Subgroups — #26199556

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 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×#15689471	C₂	1	1	yes	maximal, characteristic
3	#20252323	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242930	C₂	1	4	yes	maximal, complemented
5	#20242806	C₂	1	4	yes	maximal, complemented
6	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242069	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242055	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#3230	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094021	C₂²	1	1	yes	characteristic
11	#15689497	—	2	1	no	—
12	#15689471	—	2	1	no	—
13	#15689471	—	2	1	no	—
14	#15689471	—	2	1	no	—
15	#15689471	C₂²	1	2	yes	—
16	#15689471	C₂²	1	2	yes	—
17	#15689323	C₂²	1	4	yes	—
18	#15689273	—	2	2	no	—
19	#15689254	—	2	1	no	—
20	#15689254	C₂²	1	2	yes	complemented
21	Q₈×C₄₇⋊C₈	—	2	1	no	—
22	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
23	#15689176	C₂²	1	1	yes	characteristic
24	#15689150	—	2	1	no	—
25	#15689147	C₂²	1	1	yes	characteristic
26	C₂×#13093944	—	2	1	no	—
27	C₂×#13093944	—	2	1	no	—
28	#15689079	—	2	1	no	—
29	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	#15688875	C₂²	1	2	yes	complemented
31	#15688873	C₂²	1	2	yes	complemented
32	#15688872	C₂²	1	4	yes	complemented
33	#15688866	C₂²	1	4	yes	—
34	C₄₇×#533	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#500	—	2	1	no	—
37	C₄₇×#487	—	2	1	no	—
38	C₄₇×#484	—	2	2	no	—
39	C₄₇×#477	C₂²	1	4	yes	complemented
40	C₄₇×#431	C₂²	1	1	yes	characteristic
41	C₂²×D₁₈₈	—	2	1	no	—
42	C₂²×D₁₈₈	—	2	1	no	—
43	C₂²×D₁₈₈	—	2	1	no	—
44	C₂×#10625336	—	2	4	no	—
45	C₂×#10625329	—	2	1	no	—
46	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
48	#13094021	—	2	1	no	—
49	#13094021	—	2	1	no	—
50	#13094021	—	2	1	no	—
51	#13094021	C₂³	1	2	yes	—
52	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
56	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
57	#13093944	—	2	4	no	—
58	#13093944	—	2	2	no	—
59	#13093944	—	2	2	no	—
60	#13093944	—	2	2	no	—
61	#13093944	—	2	2	no	—
62	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
63	#13093935	—	2	2	no	—
64	C₄×C₄₇⋊C₈	—	2	1	no	—
65	C₄×C₄₇⋊C₈	—	2	2	no	—
66	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₄₇	—	2	1	no	—
71	C₄⋊Q₈×C₄₇	—	2	1	no	—
72	C₄⋊Q₈×C₄₇	—	2	1	no	—
73	C₄⋊Q₈×C₄₇	—	2	1	no	—
74	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
77	#125×C₄₇	—	2	1	no	—
78	C₂²⋊Q₈×C₄₇	—	2	1	no	—
79	C₂²⋊Q₈×C₄₇	D₄	1	2	yes	—
80	C₄×Q₈×C₄₇	—	2	1	no	—
81	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
82	#117×C₄₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
86	#107×C₄₇	—	2	4	no	—
87	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
88	#103×C₄₇	—	2	4	no	—
89	C₂²⋊C₈×C₄₇	—	2	1	no	—
90	C₄×C₈×C₄₇	—	2	1	no	—
91	C₂³×D₄₇	—	4	1	no	—
