Subgroups — #26201556

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254326	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689323	C₂	1	1	yes	maximal, characteristic
3	#20252469	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242806	C₂	1	4	yes	maximal, complemented
5	#20242738	C₂	1	4	yes	maximal, complemented
6	#20242160	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242057	C₂	1	1	yes	maximal, characteristic
8	#20242045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625336	—	2	1	no	—
11	C₂×#13094021	C₂²	1	1	yes	characteristic
12	#15689497	C₂²	1	2	yes	—
13	#15689471	C₂²	1	2	yes	—
14	#15689323	—	2	2	no	—
15	#15689323	—	2	2	no	—
16	#15689323	C₂²	1	4	yes	—
17	#15689273	—	2	2	no	—
18	Q₈×C₄₇⋊C₈	—	2	2	no	—
19	Q₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
20	#15689176	—	2	2	no	—
21	#15689176	C₂²	1	1	yes	characteristic
22	#15689176	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×#13093944	—	2	1	no	—
25	#15688873	C₂²	1	4	yes	complemented
26	#15688872	C₂²	1	2	yes	complemented
27	#15688869	C₂²	1	2	yes	complemented
28	#15688866	C₂²	1	4	yes	—
29	C₄₇×#533	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#479	—	2	2	no	—
32	C₄₇×#477	C₂²	1	2	yes	complemented
33	C₄₇×#475	C₂²	1	2	yes	complemented
34	C₄₇×#428	—	2	1	no	—
35	C₄₇×#423	C₂²	1	1	yes	characteristic
36	C₄₇×#419	—	2	1	no	—
37	C₂²×D₁₈₈	—	2	1	no	—
38	C₂²×D₁₈₈	—	2	1	no	—
39	C₂²×D₁₈₈	—	2	1	no	—
40	C₂×#10625336	—	2	4	no	—
41	C₂×#10625336	—	2	4	no	—
42	C₂×#10625336	—	4	4	no	—
43	C₂²×C₄₇⋊C₈	—	2	1	no	—
44	#13094021	—	2	1	no	—
45	#13094021	—	2	1	no	—
46	#13094021	—	2	1	no	—
47	#13094021	C₂³	1	2	yes	—
48	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
50	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
51	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
53	#13093944	—	2	4	no	—
54	#13093944	—	2	4	no	—
55	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
57	C₄×C₄₇⋊C₈	—	2	1	no	—
58	C₄×C₄₇⋊C₈	—	2	4	no	—
59	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
60	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
61	C₂×M₁₆×C₄₇	—	2	1	no	—
62	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
64	#125×C₄₇	—	2	1	no	—
65	C₂²⋊Q₈×C₄₇	—	2	1	no	—
66	C₂²⋊Q₈×C₄₇	—	2	1	no	—
67	C₄×Q₈×C₄₇	—	2	1	no	—
68	C₄×Q₈×C₄₇	—	2	1	no	—
69	C₄×Q₈×C₄₇	—	2	1	no	—
70	C₄×Q₈×C₄₇	—	2	1	no	—
71	C₄×Q₈×C₄₇	D₄	1	2	yes	—
72	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
73	#117×C₄₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₄₇	—	2	1	no	—
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	#107×C₄₇	—	2	2	no	—
79	#106×C₄₇	—	2	2	no	—
80	C₄⋊C₈×C₄₇	—	2	1	no	—
81	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
82	#103×C₄₇	—	2	2	no	—
83	#103×C₄₇	—	2	2	no	—
84	C₂³×D₄₇	—	4	1	no	—
85	C₂×Q₈×C₄₇	—	2	1	no	—
86	C₂×Q₈×C₄₇	—	2	2	no	—
87	C₂×Q₈×C₄₇	—	2	1	no	—
88	C₂×Q₈×C₄₇	—	2	1	no	—
89	C₂×Q₈×C₄₇	SD₁₆	1	4	yes	complemented
90	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₇	—	2	1	no	—
92	C₂²×C₄×C₄₇	—	2	1	no	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
96	C₂×D₁₈₈	—	2	2	no	—
97	C₂×D₁₈₈	—	2	2	no	—
98	C₂×D₁₈₈	—	2	2	no	—
99	C₂×D₁₈₈	—	4	2	no	—
100	C₂×D₁₈₈	—	4	2	no	—
101	C₂×D₁₈₈	—	4	1	no	—
102	C₂×D₁₈₈	—	4	1	no	—
103	C₂×D₁₈₈	—	4	1	no	—
104	C₂×D₁₈₈	—	8	2	no	—
105	C₂×D₁₈₈	—	8	2	no	—
106	M₁₆×C₄₇	—	4	2	no	—
107	C₂×C₈×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	2	1	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₄⋊C₄×C₄₇	—	2	1	no	—
112	C₄⋊C₄×C₄₇	—	2	1	no	—
113	C₄⋊C₄×C₄₇	—	2	1	no	—
114	C₄⋊C₄×C₄₇	—	2	1	no	—
115	C₄⋊C₄×C₄₇	—	4	1	no	—
116	C₄⋊C₄×C₄₇	—	4	1	no	—
117	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
118	C₂²⋊C₄×C₄₇	—	4	1	no	—
119	C₄²×C₄₇	—	2	2	no	—
120	C₄²×C₄₇	—	2	2	no	—
