Subgroups — #26220914

Order 12032.

1114 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic
2	#117×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244370	C₂	1	4	yes	maximal, complemented
4	#20244366	C₂	1	4	yes	maximal, complemented
5	#20243097	C₂	1	2	yes	maximal
6	C₂×#15688882	C₂	1	2	yes	maximal
7	C₄₇×#3458	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689150	—	2	4	no	—
12	#15689147	C₂²	1	4	yes	—
13	C₄⋊C₄×Dic₄₇	—	2	2	no	—
14	C₂²⋊C₄×Dic₄₇	—	2	2	no	—
15	#15688969	C₂²	1	2	yes	—
16	C₄²×Dic₄₇	C₂²	1	2	yes	—
17	#15688882	C₂²	1	4	yes	—
18	#15688882	C₂²	1	8	yes	—
19	#15688882	C₂²	1	4	yes	—
20	C₄₇×#544	—	2	2	no	—
21	C₄₇×#544	C₄	1	8	yes	complemented
22	C₄₇×#543	—	2	2	no	—
23	C₄₇×#533	—	2	4	no	—
24	C₄₇×#533	C₂²	1	4	yes	—
25	C₂×#117×C₄₇	C₂²	1	2	yes	—
26	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
27	C₂³×Dic₄₇	—	2	1	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	#13093983	—	2	2	no	—
34	#13093983	Q₈	1	4	yes	complemented
35	C₂×C₄⋊Dic₄₇	—	2	1	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	—	2	1	no	—
38	C₂×C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	4	no	—
40	C₂×C₄×Dic₄₇	Q₈	1	4	yes	complemented
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₇	—	2	8	no	—
48	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
49	#125×C₄₇	—	2	8	no	—
50	#125×C₄₇	C₂×C₄	1	8	yes	complemented
51	C₂²⋊Q₈×C₄₇	—	2	8	no	—
52	C₂²⋊Q₈×C₄₇	—	2	16	no	—
53	C₄×Q₈×C₄₇	—	2	8	no	—
54	#117×C₄₇	—	2	2	no	—
55	#117×C₄₇	—	2	2	no	—
56	#117×C₄₇	C₂×C₄	1	4	yes	—
57	#117×C₄₇	C₂×C₄	1	4	yes	—
58	#117×C₄₇	D₄	1	4	yes	—
59	#117×C₄₇	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
64	C₂×C₄²×C₄₇	—	2	2	no	—
65	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
66	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	1	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	2	1	no	—
81	C₂²×Dic₄₇	—	2	2	no	—
82	C₂²×Dic₄₇	—	4	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	2	16	no	—
86	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
88	C₂²⋊C₄×C₄₇	—	2	4	no	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
91	C₄²×C₄₇	—	2	4	no	—
92	C₄²×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
94	C₄²×C₄₇	C₂×D₄	1	4	yes	—
95	C₂²⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄×Dic₄₇	—	2	4	no	—
98	C₄×Dic₄₇	—	2	1	no	—
99	C₄×Dic₄₇	—	2	1	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	4	8	no	—
102	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
103	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
104	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
105	#10625329	—	2	4	no	—
106	#10625329	—	2	4	no	—
107	#10625329	—	4	4	no	—
108	C₂×C₄₇⋊C₈	—	2	4	no	—
109	C₂×C₄₇⋊C₈	—	4	4	no	—
110	C₂³×C₄₇	—	4	2	no	—
111	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
113	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₄₇	—	8	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	8	4	no	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	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	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	#124	1	4	yes	—
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₄₇	—	2	4	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₄₇⋊C₈	—	4	4	no	—
141	#21330	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	4	2	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	#391	1	4	yes	—
147	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₄₇	—	2	8	no	—
151	C₄×C₄₇	—	4	8	no	—
152	C₄×C₄₇	#391	1	4	yes	—
153	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
156	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	Dic₄₇	—	2	4	no	—
158	Dic₄₇	—	4	4	no	—
159	Dic₄₇	—	4	2	no	—
160	#3458	D₄₇	1	1	yes	characteristic
161	#2905	—	47	1	no	—
162	C₄×#117	—	47	1	no	—
163	#1995	—	47	4	no	—
164	#1994	—	47	4	no	—
165	#1733	—	47	2	no	—
166	C₂×#339	—	47	2	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	4	no	—
170	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
171	C₂×C₄₇	#2901	1	2	yes	—
172	C₂²×M₁₆	—	47	2	no	—
173	#544	—	2	2	no	—
174	#544	Dic₄₇	1	8	yes	complemented
175	#543	—	2	2	no	—
176	#533	—	2	4	no	—
177	#533	C₂×D₄₇	1	4	yes	—
178	C₂×#117	—	47	1	no	—
179	C₂×#117	C₂×D₄₇	1	2	yes	—
180	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₄²	—	47	1	no	—
182	#424	—	94	4	no	—
183	#423	—	47	4	no	—
184	C₄×C₄⋊C₄	—	94	2	no	—
185	C₄×C₂²⋊C₄	—	94	2	no	—
186	#376	—	47	2	no	—
187	C₄³	—	47	2	no	—
188	#339	—	47	4	no	—
189	#339	—	47	4	no	—
