Subgroups — #26186403

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253419	C₂	1	1	yes	maximal, characteristic
2	#20252469	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252466	C₂	1	1	yes	maximal, characteristic
4	#20242263	C₂	1	2	yes	maximal, complemented
5	#20242228	C₂	1	2	yes	maximal, complemented
6	#20242203	C₂	1	2	yes	maximal, complemented
7	#20242185	C₂	1	2	yes	maximal, complemented
8	#20242169	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241806	C₂	1	1	yes	maximal, characteristic
10	#20241685	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1491	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689673	C₂²	1	1	yes	characteristic
13	#15689441	C₄	1	2	yes	—
14	#15689408	C₄	1	2	yes	—
15	#15689323	C₄	1	2	yes	—
16	#15689262	C₄	1	2	yes	—
17	Q₈×C₄₇⋊C₈	—	2	2	no	—
18	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	—
19	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₄×#10625336	C₂²	1	2	yes	—
21	C₄×#10625336	C₂²	1	2	yes	—
22	#15689200	C₂²	1	2	yes	—
23	#15689196	—	2	2	no	—
24	#15689188	C₂²	1	2	yes	—
25	#15689176	—	2	2	no	—
26	#15689176	C₂²	1	1	yes	characteristic
27	#15689176	C₂²	1	1	yes	characteristic
28	#15689176	C₂²	1	1	yes	characteristic
29	#15689088	—	2	2	no	—
30	#15688875	C₂²	1	2	yes	—
31	#15688873	C₂²	1	2	yes	—
32	#15688852	C₂²	1	2	yes	—
33	#15688852	C₂²	1	2	yes	—
34	#15688845	C₂²	1	1	yes	characteristic
35	#15688845	C₂²	1	1	yes	characteristic
36	#15688844	C₂²	1	1	yes	characteristic
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#432	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₂×#10625350	—	2	1	no	—
44	C₂×#10625336	C₂×C₄	1	2	yes	—
45	C₂×#10625336	C₂×C₄	1	2	yes	—
46	C₂×#10625324	—	2	2	no	—
47	C₂×C₈×D₄₇	—	2	2	no	—
48	#13094024	C₂×C₄	1	1	yes	characteristic
49	#13094021	C₂×C₄	1	1	yes	characteristic
50	#13094012	—	2	1	no	—
51	#13093998	—	2	1	no	—
52	#13093998	C₂×C₄	1	2	yes	—
53	C₄×D₁₈₈	—	2	1	no	—
54	C₄×D₁₈₈	C₂³	1	2	yes	—
55	#13093955	C₂×C₄	1	2	yes	—
56	#13093955	C₂×C₄	1	2	yes	—
57	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
59	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
61	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
62	#13093948	C₂×C₄	1	2	yes	—
63	#13093948	C₂×C₄	1	2	yes	—
64	#13093944	C₂×C₄	1	2	yes	—
65	#13093944	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
68	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
69	(C₂×D₄₇)⋊C₈	—	2	2	no	—
70	(C₂×D₄₇)⋊C₈	—	2	2	no	—
71	C₄×C₄₇⋊C₈	—	2	1	no	—
72	C₄×C₄₇⋊C₈	—	2	2	no	—
73	C₄×C₄₇⋊C₈	—	2	2	no	—
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
79	C₂×M₁₆×C₄₇	—	2	1	no	—
80	C₄×Q₈×C₄₇	—	2	1	no	—
81	C₄×Q₈×C₄₇	—	2	1	no	—
82	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₄₇	—	2	1	no	—
87	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	C₂²⋊C₈×C₄₇	—	2	1	no	—
90	#97×C₄₇	—	2	1	no	—
91	#97×C₄₇	—	2	1	no	—
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	—	2	4	no	—
95	C₂×Q₈×C₄₇	—	2	1	no	—
96	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₄₇	—	2	1	no	—
98	C₂²×C₄×C₄₇	—	2	1	no	—
99	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₄₇	—	2	2	no	—
103	#10625350	—	8	2	no	—
104	C₂×D₁₈₈	—	2	1	no	—
105	C₂×D₁₈₈	—	2	1	no	—
106	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₄₇	—	2	2	no	—
108	C₂×Dic₉₄	—	2	1	no	—
109	M₁₆×C₄₇	—	4	2	no	—
110	C₂×C₈×C₄₇	—	2	1	no	—
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₄⋊C₄×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	—	2	1	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₄²×C₄₇	—	2	2	no	—