121	C₄²×C₄₇	—	4	1	no	—
122	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₄₇	#41	1	1	yes	characteristic
125	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
126	#10625336	—	4	8	no	—
127	#10625336	—	8	4	no	—
128	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	—	2	2	no	—
130	C₂×C₄₇⋊C₈	—	4	2	no	—
131	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
132	C₂³×C₄₇	#121	1	1	yes	characteristic
133	C₂²×D₄₇	—	4	2	no	—
134	C₂²×D₄₇	—	8	2	no	—
135	C₂²×D₄₇	—	8	2	no	—
136	C₂²×D₄₇	—	8	2	no	—
137	Q₈×C₄₇	—	2	2	no	—
138	Q₈×C₄₇	—	2	4	no	—
139	Q₈×C₄₇	—	4	2	no	—
140	Q₈×C₄₇	—	8	1	no	—
141	Q₈×C₄₇	C₂×SD₁₆	1	4	yes	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	4	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	—	4	1	no	—
148	C₂×C₄×C₄₇	—	4	1	no	—
149	C₂×C₄×C₄₇	—	8	1	no	—
150	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₄₇	#135	1	2	yes	—
155	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
158	D₁₈₈	—	4	2	no	—
159	D₁₈₈	—	4	2	no	—
160	D₁₈₈	—	4	4	no	—
161	D₁₈₈	—	8	2	no	—
162	D₁₈₈	—	8	2	no	—
163	D₁₈₈	—	8	2	no	—
164	D₁₈₈	—	8	2	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₄₇⋊C₈	—	2	4	no	—
167	C₄₇⋊C₈	—	4	2	no	—
168	C₄₇⋊C₈	—	4	2	no	—
169	#19396	—	47	1	no	maximal, Sylow
170	C₂²×C₄₇	—	2	1	no	—
171	C₂²×C₄₇	—	2	1	no	—
172	C₂²×C₄₇	#465	1	2	yes	—
173	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
174	C₂×D₄₇	—	8	2	no	—
175	C₂×D₄₇	—	8	2	no	—
176	C₂×D₄₇	—	8	2	no	—
177	C₂×D₄₇	—	16	4	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	2	4	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	4	2	no	—
183	C₄×C₄₇	—	8	1	no	—
184	C₄×C₄₇	—	8	1	no	—
185	C₄×C₄₇	#465	1	2	yes	—
186	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
187	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
188	C₄×C₄₇	#577	1	2	yes	—
189	#3208	—	47	1	no	—
190	#3078	D₄₇	1	1	yes	characteristic, complemented
191	C₂×#460	—	47	1	no	—
192	#2982	—	47	1	no	—
193	#1678	—	47	4	no	—
194	#1666	—	47	4	no	—
195	#1570	—	47	1	no	—
196	#1545	—	47	1	no	—
197	#1542	—	47	1	no	—
198	C₂×C₄₇	—	2	2	no	—
199	C₂×C₄₇	#3043	1	1	yes	characteristic
200	C₂×C₄₇	C₂×#465	1	1	yes	characteristic
201	C₂×C₄₇	#3068	1	1	yes	characteristic
202	D₄₇	—	16	2	no	—
203	C₂²×SD₁₆	—	94	1	no	—
204	#533	C₂×D₄₇	1	1	yes	characteristic
205	C₂×#127	—	47	1	no	—
206	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
207	#492	—	47	2	no	—
208	#486	—	47	2	no	—
209	#479	—	2	2	no	—
210	#477	C₂×D₄₇	1	2	yes	complemented
211	#475	C₂×D₄₇	1	2	yes	complemented
212	#460	—	47	4	no	—
213	#460	—	94	2	no	—
214	#460	—	94	2	no	—
215	#450	—	94	2	no	—
216	C₈×Q₈	—	47	4	no	—
217	C₈×Q₈	—	94	2	no	—
218	#431	—	47	1	no	—
219	#431	—	47	1	no	—
220	#431	—	94	2	no	—
221	#428	—	2	1	no	—
222	#423	C₂×D₄₇	1	1	yes	characteristic
223	C₂×C₄⋊C₈	—	47	1	no	—
224	#419	—	2	1	no	—
225	C₂×#102	—	94	1	no	—
226	#334	—	47	4	no	—
227	#333	—	47	2	no	—
228	#332	—	47	2	no	—
229	#331	—	47	4	no	—
230	C₄₇	#19396	1	1	yes	characteristic, complemented, Sylow
231	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂²×D₄	—	94	1	no	—
233	C₂²×D₄	—	94	1	no	—
234	C₂²×D₄	—	94	1	no	—
235	C₂×SD₁₆	—	94	4	no	—
236	C₂×SD₁₆	—	94	4	no	—
237	C₂×SD₁₆	—	188	4	no	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂²×C₈	—	94	1	no	—
240	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
242	#127	—	47	2	no	—
243	#127	—	94	1	no	—
244	#127	—	94	1	no	—
245	#127	—	94	1	no	—
246	#125	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	C₂²⋊D₄₇	1	2	yes	—
254	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
255	#117	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
260	#107	—	2	2	no	—
261	#106	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	47	2	no	—
264	C₄⋊C₈	—	47	2	no	—
