yes	complemented
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
107	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₄₇	#127	1	2	yes	—
111	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₄₇	#127	1	2	yes	—
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	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×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	8	no	—
133	C₂×Dic₄₇	—	8	8	no	—
134	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
135	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
137	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
138	#20802	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	#384	1	4	yes	—
143	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₄₇	#577	1	4	yes	—
147	C₄×C₄₇	—	8	2	no	—
148	C₄×C₄₇	—	8	2	no	—
149	C₄×C₄₇	#384	1	4	yes	—
150	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	Dic₄₇	—	2	8	no	—
155	Dic₄₇	—	8	4	no	—
156	C₂×#428	—	47	2	no	—
157	C₂×#427	D₄₇	1	1	yes	characteristic
158	C₂²×#106	—	47	1	no	—
159	C₂×#405	—	47	1	no	—
160	#2317	—	47	2	no	—
161	#1847	—	47	4	no	—
162	#1846	—	47	2	no	—
163	#1844	—	47	2	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
166	C₂×C₄₇	#2912	1	2	yes	—
167	C₂×C₄₇	#3154	1	2	yes	—
168	C₂×C₄₇	#3163	1	2	yes	—
169	C₂²×M₁₆	—	47	1	no	—
170	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#117	—	47	1	no	—
172	C₂×#117	C₂×D₄₇	1	2	yes	—
173	C₂²×C₄⋊C₄	—	47	2	no	—
174	#428	—	94	8	no	—
175	#427	—	2	4	no	—
176	C₂×#107	—	47	4	no	—
177	C₂×#107	C₂×D₄₇	1	2	yes	—
178	C₂×#106	—	47	2	no	—
179	C₂×#106	—	47	2	no	—
180	C₂×#106	—	47	4	no	—
181	C₂×#106	—	94	4	no	—
182	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
184	#405	—	47	8	no	—
185	C₂×#97	—	47	2	no	—
186	C₂×C₄×C₈	—	47	2	no	—
187	#389	—	94	4	no	—
188	#385	—	94	4	no	—
189	#384	—	47	4	no	—
190	#382	—	47	4	no	—
191	C₄₇	#20802	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	94	2	no	—
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂×M₁₆	—	47	4	no	—
195	C₂×M₁₆	—	47	4	no	—
196	C₂×M₁₆	—	47	4	no	—
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	Dic₉₄	1	4	yes	—
199	C₂²×C₈	C₄×D₄₇	1	4	yes	—
200	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	#117	—	2	4	no	—
204	#117	—	2	4	no	—
205	#117	—	47	4	no	—
206	#117	—	47	4	no	—
207	C₂×C₄⋊C₄	—	47	2	no	—
208	C₂×C₄⋊C₄	—	47	4	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	—	94	4	no	—
211	C₂×C₄⋊C₄	—	94	8	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
214	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	47	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	47	2	no	—
219	#107	—	2	4	no	—
220	#107	—	94	8	no	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	—	94	8	no	—
224	#106	—	94	4	no	—
225	#106	—	94	4	no	—
226	#97	—	47	8	no	—
227	C₄×C₈	—	47	8	no	—
228	#95	—	94	8	no	—
229	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	47	4	no	—
233	C₂²×C₄	—	94	4	no	—
234	C₂²×C₄	—	376	4	no	—
235	C₂²×C₄	—	188	4	no	—
236	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	D₄×D₄₇	1	2	yes	—
240	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
241	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	M₁₆	—	47	16	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	4	no	—
248	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
249	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
250	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
252	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
253	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	47	8	no	—
260	C₄⋊C₄	—	94	4	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	376	8	no	—
263	C₄⋊C₄	—	188	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	47	8	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	47	4	no	—
268	C₄²	—	47	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
271	C₂³	#13094020	1	2	yes	—
272	C₂³	#13094023	1	2	yes	—
273	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	47	8	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	—	376	8	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	—	188	8	no	—
283	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
284	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
285	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
286	C₂×C₄	#13094020	1	2	yes	—
287	C₂×C₄	#13094022	1	4	yes	—
288	C₂×C₄	#13094023	1	2	yes	—
289	C₂×C₄	#13094023	1	4	yes	—
290	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
296	C₈	—	47	8	no	—
297	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
298	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#15688992	1	4	yes	—
303	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
304	C₂²	C₂×#13094020	1	1	yes	characteristic
305	C₂²	C₂×#13094023	1	1	yes	characteristic
306	C₂²	#15689930	1	4	yes	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	94	8	no	—
310	C₄	—	376	4	no	—
311	C₄	#15688992	1	4	yes	—
312	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
313	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
314	C₄	C₂×#13094022	1	1	yes	characteristic
315	C₄	C₂×#13094023	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×#15688992	1	1	yes	characteristic
318	C₂	#20251980	1	2	yes	—
319	C₂	#20253865	1	2	yes	—
320	C₂	#20253920	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#20802
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
5. #20243555	C₂	semidirect
6. #20243549	C₂	semidirect
7. #20243543	C₂	semidirect
18. #15689078	C₄	semidirect
50. #13093934	C₂×C₄	semidirect
51. C₈×Dic₄₇	C₂×C₄	semidirect
83. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
104. #10625329	C₄⋊C₄	semidirect
191. C₄₇	#20802	semidirect
248. C₂×C₈	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689166	2
C₂²×#13093956	2
C₂×#15689078	2
#20246008	2
#20243555	2
#20243549	2
#20243543	2
C₂×C₄₇×#427	2
#20802	47