Subgroups — #26183502

Order 12032.

1030 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252477	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245753	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241913	C₂	1	3	yes	maximal, complemented
4	#20241896	C₂	1	3	yes	maximal, complemented
5	#20241883	C₂	1	3	yes	maximal, complemented
6	#20241874	C₂	1	3	yes	maximal, complemented
7	C₄₇×#3522	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₄₇	—	2	1	no	—
9	#15689254	—	2	3	no	—
10	Q₈×C₄₇⋊C₈	—	2	1	no	—
11	#15689180	C₂²	1	3	yes	—
12	#15689176	C₂²	1	3	yes	—
13	C₄⋊C₄×Dic₄₇	—	2	3	no	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	3	yes	—
15	#15688969	C₂²	1	3	yes	—
16	#15688860	C₂²	1	1	yes	characteristic
17	#15688860	C₂²	1	3	yes	—
18	#15688857	C₂²	1	3	yes	—
19	#15688857	C₂²	1	3	yes	—
20	#15688857	C₂²	1	6	yes	—
21	C₄₇×#559	C₄	1	3	yes	complemented
22	Q₈²×C₄₇	C₄	1	1	yes	characteristic, complemented
23	C₄₇×#554	C₄	1	3	yes	complemented
24	C₄₇×#550	C₄	1	1	yes	characteristic, complemented
25	C₄₇×#548	—	2	1	no	—
26	C₄₇×#547	—	2	3	no	—
27	C₄₇×#542	—	2	3	no	—
28	C₄₇×#532	C₂²	1	3	yes	—
29	C₄₇×#529	C₂²	1	3	yes	—
30	C₄₇×#520	—	2	1	no	—
31	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	Q₈×Dic₄₇	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	—	2	3	no	—
34	C₂×C₄×Dic₄₇	—	2	3	no	—
35	C₂×C₄×Dic₄₇	—	2	3	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	3	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	3	no	—
38	C₂²⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
39	C₂²⋊(C₄₇⋊C₈)	C₂³	1	3	yes	—
40	#13093935	—	2	3	no	—
41	C₄×C₄₇⋊C₈	—	2	3	no	—
42	C₄₇⋊#95	—	2	3	no	—
43	C₄₇⋊#95	C₂³	1	1	yes	characteristic
44	C₄₇⋊#95	C₂³	1	3	yes	—
45	C₂×#41×C₄₇	—	2	1	no	—
46	C₂×#41×C₄₇	—	2	3	no	—
47	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₇	—	2	3	no	—
49	C₄⋊Q₈×C₄₇	C₂×C₄	1	3	yes	—
50	#127×C₄₇	C₂×C₄	1	3	yes	—
51	#126×C₄₇	—	2	3	no	—
52	#125×C₄₇	C₂×C₄	1	3	yes	—
53	#124×C₄₇	—	2	3	no	—
54	#124×C₄₇	—	2	3	no	—
55	#124×C₄₇	C₂×C₄	1	3	yes	—
56	#123×C₄₇	—	2	3	no	—
57	#123×C₄₇	—	2	6	no	—
58	C₂²⋊Q₈×C₄₇	—	2	3	no	—
59	C₂²⋊Q₈×C₄₇	—	2	6	no	—
60	C₂²⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
61	#121×C₄₇	—	2	3	no	—
62	#121×C₄₇	—	2	3	no	—
63	#121×C₄₇	C₂³	1	3	yes	—
64	C₄×Q₈×C₄₇	—	2	1	no	—
65	C₄×Q₈×C₄₇	C₂×C₄	1	3	yes	—
66	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈×C₄₇	C₂×C₄	1	3	yes	—
68	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
69	C₄×Q₈×C₄₇	D₄	1	3	yes	—
70	C₄×D₄×C₄₇	—	2	3	no	—
71	C₄×D₄×C₄₇	—	2	3	no	—
72	C₄×D₄×C₄₇	—	2	3	no	—
73	C₄×D₄×C₄₇	C₂×C₄	1	3	yes	—
74	C₄×D₄×C₄₇	C₂×C₄	1	6	yes	—
75	#117×C₄₇	—	2	3	no	—
76	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
77	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
78	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
79	#41×C₄₇	—	4	4	no	—
80	#41×C₄₇	—	8	3	no	—
81	#41×C₄₇	—	8	3	no	—
82	C₂×Q₈×C₄₇	—	2	4	no	—
83	C₂×Q₈×C₄₇	—	2	3	no	—
84	C₂×Q₈×C₄₇	C₂²⋊C₄	1	3	yes	—