92	C₂×Q₈×C₄₇	—	2	1	no	—
93	C₂×Q₈×C₄₇	—	2	2	no	—
94	C₂×Q₈×C₄₇	—	4	1	no	—
95	C₂×Q₈×C₄₇	—	8	2	no	—
96	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₇	—	4	1	no	—
98	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
101	C₂×D₁₈₈	—	2	2	no	—
102	C₂×D₁₈₈	—	2	2	no	—
103	C₂×D₁₈₈	—	2	2	no	—
104	C₂×D₁₈₈	—	4	2	no	—
105	C₂×D₁₈₈	—	4	2	no	—
106	C₂×D₁₈₈	—	4	1	no	—
107	C₂×D₁₈₈	—	4	1	no	—
108	C₂×D₁₈₈	—	4	1	no	—
109	C₂×D₁₈₈	—	8	2	no	—
110	C₂×D₁₈₈	—	8	2	no	—
111	C₂×C₈×C₄₇	—	2	2	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	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	2	2	no	—
117	C₄⋊C₄×C₄₇	—	2	2	no	—
118	C₄⋊C₄×C₄₇	—	2	2	no	—
119	C₄⋊C₄×C₄₇	—	4	1	no	—
120	C₄⋊C₄×C₄₇	—	4	1	no	—
121	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
122	C₂²⋊C₄×C₄₇	—	2	2	no	—
123	C₄²×C₄₇	—	2	2	no	—
124	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₄₇	#41	1	1	yes	characteristic
127	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
128	#10625336	—	8	4	no	—
129	#10625329	—	4	2	no	—
130	C₂×C₄₇⋊C₈	—	2	2	no	—
131	C₂×C₄₇⋊C₈	—	2	2	no	—
132	C₂×C₄₇⋊C₈	—	2	2	no	—
133	C₂×C₄₇⋊C₈	—	2	2	no	—
134	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
138	C₂³×C₄₇	#121	1	1	yes	characteristic
139	C₂²×D₄₇	—	4	2	no	—
140	C₂²×D₄₇	—	8	2	no	—
141	C₂²×D₄₇	—	8	2	no	—
142	C₂²×D₄₇	—	8	2	no	—
143	Q₈×C₄₇	—	2	4	no	—
144	Q₈×C₄₇	—	4	2	no	—
145	Q₈×C₄₇	—	8	2	no	—
146	Q₈×C₄₇	—	8	2	no	—
147	C₂×C₄×C₄₇	—	2	2	no	—
148	C₂×C₄×C₄₇	—	2	2	no	—
149	C₂×C₄×C₄₇	—	2	2	no	—
150	C₂×C₄×C₄₇	—	2	2	no	—
151	C₂×C₄×C₄₇	—	4	2	no	—
152	C₂×C₄×C₄₇	—	8	2	no	—
153	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
154	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
155	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
156	C₂×C₄×C₄₇	#135	1	2	yes	—
157	C₂×C₄×C₄₇	#137	1	2	yes	—
158	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
161	D₁₈₈	—	4	2	no	—
162	D₁₈₈	—	4	2	no	—
163	D₁₈₈	—	4	4	no	—
164	D₁₈₈	—	8	2	no	—
165	D₁₈₈	—	8	2	no	—
166	D₁₈₈	—	8	2	no	—
167	D₁₈₈	—	8	2	no	—
168	C₈×C₄₇	—	4	2	no	—
169	C₄₇⋊C₈	—	2	1	no	—
170	C₄₇⋊C₈	—	2	1	no	—
171	C₄₇⋊C₈	—	2	1	no	—
172	C₄₇⋊C₈	—	2	1	no	—
173	C₄₇⋊C₈	—	4	2	no	—
174	C₄₇⋊C₈	—	4	2	no	—
175	#19212	—	47	1	no	maximal, Sylow
176	C₂²×C₄₇	—	2	1	no	—
177	C₂²×C₄₇	—	2	1	no	—
178	C₂²×C₄₇	#464	1	2	yes	—
179	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
180	C₂×D₄₇	—	8	2	no	—
181	C₂×D₄₇	—	8	2	no	—
182	C₂×D₄₇	—	8	2	no	—
183	C₂×D₄₇	—	16	4	no	—
184	C₄×C₄₇	—	2	1	no	—
185	C₄×C₄₇	—	2	1	no	—
186	C₄×C₄₇	—	4	2	no	—
187	C₄×C₄₇	—	4	2	no	—
188	C₄×C₄₇	—	8	2	no	—
189	C₄×C₄₇	#464	1	2	yes	—
190	C₄×C₄₇	#472	1	2	yes	—
191	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
192	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
193	C₄×C₄₇	#577	1	2	yes	—
194	#3230	D₄₇	1	1	yes	characteristic, complemented
195	#3208	—	47	1	no	—