190	#339	—	47	8	no	—
191	C₄₇	#21330	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	94	1	no	—
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂×M₁₆	—	47	4	no	—
195	C₂×M₁₆	—	94	2	no	—
196	C₂×M₁₆	—	94	4	no	—
197	C₂×M₁₆	—	94	4	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₄⋊Q₈	—	2	8	no	—
200	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	complemented
201	#125	—	2	8	no	—
202	#125	C₂×Dic₄₇	1	8	yes	complemented
203	C₂²⋊Q₈	—	2	8	no	—
204	C₂²⋊Q₈	—	2	16	no	—
205	C₄×Q₈	—	2	8	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	—	47	4	no	—
209	#117	—	94	2	no	—
210	#117	C₂×Dic₄₇	1	4	yes	—
211	#117	C₂×Dic₄₇	1	4	yes	—
212	#117	C₂²⋊D₄₇	1	4	yes	—
213	#117	C₂²×D₄₇	1	4	yes	—
214	C₂×C₄⋊C₄	—	94	1	no	—
215	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
216	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₂²⋊C₄	—	94	1	no	—
219	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	47	4	no	—
222	C₂×C₄²	—	47	2	no	—
223	C₂×C₄²	—	47	1	no	—
224	C₂×C₄²	—	47	1	no	—
225	C₂×C₄²	—	94	2	no	—
226	C₂×C₄²	—	94	1	no	—
227	C₂×C₄²	—	94	4	no	—
228	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
229	C₄⋊C₈	—	94	8	no	—
230	#95	—	94	4	no	—
231	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂×Q₈	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	94	1	no	—
235	C₂²×C₄	—	94	4	no	—
236	C₂²×C₄	—	94	1	no	—
237	C₂²×C₄	—	94	2	no	—
238	C₂²×C₄	—	188	4	no	—
239	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
240	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
241	C₂²×C₄	#10625354	1	2	yes	—
242	C₂²×C₄	#10625354	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	M₁₆	—	94	4	no	—
249	M₁₆	—	94	4	no	—
250	M₁₆	—	188	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	188	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	94	4	no	—
257	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
258	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	94	4	no	—
262	C₂²⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	47	4	no	—
268	C₄²	—	94	4	no	—
269	C₄²	—	94	1	no	—
270	C₄²	—	94	1	no	—
271	C₄²	—	94	2	no	—
272	C₄²	—	188	8	no	—
273	C₄²	C₂²×Dic₄₇	1	4	yes	—
274	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
275	C₂³	—	4	2	no	—
276	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
277	C₂³	#13094000	1	2	yes	—
278	C₂³	#13094000	1	2	yes	—
279	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
280	Q₈	—	8	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	8	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	188	8	no	—
289	C₂×C₄	—	188	4	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
292	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
293	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
294	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
295	C₂×C₄	#13094000	1	2	yes	—
296	C₂×C₄	#13094000	1	2	yes	—
297	C₂×C₄	#13094013	1	4	yes	—
298	C₂×C₄	#13094022	1	4	yes	—
299	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
304	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
305	C₈	—	188	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	#15689030	1	4	yes	—
311	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
312	C₂²	C₂×#13094000	1	1	yes	characteristic
313	C₂²	C₂×#13094000	1	1	yes	characteristic
314	C₄	—	2	8	no	—
315	C₄	—	4	8	no	—
316	C₄	—	94	4	no	—
317	C₄	—	188	2	no	—
318	C₄	—	188	4	no	—
319	C₄	#15689030	1	4	yes	—
320	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
321	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
322	C₄	C₂×#13094013	1	1	yes	characteristic
323	C₄	C₂×#13094022	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	4	no	—
327	C₂	C₂×#15689030	1	1	yes	characteristic
328	C₂	#20251935	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3458	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3458	D₄₇
hypocenter	C₄₇	#21330
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #3458

Complements

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

N	H ≅ G/N	type
3. #20244370	C₂	semidirect
4. #20244366	C₂	semidirect
21. C₄₇×#544	C₄	semidirect
34. #13093983	Q₈	semidirect
40. C₂×C₄×Dic₄₇	Q₈	semidirect
48. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
50. #125×C₄₇	C₂×C₄	semidirect
174. #544	Dic₄₇	semidirect
191. C₄₇	#21330	semidirect
200. C₄⋊Q₈	C₂×Dic₄₇	semidirect
202. #125	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
#117×Dic₄₇	2
#20244370	2
#20244366	2
#20243097	2
C₂×#15688882	2
C₄₇×#3458	2
#21330	47