Subgroups — #26221210

Order 12032.

826 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245997	C₂	1	1	yes	maximal, characteristic
2	#20245744	C₂	1	1	yes	maximal, characteristic
3	#20244158	C₂	1	2	yes	maximal, complemented
4	#20244152	C₂	1	2	yes	maximal, complemented
5	#20243589	C₂	1	2	yes	maximal
6	#20243583	C₂	1	2	yes	maximal
7	#20243285	C₂	1	2	yes	maximal, complemented
8	C₄×#13093956	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₇×#433	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	#15689016	—	2	2	no	—
17	#15689009	—	2	2	no	—
18	#15688993	C₂²	1	1	yes	characteristic
19	#15688989	C₂²	1	1	yes	characteristic
20	#15688983	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₄₇	—	2	2	no	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
25	C₂²⋊C₄×Dic₄₇	—	2	2	no	—
26	#15688886	C₂²	1	2	yes	—
27	#15688886	C₂²	1	2	yes	—
28	#15688886	C₂²	1	2	yes	—
29	#15688886	C₂²	1	2	yes	—
30	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#433	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
34	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂³×Dic₄₇	—	2	1	no	—
37	#13093983	—	2	4	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄⋊Dic₄₇	—	2	4	no	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
46	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	1	no	—
48	C₂×C₄×Dic₄₇	—	2	2	no	—
49	C₂×C₄×Dic₄₇	—	2	2	no	—
50	C₂×C₄×Dic₄₇	—	2	2	no	—
51	#13093958	—	2	4	no	—
52	#13093956	—	2	4	no	—
53	#13093956	—	2	4	no	—
54	#13093956	—	2	4	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
63	#117×C₄₇	C₂×C₄	1	4	yes	—
64	#117×C₄₇	Q₈	1	4	yes	—
65	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
70	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
71	#97×C₄₇	C₂×C₄	1	4	yes	complemented
72	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
73	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	—	2	2	no	—
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	4	no	—
85	C₂²×Dic₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	—	2	2	no	—
87	C₂²×Dic₄₇	—	8	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
95	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
99	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
100	C₂²⋊Dic₄₇	—	2	4	no	—
101	C₂²⋊Dic₄₇	—	8	2	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	8	2	no	—
110	C₄⋊Dic₄₇	#41	1	4	yes	—
111	C₄⋊Dic₄₇	#41	1	4	yes	—
112	C₄×Dic₄₇	—	2	4	no	—
113	C₄×Dic₄₇	—	8	2	no	—
114	C₄×Dic₄₇	—	8	2	no	—
115	C₄×Dic₄₇	—	8	2	no	—
116	C₂³×C₄₇	—	2	1	no	—
117	C₂³×C₄₇	—	2	1	no	—
118	C₂³×C₄₇	—	2	2	no	—
119	C₂³×C₄₇	—	2	2	no	—
120	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	2	4	no	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
137	C₂×Dic₄₇	—	2	4	no	—
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	8	4	no	—
150	C₈×C₄₇	—	2	4	no	—
151	C₈×C₄₇	—	2	4	no	—
152	#21369	—	47	1	no	maximal, Sylow
153	C₂²×C₄₇	—	2	4	no	—
154	C₂²×C₄₇	—	2	2	no	—
155	C₂²×C₄₇	—	4	1	no	—
156	C₂²×C₄₇	—	4	2	no	—
157	C₂²×C₄₇	—	4	1	no	—
158	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₄₇	#525	1	1	yes	characteristic
160	C₂²×C₄₇	#525	1	1	yes	characteristic
161	C₂²×C₄₇	#577	1	2	yes	—
162	C₂²×C₄₇	#577	1	2	yes	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	—	2	4	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	C₄×#41	1	2	yes	—
167	C₄×C₄₇	#533	1	2	yes	—
168	Dic₄₇	—	4	4	no	—
169	Dic₄₇	—	4	4	no	—
170	Dic₄₇	—	8	2	no	—
171	Dic₄₇	—	8	2	no	—
172	C₂×#433	D₄₇	1	1	yes	characteristic
173	#2314	—	47	1	no	—
174	#2274	—	47	1	no	—
175	#1949	—	47	2	no	—
176	#1948	—	47	2	no	—
177	#1854	—	47	2	no	—
178	#1853	—	47	2	no	—
179	#1776	—	47	2	no	—
180	C₄×#106	—	47	2	no	—
181	C₂×C₄₇	—	4	2	no	—
182	C₂×C₄₇	#2296	1	1	yes	characteristic
183	C₂×C₄₇	#2912	1	2	yes	—
184	C₂×C₄₇	#3123	1	1	yes	characteristic
185	C₂×C₄₇	#3125	1	2	yes	—
186	C₂×C₄₇	#3126	1	1	yes	characteristic
187	C₂×#117	—	47	1	no	—
188	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
189	C₂²×C₄⋊C₄	—	47	1	no	—
190	#433	Dic₄₇	1	8	yes	complemented
191	C₂×#107	—	47	2	no	—
192	C₂×#106	—	47	2	no	—
193	C₂×#106	—	47	2	no	—
194	C₂×#106	—	47	2	no	—
195	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
196	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
197	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
199	#388	—	94	2	no	—