196	C₂×#486	—	47	1	no	—
197	#2960	—	47	1	no	—
198	#1698	—	47	4	no	—
199	#1678	—	47	4	no	—
200	#1564	—	47	1	no	—
201	#1548	—	47	1	no	—
202	#1545	—	47	1	no	—
203	C₂×C₄₇	—	2	2	no	—
204	C₂×C₄₇	C₂×#464	1	1	yes	characteristic
205	C₂×C₄₇	#3057	1	1	yes	characteristic
206	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
207	D₄₇	—	16	2	no	—
208	#533	C₂×D₄₇	1	1	yes	characteristic
209	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
210	C₂×#127	—	47	1	no	—
211	#500	—	2	1	no	—
212	#492	—	94	1	no	—
213	#487	—	2	1	no	—
214	#486	—	47	2	no	—
215	#486	—	47	2	no	—
216	#486	—	94	1	no	—
217	#486	—	94	1	no	—
218	#486	—	94	1	no	—
219	#484	—	2	2	no	—
220	#477	C₂×D₄₇	1	4	yes	complemented
221	#460	—	47	4	no	—
222	#450	—	94	2	no	—
223	#446	—	47	2	no	—
224	#446	—	94	1	no	—
225	C₈×Q₈	—	47	2	no	—
226	C₈×Q₈	—	94	1	no	—
227	#431	—	47	1	no	—
228	#431	C₂×D₄₇	1	1	yes	characteristic
229	#424	—	94	1	no	—
230	#423	—	47	1	no	—
231	C₂×#102	—	94	1	no	—
232	C₂×#102	—	94	1	no	—
233	#405	—	94	1	no	—
234	C₂×C₄×C₈	—	47	1	no	—
235	#335	—	47	2	no	—
236	#334	—	47	2	no	—
237	#333	—	47	4	no	—
238	#331	—	47	4	no	—
239	C₄₇	#19212	1	1	yes	characteristic, complemented, Sylow
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×D₄	—	94	1	no	—
242	C₂²×D₄	—	94	1	no	—
243	C₂²×D₄	—	94	1	no	—
244	C₂×SD₁₆	—	94	4	no	—
245	C₂×M₁₆	—	94	1	no	—
246	C₂²×C₈	—	47	1	no	—
247	C₂²×C₈	—	47	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
252	C₄⋊Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
253	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
254	#127	—	47	2	no	—
255	#127	—	94	1	no	—
256	#127	—	94	1	no	—
257	#127	—	94	1	no	—
258	#125	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
263	#117	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
267	#107	—	2	4	no	—
268	C₄⋊C₈	—	47	2	no	—
269	C₄⋊C₈	—	47	2	no	—
270	C₄⋊C₈	—	94	2	no	—
271	C₄⋊C₈	—	94	2	no	—
272	C₄⋊C₈	—	94	1	no	—
273	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
274	#103	—	2	4	no	—
275	#102	—	94	2	no	—
276	#102	—	94	2	no	—
277	#102	—	94	2	no	—
278	#102	—	94	2	no	—
279	#102	—	94	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	94	1	no	—
282	#97	—	94	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	—	47	1	no	—
286	C₄×C₈	—	47	1	no	—
287	C₄×C₈	—	47	1	no	—
288	C₄×C₈	—	94	2	no	—
289	C₄×C₈	—	94	1	no	—
290	C₂⁴	—	188	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×Q₈	C₂×C₂²⋊D₄₇	1	2	yes	—
296	C₂×D₄	—	94	2	no	—
297	C₂×D₄	—	94	2	no	—
298	C₂×D₄	—	94	2	no	—
299	C₂×D₄	—	376	2	no	—
300	C₂×D₄	—	376	2	no	—
301	C₂×D₄	—	188	1	no	—
302	C₂×D₄	—	188	1	no	—
303	C₂×D₄	—	188	1	no	—
304	C₂×D₄	—	188	2	no	—
305	C₂×D₄	—	188	2	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
308	C₂²×C₄	#10625352	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
310	SD₁₆	—	376	4	no	—