85	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Q₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×D₄×C₄₇	—	2	3	no	—
88	C₂×D₄×C₄₇	—	2	3	no	—
89	C₂×D₄×C₄₇	—	2	3	no	—
90	C₂×D₄×C₄₇	C₂²×C₄	1	3	yes	—
91	C₂²×C₄×C₄₇	—	2	3	no	—
92	C₂²×C₄×C₄₇	—	2	6	no	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
95	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₂²×Dic₄₇	—	2	1	no	—
97	C₂²×Dic₄₇	—	2	3	no	—
98	C₄⋊C₄×C₄₇	—	2	3	no	—
99	C₄⋊C₄×C₄₇	—	2	3	no	—
100	C₄⋊C₄×C₄₇	—	2	3	no	—
101	C₄⋊C₄×C₄₇	—	2	3	no	—
102	C₄⋊C₄×C₄₇	—	2	1	no	—
103	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
104	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	3	yes	—
105	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
106	C₄⋊C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₄⋊C₄×C₄₇	C₂²×C₄	1	3	yes	—
108	C₂²⋊C₄×C₄₇	—	2	1	no	—
109	C₂²⋊C₄×C₄₇	—	2	3	no	—
110	C₂²⋊C₄×C₄₇	—	2	3	no	—
111	C₂²⋊C₄×C₄₇	—	2	3	no	—
112	C₂²⋊C₄×C₄₇	—	2	6	no	—
113	C₂²⋊C₄×C₄₇	—	2	3	no	—
114	C₄²×C₄₇	—	2	3	no	—
115	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
116	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
117	C₄²×C₄₇	C₂×D₄	1	3	yes	—
118	C₄²×C₄₇	C₂×D₄	1	3	yes	—
119	C₄⋊Dic₄₇	—	2	12	no	—
120	C₄×Dic₄₇	—	2	12	no	—
121	C₄×Dic₄₇	—	8	3	no	—
122	C₂×C₄₇⋊C₈	—	2	1	no	—
123	C₂×C₄₇⋊C₈	—	2	3	no	—
124	C₂³×C₄₇	—	2	3	no	—
125	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₄₇	—	2	6	no	—
127	Q₈×C₄₇	—	4	1	no	—
128	Q₈×C₄₇	—	8	3	no	—
129	Q₈×C₄₇	#104	1	4	yes	—
130	D₄×C₄₇	—	4	3	no	—
131	D₄×C₄₇	—	4	3	no	—
132	D₄×C₄₇	—	4	3	no	—
133	D₄×C₄₇	—	4	6	no	—
134	D₄×C₄₇	—	8	3	no	—
135	C₂×C₄×C₄₇	—	2	3	no	—
136	C₂×C₄×C₄₇	—	2	3	no	—
137	C₂×C₄×C₄₇	—	2	6	no	—
138	C₂×C₄×C₄₇	—	2	1	no	—
139	C₂×C₄×C₄₇	—	2	3	no	—
140	C₂×C₄×C₄₇	—	2	1	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	6	no	—
143	C₂×C₄×C₄₇	—	8	6	no	—
144	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
145	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
146	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
147	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
148	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	6	no	—
152	C₂×Dic₄₇	—	4	1	no	—
153	C₂×Dic₄₇	—	4	3	no	—
154	C₂×Dic₄₇	—	4	3	no	—
155	C₂×Dic₄₇	—	4	1	no	—
156	C₄₇⋊C₈	—	4	1	no	—
157	C₄₇⋊C₈	—	4	3	no	—
158	#17550	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	4	3	no	—
162	C₂²×C₄₇	—	4	3	no	—
163	C₂²×C₄₇	—	4	3	no	—
164	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄₇	#576	1	1	yes	characteristic
166	C₂²×C₄₇	#577	1	1	yes	characteristic
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	3	no	—
170	C₄×C₄₇	—	2	3	no	—
171	C₄×C₄₇	—	2	3	no	—
172	C₄×C₄₇	—	4	1	no	—
173	C₄×C₄₇	—	4	3	no	—
174	C₄×C₄₇	—	4	1	no	—
175	C₄×C₄₇	C₂×#104	1	6	yes	—
176	Dic₄₇	—	4	6	no	—
177	Dic₄₇	—	4	2	no	—
178	#3522	D₄₇	1	1	yes	characteristic
179	#2984	—	47	1	no	—
180	#2276	—	47	1	no	—
181	#1509	—	47	3	no	—
182	#1505	—	47	3	no	—
183	#1502	—	47	3	no	—
184	#1500	—	47	3	no	—
185	C₂×C₄₇	—	2	1	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	8	3	no	—
188	C₂×C₄₇	#2884	1	1	yes	characteristic