200	#387	—	94	2	no	—
201	#384	—	47	1	no	—
202	#382	—	47	1	no	—
203	#380	—	47	2	no	—
204	C₄×C₄⋊C₄	—	47	2	no	—
205	C₄×C₄⋊C₄	—	47	2	no	—
206	C₄×C₄⋊C₄	—	47	2	no	—
207	C₄×C₄⋊C₄	—	94	2	no	—
208	C₄×C₂²⋊C₄	—	94	2	no	—
209	#340	—	47	2	no	—
210	#340	—	47	2	no	—
211	#340	—	47	2	no	—
212	#340	—	47	2	no	—
213	C₄₇	#21369	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₄	—	94	1	no	—
215	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
218	#117	—	94	4	no	—
219	#117	Dic₉₄	1	4	yes	—
220	#117	C₂×Dic₄₇	1	4	yes	—
221	C₂×C₄⋊C₄	—	47	2	no	—
222	C₂×C₄⋊C₄	—	47	2	no	—
223	C₂×C₄⋊C₄	—	47	2	no	—
224	C₂×C₄⋊C₄	—	47	2	no	—
225	C₂×C₄⋊C₄	—	94	4	no	—
226	C₂×C₄⋊C₄	—	94	1	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
229	C₂×C₂²⋊C₄	—	94	1	no	—
230	C₂×C₂²⋊C₄	—	94	2	no	—
231	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
232	C₂×C₄²	—	94	1	no	—
233	C₂×C₄²	—	94	2	no	—
234	C₂×C₄²	—	94	2	no	—
235	C₂×C₄²	—	94	2	no	—
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	#107	—	94	4	no	—
239	#106	—	94	4	no	—
240	#106	—	94	4	no	—
241	#106	—	94	4	no	—
242	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
243	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
244	#97	C₂×Dic₄₇	1	4	yes	complemented
245	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
246	#95	—	94	2	no	—
247	#95	—	94	2	no	—
248	#95	—	94	2	no	—
249	#95	—	94	2	no	—
250	#95	—	94	2	no	—
251	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	94	4	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	—	94	2	no	—
257	C₂²×C₄	—	94	2	no	—
258	C₂²×C₄	—	94	2	no	—
259	C₂²×C₄	—	376	2	no	—
260	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
261	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
262	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
263	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
271	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
272	C₄⋊C₄	—	47	4	no	—
273	C₄⋊C₄	—	47	4	no	—
274	C₄⋊C₄	—	94	4	no	—
275	C₄⋊C₄	—	94	2	no	—
276	C₄⋊C₄	—	94	2	no	—
277	C₄⋊C₄	—	94	2	no	—
278	C₄⋊C₄	—	94	2	no	—
279	C₄⋊C₄	—	94	2	no	—
280	C₄⋊C₄	—	94	2	no	—
281	C₄⋊C₄	—	376	2	no	—
282	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
283	C₂²⋊C₄	—	94	4	no	—
284	C₂²⋊C₄	—	376	2	no	—
285	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
286	C₄²	—	94	4	no	—
287	C₄²	—	376	2	no	—
288	C₄²	—	376	2	no	—
289	C₄²	—	376	2	no	—
290	C₄²	C₂×Dic₉₄	1	2	yes	—
291	C₄²	C₂×Dic₉₄	1	2	yes	—
292	C₄²	C₂²×Dic₄₇	1	2	yes	—
293	C₄²	C₂²×Dic₄₇	1	2	yes	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
299	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	94	4	no	—
310	C₂×C₄	—	376	4	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	—	188	2	no	—
321	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
322	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
323	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
324	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
326	C₂×C₄	#41×D₄₇	1	2	yes	—
327	C₂×C₄	#41×D₄₇	1	2	yes	—
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
336	C₂²	#15689635	1	1	yes	characteristic
337	C₂²	#15689635	1	1	yes	characteristic
338	C₂²	#15689935	1	2	yes	—
339	C₂²	#15689935	1	2	yes	—
340	C₄	—	2	4	no	—
341	C₄	—	2	4	no	—
342	C₄	—	2	2	no	—
343	C₄	—	376	2	no	—
344	C₄	—	376	2	no	—
345	C₄	—	188	4	no	—
346	C₄	—	188	4	no	—
347	C₄	#41×Dic₄₇	1	2	yes	—
348	C₄	#15689676	1	2	yes	—
349	C₂	—	4	2	no	—
350	C₂	#20245872	1	1	yes	characteristic
351	C₂	#20251983	1	2	yes	—
352	C₂	#20253641	1	1	yes	characteristic
353	C₂	#20253667	1	2	yes	—
354	C₂	#20253680	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#433	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#433	D₄₇
hypocenter	C₄₇	#21369
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #20244158	C₂	semidirect
4. #20244152	C₂	semidirect
7. #20243285	C₂	semidirect
8. C₄×#13093956	C₂	semidirect
31. C₄₇×#433	C₄	semidirect
69. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
71. #97×C₄₇	C₂×C₄	semidirect
72. C₄×C₈×C₄₇	C₂×C₄	semidirect
190. #433	Dic₄₇	semidirect
213. C₄₇	#21369	semidirect
242. C₄⋊C₈	C₂×Dic₄₇	semidirect
244. #97	C₂×Dic₄₇	semidirect
245. C₄×C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20245997	2
#20245744	2
#20244158	2
#20244152	2
#20243589	2
#20243583	2
#20243285	2
C₄×#13093956	2
C₂×C₄₇×#433	2
#21369	47