119	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₄₇	#41	1	1	yes	characteristic
122	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
123	#10625336	—	2	4	no	—
124	#10625336	—	2	4	no	—
125	(C₂×D₄₇)⋊C₄	—	2	2	no	—
126	(C₂×D₄₇)⋊C₄	—	2	2	no	—
127	C₄⋊Dic₄₇	—	2	1	no	—
128	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
129	C₂×C₄₇⋊C₈	—	2	2	no	—
130	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
132	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
133	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
134	#10625324	—	8	2	no	—
135	C₈×D₄₇	—	8	2	no	—
136	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₄₇	—	2	2	no	—
138	Q₈×C₄₇	—	2	4	no	—
139	Q₈×C₄₇	—	2	2	no	—
140	Q₈×C₄₇	#131	1	4	yes	—
141	C₂×C₄×C₄₇	—	2	4	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	2	no	—
144	C₂×C₄×C₄₇	—	2	2	no	—
145	C₂×C₄×C₄₇	—	2	2	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	8	1	no	—
148	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₄₇	—	8	2	no	—
155	C₂×Dic₄₇	—	2	2	no	—
156	D₁₈₈	—	2	2	no	—
157	D₁₈₈	—	2	2	no	—
158	D₁₈₈	—	8	1	no	—
159	D₁₈₈	—	8	1	no	—
160	C₄×D₄₇	—	4	2	no	—
161	C₄×D₄₇	—	4	2	no	—
162	Dic₉₄	—	8	1	no	—
163	C₈×C₄₇	—	2	2	no	—
164	C₈×C₄₇	—	4	1	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₄₇⋊C₈	—	2	2	no	—
167	C₄₇⋊C₈	—	2	2	no	—
168	C₄₇⋊C₈	—	2	2	no	—
169	C₄₇⋊C₈	—	2	2	no	—
170	C₄₇⋊C₈	—	4	2	no	—
171	#17923	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	—	2	1	no	—
174	C₂²×C₄₇	—	2	1	no	—
175	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₂×D₄₇	—	4	2	no	—
179	C₄×C₄₇	—	2	4	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	2	1	no	—
184	C₄×C₄₇	—	4	2	no	—
185	C₄×C₄₇	C₂×#131	1	2	yes	—
186	C₄×C₄₇	#577	1	2	yes	—
187	Dic₄₇	—	4	2	no	—
188	#3103	—	47	1	no	—
189	#2982	—	47	1	no	—
190	#2981	—	47	1	no	—
191	#1588	—	47	2	no	—
192	#1583	—	47	2	no	—
193	SD₁₆⋊C₈	—	47	2	no	—
194	C₈×SD₁₆	—	47	2	no	—
195	#1572	—	47	1	no	—
196	#1491	D₄₇	1	1	yes	characteristic, complemented
197	#1484	—	47	1	no	—
198	#1453	—	47	1	no	—
199	C₂×C₄₇	—	4	1	no	—
200	C₂×C₄₇	#2929	1	1	yes	characteristic
201	C₂×C₄₇	#2949	1	1	yes	characteristic
202	C₂×C₄₇	#2956	1	1	yes	characteristic
203	D₄₇	—	8	2	no	—
204	#532	—	47	1	no	—
205	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
206	#481	—	47	2	no	—
207	#475	—	47	2	no	—
208	#460	—	47	2	no	—
209	#448	—	47	2	no	—
210	C₈×Q₈	—	47	2	no	—
211	C₈×Q₈	—	47	2	no	—
212	C₈×Q₈	—	94	2	no	—
213	C₄×SD₁₆	—	47	2	no	—
214	C₄×SD₁₆	—	47	2	no	—
215	#436	—	47	2	no	—
216	#435	—	94	2	no	—
217	C₈×D₄	—	47	2	no	—
218	#432	C₂×D₄₇	1	1	yes	characteristic
219	#431	—	47	1	no	—
220	#431	—	47	1	no	—
221	#431	—	47	1	no	—
222	#431	—	94	2	no	—
223	#408	—	94	2	no	—
224	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
225	#335	—	47	2	no	—
226	#334	—	47	2	no	—
227	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
228	Q₈⋊C₈	C₂×D₄₇	1	2	yes	—
229	D₄⋊C₈	—	47	2	no	—
230	D₄⋊C₈	—	47	2	no	—
231	#322	—	47	1	no	—
232	#322	—	47	1	no	—
233	#322	—	47	1	no	—
234	C₈²	—	47	1	no	—
235	C₄₇	#17923	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	94	1	no	—
237	C₂²×Q₈	C₄×D₄₇	1	1	yes	characteristic
238	C₂×SD₁₆	—	47	2	no	—
239	C₂×SD₁₆	—	47	2	no	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	94	2	no	—
242	C₂²×C₈	—	94	2	no	—
243	C₄⋊Q₈	—	47	1	no	—
244	#127	—	47	1	no	—
245	#124	—	94	1	no	—
246	#121	—	47	2	no	—
247	#121	—	94	1	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	C₄×D₄₇	1	2	yes	—
251	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
252	C₄×D₄	—	47	2	no	—