311	M₁₆	—	188	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	47	1	no	—
314	C₂×C₈	—	47	1	no	—
315	C₂×C₈	—	47	1	no	—
316	C₂×C₈	—	47	1	no	—
317	C₂×C₈	—	94	2	no	—
318	C₂×C₈	—	94	2	no	—
319	C₂×C₈	—	94	2	no	—
320	C₂×C₈	—	94	2	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	D₄×D₄₇	1	1	yes	characteristic
334	C₄²	#10625352	1	1	yes	characteristic
335	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
336	C₄²	C₂³×D₄₇	1	1	yes	characteristic
337	C₂³	—	376	2	no	—
338	C₂³	—	376	2	no	—
339	C₂³	—	376	2	no	—
340	C₂³	—	188	2	no	—
341	C₂³	#13093996	1	1	yes	characteristic
342	Q₈	—	2	4	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	2	no	—
345	Q₈	—	8	2	no	—
346	D₄	—	376	2	no	—
347	D₄	—	376	2	no	—
348	D₄	—	376	2	no	—
349	D₄	—	376	2	no	—
350	D₄	—	188	2	no	—
351	D₄	—	188	2	no	—
352	D₄	—	188	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	#13093996	1	1	yes	characteristic
360	C₂×C₄	#13093996	1	1	yes	characteristic
361	C₂×C₄	#13093996	1	1	yes	characteristic
362	C₂×C₄	#13094047	1	2	yes	—
363	C₂×C₄	#13094056	1	2	yes	—
364	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
365	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
367	C₈	—	4	2	no	—
368	C₈	—	94	1	no	—
369	C₈	—	94	1	no	—
370	C₈	—	94	1	no	—
371	C₈	—	94	1	no	—
372	C₈	—	188	2	no	—
373	C₈	—	188	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	376	2	no	—
377	C₂²	—	376	2	no	—
378	C₂²	—	376	2	no	—
379	C₂²	—	752	4	no	—
380	C₂²	#15689347	1	2	yes	—
381	C₂²	C₂×#13093996	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	—	8	2	no	—
387	C₄	#15689347	1	2	yes	—
388	C₄	#15689391	1	2	yes	—
389	C₄	C₂×#13094047	1	1	yes	characteristic
390	C₄	C₂×#13094056	1	1	yes	characteristic
391	C₄	#15689931	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	752	2	no	—
394	C₂	C₂×#15689347	1	1	yes	characteristic
395	C₂	#20253029	1	1	yes	characteristic
396	C₂	C₂×#15689391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093996
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3230	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂×#13094021	C₂²
hypercenter	#3230	D₄₇
hypocenter	C₄₇	#19212
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3230

Complements

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

N	H ≅ G/N	type
3. #20252323	C₂	semidirect
4. #20242930	C₂	semidirect
5. #20242806	C₂	semidirect
6. #20242134	C₂	semidirect
7. #20242069	C₂	semidirect
9. C₄₇×#3230	C₂	semidirect
20. #15689254	C₂²	semidirect
22. Q₈×C₄₇⋊C₈	C₂²	semidirect
30. #15688875	C₂²	semidirect
31. #15688873	C₂²	semidirect
32. #15688872	C₂²	semidirect
39. C₄₇×#477	C₂²	semidirect
194. #3230	D₄₇	semidirect
220. #477	C₂×D₄₇	semidirect
239. C₄₇	#19212	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×#15689471	2
#20252323	2
#20242930	2
#20242806	2
#20242134	2
#20242069	2
#20242055	2
C₄₇×#3230	2
#19212	47