189	C₂×C₄₇	#2897	1	1	yes	characteristic
190	C₂×C₄₇	C₂²×#104	1	1	yes	characteristic
191	#559	Dic₄₇	1	3	yes	complemented
192	Q₈²	Dic₄₇	1	1	yes	characteristic, complemented
193	#554	Dic₄₇	1	3	yes	complemented
194	#550	Dic₄₇	1	1	yes	characteristic, complemented
195	#548	—	2	1	no	—
196	#547	—	2	3	no	—
197	#542	—	2	3	no	—
198	#532	C₂×D₄₇	1	3	yes	—
199	#529	C₂×D₄₇	1	3	yes	—
200	#520	—	2	1	no	—
201	C₂×C₄×Q₈	—	94	1	no	—
202	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
203	#446	—	94	3	no	—
204	C₈×Q₈	—	94	1	no	—
205	#432	—	47	3	no	—
206	#431	—	47	3	no	—
207	C₄×C₄⋊C₄	—	47	3	no	—
208	C₄×C₄⋊C₄	—	94	3	no	—
209	#376	—	47	3	no	—
210	#328	—	47	1	no	—
211	#328	—	47	3	no	—
212	#327	—	47	3	no	—
213	#327	—	47	3	no	—
214	#327	—	47	6	no	—
215	C₄₇	#17550	1	1	yes	characteristic, complemented, Sylow
216	C₂×#41	—	2	1	no	—
217	C₂×#41	—	2	3	no	—
218	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	3	no	—
220	C₄⋊Q₈	C₂×Dic₄₇	1	3	yes	—
221	#127	C₂×Dic₄₇	1	3	yes	—
222	#126	—	2	3	no	—
223	#125	C₂×Dic₄₇	1	3	yes	—
224	#124	—	2	3	no	—
225	#124	—	2	3	no	—
226	#124	C₂×Dic₄₇	1	3	yes	—
227	#123	—	2	3	no	—
228	#123	—	2	6	no	—
229	C₂²⋊Q₈	—	2	3	no	—
230	C₂²⋊Q₈	—	2	6	no	—
231	C₂²⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
232	#121	—	2	3	no	—
233	#121	—	2	3	no	—
234	#121	C₂²×D₄₇	1	3	yes	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×Q₈	—	94	8	no	—
237	C₄×Q₈	C₂×Dic₄₇	1	3	yes	—
238	C₄×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
239	C₄×Q₈	C₂×Dic₄₇	1	3	yes	—
240	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
241	C₄×Q₈	C₂²⋊D₄₇	1	3	yes	—
242	C₄×D₄	—	2	3	no	—
243	C₄×D₄	—	2	3	no	—
244	C₄×D₄	—	2	3	no	—
245	C₄×D₄	C₂×Dic₄₇	1	3	yes	—
246	C₄×D₄	C₂×Dic₄₇	1	6	yes	—
247	#117	—	2	3	no	—
248	C₂×C₄⋊C₄	—	2	3	no	—
249	C₂×C₄⋊C₄	—	94	3	no	—
250	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
251	C₂×C₄²	—	94	3	no	—
252	C₂×C₄²	—	94	3	no	—
253	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
254	C₄⋊C₈	—	94	3	no	—
255	C₄⋊C₈	—	94	3	no	—
256	C₂²⋊C₈	—	47	1	no	—
257	C₂²⋊C₈	—	47	3	no	—
258	#97	—	94	3	no	—
259	C₄×C₈	—	94	3	no	—
260	#95	—	47	1	no	—
261	#95	—	47	3	no	—
262	#95	—	94	3	no	—
263	#41	—	4	4	no	—
264	#41	—	8	3	no	—
265	#41	—	8	3	no	—
266	C₂×Q₈	—	2	4	no	—
267	C₂×Q₈	—	2	3	no	—
268	C₂×Q₈	C₂²⋊Dic₄₇	1	3	yes	—
269	C₂×Q₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
270	C₂×Q₈	C₂²×Dic₄₇	1	1	yes	characteristic
271	C₂×D₄	—	2	3	no	—
272	C₂×D₄	—	2	3	no	—
273	C₂×D₄	—	2	3	no	—
274	C₂×D₄	C₂²×Dic₄₇	1	3	yes	—
275	C₂²×C₄	—	2	3	no	—
276	C₂²×C₄	—	2	6	no	—
277	C₂²×C₄	—	94	1	no	—
278	C₂²×C₄	—	94	3	no	—
279	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
280	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
281	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
282	C₂×C₈	—	94	1	no	—
283	C₂×C₈	—	94	3	no	—
284	C₄⋊C₄	—	2	3	no	—
285	C₄⋊C₄	—	2	3	no	—
286	C₄⋊C₄	—	2	3	no	—
287	C₄⋊C₄	—	2	3	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	94	12	no	—
290	C₄⋊C₄	C₂²⋊Dic₄₇	1	3	yes	—
291	C₄⋊C₄	C₂²⋊Dic₄₇	1	3	yes	—