265	C₄⋊C₈	—	47	2	no	—
266	C₄⋊C₈	—	94	4	no	—
267	C₄⋊C₈	—	94	1	no	—
268	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
269	#103	—	2	2	no	—
270	#103	—	2	2	no	—
271	#102	—	94	4	no	—
272	#102	—	94	4	no	—
273	C₂²⋊C₈	—	47	2	no	—
274	C₂²⋊C₈	—	94	1	no	—
275	C₄×C₈	—	47	2	no	—
276	C₄×C₈	—	94	4	no	—
277	C₄×C₈	—	94	1	no	—
278	C₂⁴	—	188	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	#10625336	1	4	yes	complemented
284	C₂×Q₈	C₂×C₂²⋊D₄₇	1	2	yes	—
285	C₂×D₄	—	94	2	no	—
286	C₂×D₄	—	94	2	no	—
287	C₂×D₄	—	94	2	no	—
288	C₂×D₄	—	376	2	no	—
289	C₂×D₄	—	376	2	no	—
290	C₂×D₄	—	188	1	no	—
291	C₂×D₄	—	188	1	no	—
292	C₂×D₄	—	188	1	no	—
293	C₂×D₄	—	188	2	no	—
294	C₂×D₄	—	188	2	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
298	C₂²×C₄	#10625352	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	SD₁₆	—	376	4	no	—
301	SD₁₆	—	188	8	no	—
302	M₁₆	—	4	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	47	4	no	—
305	C₂×C₈	—	94	2	no	—
306	C₂×C₈	—	94	2	no	—
307	C₂×C₈	—	188	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
318	C₂²⋊C₄	—	4	1	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	D₄×D₄₇	1	1	yes	characteristic
323	C₄²	#10625352	1	1	yes	characteristic
324	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
325	C₄²	C₂³×D₄₇	1	1	yes	characteristic
326	C₂³	—	376	2	no	—
327	C₂³	—	376	2	no	—
328	C₂³	—	376	2	no	—
329	C₂³	—	188	2	no	—
330	C₂³	#13093996	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	8	1	no	—
335	Q₈	C₂×#10625336	1	4	yes	—
336	D₄	—	376	2	no	—
337	D₄	—	376	2	no	—
338	D₄	—	376	2	no	—
339	D₄	—	376	2	no	—
340	D₄	—	188	2	no	—
341	D₄	—	188	2	no	—
342	D₄	—	188	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	#13093996	1	1	yes	characteristic
352	C₂×C₄	#13093996	1	1	yes	characteristic
353	C₂×C₄	#13093996	1	1	yes	characteristic
354	C₂×C₄	C₂×#10625336	1	2	yes	—
355	C₂×C₄	#13094044	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
357	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	94	4	no	—
361	C₈	—	188	2	no	—
362	C₈	—	188	2	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	376	2	no	—
366	C₂²	—	376	2	no	—
367	C₂²	—	376	2	no	—
368	C₂²	—	752	4	no	—
369	C₂²	#15689354	1	2	yes	—
370	C₂²	C₂×#13093996	1	1	yes	characteristic
371	C₄	—	2	4	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	#15689354	1	2	yes	—
379	C₄	C₂²×#10625336	1	1	yes	characteristic
380	C₄	C₂×#13094044	1	1	yes	characteristic
381	C₄	#15689932	1	2	yes	—
382	C₂	—	2	2	no	—
383	C₂	—	752	2	no	—
384	C₂	#20252911	1	1	yes	characteristic
385	C₂	C₂×#15689354	1	1	yes	characteristic
386	C₂	#20253118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093996
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3078	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂×#13094021	C₂²
hypercenter	#3078	D₄₇
hypocenter	C₄₇	#19396
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3078

Complements

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

N	H ≅ G/N	type
3. #20252469	C₂	semidirect
4. #20242806	C₂	semidirect
5. #20242738	C₂	semidirect
6. #20242160	C₂	semidirect
8. #20242045	C₂	semidirect
9. C₄₇×#3078	C₂	semidirect
19. Q₈×C₄₇⋊C₈	C₂²	semidirect
25. #15688873	C₂²	semidirect
26. #15688872	C₂²	semidirect
27. #15688869	C₂²	semidirect
32. C₄₇×#477	C₂²	semidirect
33. C₄₇×#475	C₂²	semidirect
89. C₂×Q₈×C₄₇	SD₁₆	semidirect
190. #3078	D₄₇	semidirect
210. #477	C₂×D₄₇	semidirect
211. #475	C₂×D₄₇	semidirect
230. C₄₇	#19396	semidirect
283. C₂×Q₈	#10625336	semidirect

Maximal subgroups

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

subgroup	index
#20254326	2
C₂×#15689323	2
#20252469	2
#20242806	2
#20242738	2
#20242160	2
#20242057	2
#20242045	2
C₄₇×#3078	2
#19396	47