253	C₄×D₄	—	94	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄²	—	2	1	no	—
257	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
258	#106	—	47	2	no	—
259	#106	—	47	2	no	—
260	C₄⋊C₈	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	47	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	—	94	2	no	—
265	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
266	#103	—	47	2	no	—
267	#103	—	47	2	no	—
268	#102	—	47	2	no	—
269	#102	—	47	2	no	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	—	47	2	no	—
272	C₂²⋊C₈	—	47	2	no	—
273	C₂²⋊C₈	—	94	2	no	—
274	C₂²⋊C₈	—	94	2	no	—
275	C₂²⋊C₈	—	94	1	no	—
276	#97	—	2	1	no	—
277	#97	—	2	1	no	—
278	C₄×C₈	—	47	1	no	—
279	C₄×C₈	—	47	1	no	—
280	C₄×C₈	—	47	1	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	94	2	no	—
283	C₄×C₈	—	94	2	no	—
284	C₄×C₈	—	94	1	no	—
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	#41	—	376	2	no	—
288	C₂×Q₈	—	2	4	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	94	1	no	—
291	C₂×Q₈	C₂×C₄×D₄₇	1	2	yes	—
292	C₂×D₄	—	47	2	no	—
293	C₂×D₄	—	94	2	no	—
294	C₂×D₄	—	94	1	no	—
295	C₂×D₄	—	94	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	94	2	no	—
299	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
301	C₂²×C₄	#10625352	1	1	yes	characteristic
302	SD₁₆	—	94	4	no	—
303	SD₁₆	—	94	4	no	—
304	M₁₆	—	4	2	no	—
305	M₁₆	—	376	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	47	2	no	—
310	C₂×C₈	—	47	2	no	—
311	C₂×C₈	—	47	2	no	—
312	C₂×C₈	—	47	2	no	—
313	C₂×C₈	—	94	2	no	—
314	C₂×C₈	—	376	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	47	2	no	—
319	C₄⋊C₄	—	94	1	no	—
320	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
326	C₄²	D₄×D₄₇	1	1	yes	characteristic
327	C₄²	#10625352	1	1	yes	characteristic
328	C₄²	C₂³×D₄₇	1	1	yes	characteristic
329	C₂³	—	94	2	no	—
330	C₂³	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	376	1	no	—
334	Q₈	#13094032	1	4	yes	—
335	D₄	—	94	2	no	—
336	D₄	—	94	2	no	—
337	D₄	—	376	1	no	—
338	D₄	—	376	1	no	—
339	D₄	—	376	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	1	no	—
347	C₂×C₄	—	94	2	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
351	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
352	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	94	2	no	—
360	C₈	—	94	2	no	—
361	C₈	—	94	2	no	—
362	C₈	—	94	2	no	—
363	C₈	—	188	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	188	2	no	—
368	C₂²	—	188	2	no	—
369	C₂²	—	188	2	no	—
370	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	188	2	no	—
378	C₄	C₂×#13094032	1	2	yes	—
379	C₄	#15689932	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	376	2	no	—
382	C₂	#20252078	1	1	yes	characteristic
383	C₂	#20252243	1	1	yes	characteristic
384	C₂	#20252294	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1491	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1491	D₄₇
hypocenter	C₄₇	#17923
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1491

Complements

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

N	H ≅ G/N	type
2. #20252469	C₂	semidirect
4. #20242263	C₂	semidirect
5. #20242228	C₂	semidirect
6. #20242203	C₂	semidirect
7. #20242185	C₂	semidirect
8. #20242169	C₂	semidirect
10. #20241685	C₂	semidirect
11. C₄₇×#1491	C₂	semidirect
196. #1491	D₄₇	semidirect
235. C₄₇	#17923	semidirect

Maximal subgroups

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

subgroup	index
#20253419	2
#20252469	2
#20252466	2
#20242263	2
#20242228	2
#20242203	2
#20242185	2
#20242169	2
#20241806	2
#20241685	2
C₄₇×#1491	2
#17923	47