292	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
293	C₄⋊C₄	C₂²×Dic₄₇	1	1	yes	characteristic
294	C₄⋊C₄	C₂²×Dic₄₇	1	3	yes	—
295	C₂²⋊C₄	—	2	1	no	—
296	C₂²⋊C₄	—	2	3	no	—
297	C₂²⋊C₄	—	2	3	no	—
298	C₂²⋊C₄	—	2	3	no	—
299	C₂²⋊C₄	—	2	6	no	—
300	C₂²⋊C₄	—	2	3	no	—
301	C₄²	—	2	3	no	—
302	C₄²	—	94	12	no	—
303	C₄²	—	376	3	no	—
304	C₄²	C₂²×Dic₄₇	1	3	yes	—
305	C₄²	C₂²×Dic₄₇	1	3	yes	—
306	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
307	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
308	C₂³	—	2	3	no	—
309	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
310	Q₈	—	2	6	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	8	3	no	—
313	Q₈	#13093951	1	4	yes	—
314	D₄	—	4	3	no	—
315	D₄	—	4	3	no	—
316	D₄	—	4	3	no	—
317	D₄	—	4	6	no	—
318	D₄	—	8	3	no	—
319	C₂×C₄	—	2	3	no	—
320	C₂×C₄	—	2	3	no	—
321	C₂×C₄	—	2	6	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	3	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	6	no	—
327	C₂×C₄	—	8	6	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	6	no	—
330	C₂×C₄	—	188	1	no	—
331	C₂×C₄	—	188	3	no	—
332	C₂×C₄	—	188	3	no	—
333	C₂×C₄	—	188	1	no	—
334	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
335	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
336	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
337	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
338	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
340	C₈	—	188	1	no	—
341	C₈	—	188	3	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	3	no	—
345	C₂²	—	4	3	no	—
346	C₂²	—	4	3	no	—
347	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
348	C₂²	#15689926	1	1	yes	characteristic
349	C₂²	#15689934	1	1	yes	characteristic
350	C₄	—	2	3	no	—
351	C₄	—	2	3	no	—
352	C₄	—	2	3	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	3	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	188	6	no	—
359	C₄	—	188	2	no	—
360	C₄	C₂×#13093951	1	6	yes	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	8	3	no	—
364	C₂	#20251852	1	1	yes	characteristic
365	C₂	#20251914	1	1	yes	characteristic
366	C₂	C₂²×#13093951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3522	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3522	D₄₇
hypocenter	C₄₇	#17550
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ #3522

Complements

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

N	H ≅ G/N	type
1. #20252477	C₂	semidirect
2. #20245753	C₂	semidirect
3. #20241913	C₂	semidirect
4. #20241896	C₂	semidirect
5. #20241883	C₂	semidirect
6. #20241874	C₂	semidirect
21. C₄₇×#559	C₄	semidirect
22. Q₈²×C₄₇	C₄	semidirect
23. C₄₇×#554	C₄	semidirect
24. C₄₇×#550	C₄	semidirect
191. #559	Dic₄₇	semidirect
192. Q₈²	Dic₄₇	semidirect
193. #554	Dic₄₇	semidirect
194. #550	Dic₄₇	semidirect
215. C₄₇	#17550	semidirect

Maximal subgroups

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

subgroup	index
#20252477	2
#20245753	2
#20241913	2
#20241896	2
#20241883	2
#20241874	2
C₄₇×#3522	2
#17550	47