Subgroups — #26107765

Order 12032.

1030 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	#20243368	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688869	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	#15688869	—	2	8	no	—
12	C₂³×Q₈×C₄₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
14	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
17	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₂²×Q₈×C₄₇	—	2	8	no	—
28	C₂²×Q₈×C₄₇	—	2	2	no	—
29	C₂²×Q₈×C₄₇	—	2	2	no	—
30	C₂²×Q₈×C₄₇	—	4	4	no	—
31	C₂²×Q₈×C₄₇	—	4	2	no	—
32	C₂²×Q₈×C₄₇	—	4	2	no	—
33	C₂³×C₄×C₄₇	—	2	2	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₄₇	—	2	8	no	—
38	C₄⋊Q₈×C₄₇	—	2	8	no	—
39	C₄⋊Q₈×C₄₇	—	2	8	no	—
40	C₄⋊Q₈×C₄₇	—	2	4	no	—
41	C₄⋊Q₈×C₄₇	D₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	—	2	4	no	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₄₇	—	2	8	no	—
55	C₂×Q₈×C₄₇	—	2	8	no	—
56	C₂×Q₈×C₄₇	—	4	4	no	—
57	C₂×Q₈×C₄₇	—	4	16	no	—
58	C₂×Q₈×C₄₇	—	4	16	no	—
59	C₂×Q₈×C₄₇	—	4	4	no	—
60	C₂×Q₈×C₄₇	—	4	2	no	—
61	C₂×Q₈×C₄₇	—	4	2	no	—
62	C₂×Q₈×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
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	8	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	4	4	no	—
73	C₂²×C₄×C₄₇	—	4	4	no	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	D₈	1	2	yes	—
78	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
79	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	8	no	—
88	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	C₂×D₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	#101	1	2	yes	—
104	C₂³×C₄₇	#102	1	2	yes	—
105	C₂³×C₄₇	#103	1	2	yes	—
106	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	Q₈×C₄₇	—	4	16	no	—
108	Q₈×C₄₇	—	4	8	no	—
109	Q₈×C₄₇	—	4	8	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	4	16	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	#101	1	2	yes	—
128	C₂×C₄×C₄₇	#102	1	2	yes	—
129	C₂×C₄×C₄₇	#103	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	#120	1	2	yes	—
132	C₂×C₄×C₄₇	#120	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₄₇	#137	1	2	yes	—
139	C₄₇⋊C₈	—	8	2	no	—
140	C₄₇⋊C₈	—	8	2	no	—
141	#8340	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	#332	1	4	yes	complemented
149	C₂²×C₄₇	#379	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
154	C₂²×C₄₇	#418	1	1	yes	characteristic
155	C₂²×C₄₇	#419	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	4	8	no	—
161	C₄×C₄₇	#447	1	2	yes	—
162	C₄×C₄₇	#448	1	2	yes	—
163	C₄×C₄₇	#450	1	2	yes	—
164	C₄×C₄₇	#451	1	2	yes	—
165	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
166	#1797	—	47	1	no	—
167	#1797	—	47	1	no	—
168	C₂×#332	—	47	4	no	—
169	C₂×C₄₇	—	2	4	no	—
170	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
171	C₂×C₄₇	#1546	1	1	yes	characteristic
172	C₂×C₄₇	#1549	1	1	yes	characteristic
173	C₂×C₄₇	#1556	1	1	yes	characteristic
174	C₂×C₄₇	#1784	1	1	yes	characteristic
175	C₂×C₄₇	#1786	1	1	yes	characteristic
176	C₂×C₄₇	#1787	1	1	yes	characteristic
177	C₂³×Q₈	—	2	1	no	—
178	C₂×C₄⋊Q₈	—	2	8	no	—
179	C₂×C₄⋊Q₈	—	2	1	no	—
180	C₂×C₄⋊Q₈	—	2	1	no	—
181	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
182	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
183	C₂²×C₄⋊C₄	—	2	1	no	—
184	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	47	2	no	—
187	C₂×C₄⋊C₈	—	47	2	no	—
188	C₂×C₂²⋊C₈	—	94	1	no	—
189	C₂×C₂²⋊C₈	—	94	1	no	—
190	#336	—	94	1	no	—
191	#336	—	94	1	no	—
192	#332	—	94	8	no	—
193	C₄₇	#8340	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	—	2	8	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×Q₈	—	4	4	no	—
198	C₂²×Q₈	—	4	2	no	—
199	C₂²×Q₈	—	4	2	no	—
200	C₂³×C₄	—	2	2	no	—
201	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂²×C₈	—	94	2	no	—
205	C₂²×C₈	—	94	2	no	—
206	C₄⋊Q₈	—	2	8	no	—
207	C₄⋊Q₈	—	2	8	no	—
208	C₄⋊Q₈	—	2	8	no	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	C₂²⋊D₄₇	1	8	yes	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₄⋊C₈	—	94	4	no	—
223	C₄⋊C₈	—	94	4	no	—
224	C₂²⋊C₈	—	188	2	no	—
225	C₂²⋊C₈	—	188	2	no	—
226	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	16	no	—
231	C₂×Q₈	—	4	16	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
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	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	#10625334	1	2	yes	—
248	C₂²×C₄	#10625335	1	2	yes	—
249	C₂²×C₄	#10625336	1	2	yes	—
250	C₂²×C₄	#10625337	1	2	yes	—
251	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₈	—	188	2	no	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	—	188	2	no	—
260	C₂×C₈	—	188	2	no	—
261	C₂×C₈	—	188	2	no	—
262	C₂×C₈	—	188	2	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	#13093943	1	2	yes	—
277	C₂³	#13093945	1	2	yes	—
278	C₂³	#13093948	1	2	yes	—
279	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
280	Q₈	—	4	16	no	—
281	Q₈	—	4	8	no	—
282	Q₈	—	4	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	16	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	#13093943	1	2	yes	—
301	C₂×C₄	#13093945	1	2	yes	—
302	C₂×C₄	#13093948	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
304	C₂×C₄	#13093993	1	2	yes	—
305	C₂×C₄	#13093993	1	2	yes	—
306	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
307	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
308	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
309	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
310	C₂×C₄	#13094050	1	2	yes	—
311	C₂×C₄	#13094055	1	2	yes	—
312	C₈	—	376	2	no	—
313	C₈	—	376	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	#15688869	1	4	yes	complemented
321	C₂²	#15688979	1	1	yes	characteristic
322	C₂²	C₂×#13093943	1	1	yes	characteristic
323	C₂²	C₂×#13093943	1	1	yes	characteristic
324	C₂²	C₂×#13093945	1	1	yes	characteristic
325	C₂²	C₂×#13093948	1	1	yes	characteristic
326	C₂²	#15689128	1	1	yes	characteristic
327	C₂²	#15689132	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	8	no	—
333	C₄	#15689257	1	2	yes	—
334	C₄	#15689262	1	2	yes	—
335	C₄	#15689274	1	2	yes	—
336	C₄	#15689279	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	C₂×#15688869	1	1	yes	characteristic
339	C₂	#20242060	1	1	yes	characteristic
340	C₂	#20242073	1	1	yes	characteristic
341	C₂	#20242105	1	1	yes	characteristic
342	C₂	#20243312	1	1	yes	characteristic
343	C₂	#20243319	1	1	yes	characteristic
344	C₂	#20243323	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#8340
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#15688869	C₂	semidirect
148. C₂²×C₄₇	#332	semidirect
193. C₄₇	#8340	semidirect
320. C₂²	#15688869	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₂×#15688869	2
C₂²×C₄⋊Q₈×C₄₇	2
#8340	47