Subgroups — #26125217

Order 12032.

878 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	#20243127	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	#20243107	C₂	1	2	yes	maximal
4	#20243097	C₂	1	2	yes	maximal
5	C₂×C₄₇×#544	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15688892	—	2	1	no	—
9	#15688892	—	2	1	no	—
10	#15688890	—	2	2	no	—
11	#15688885	—	2	4	no	—
12	#15688882	—	2	4	no	—
13	C₄₇×#544	—	2	4	no	—
14	C₄₇×#544	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	complemented
17	C₂×#125×C₄₇	—	2	1	no	—
18	C₂×#125×C₄₇	C₄	1	2	yes	complemented
19	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	Q₈	1	4	yes	complemented
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093983	—	2	2	no	—
33	#13093983	Q₈	1	4	yes	complemented
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂²×Q₈×C₄₇	—	2	2	no	—
38	C₂³×C₄×C₄₇	—	2	1	no	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₇	—	2	8	no	—
41	C₄⋊Q₈×C₄₇	—	2	4	no	—
42	C₄⋊Q₈×C₄₇	—	2	4	no	—
43	#125×C₄₇	—	2	8	no	—
44	#125×C₄₇	—	2	4	no	—
45	#125×C₄₇	—	2	4	no	—
46	C₂²⋊Q₈×C₄₇	—	2	16	no	—
47	#117×C₄₇	—	2	2	no	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	D₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
62	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
63	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
64	C₂×C₄²×C₄₇	—	2	1	no	—
65	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
66	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₇	—	2	8	no	—
68	C₂×Q₈×C₄₇	—	8	4	no	—
69	C₂²×C₄×C₄₇	—	2	1	no	—
70	C₂²×C₄×C₄₇	—	2	1	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	8	2	no	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₇	#41	1	2	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×Dic₄₇	—	4	2	no	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	8	no	—
87	C₄⋊C₄×C₄₇	—	2	8	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	1	no	—
90	C₄⋊C₄×C₄₇	—	2	1	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
95	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
96	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₄₇	—	2	4	no	—
98	C₂²⋊C₄×C₄₇	—	2	4	no	—
99	C₄²×C₄₇	—	2	2	no	—
100	C₄²×C₄₇	—	2	2	no	—
101	C₄²×C₄₇	—	2	4	no	—
102	C₂²⋊Dic₄₇	—	2	4	no	—
103	C₂²⋊Dic₄₇	—	4	2	no	—
104	C₄⋊Dic₄₇	—	2	4	no	—
105	C₄⋊Dic₄₇	—	4	2	no	—
106	C₄×Dic₄₇	—	2	4	no	—
107	C₄×Dic₄₇	—	4	2	no	—
108	#10625329	—	2	4	no	—
109	#10625329	—	2	4	no	—
110	#10625329	—	4	2	no	—
111	#10625329	—	4	2	no	—
112	C₂×C₄₇⋊C₈	—	2	4	no	—
113	C₂×C₄₇⋊C₈	—	4	2	no	—
114	C₂×C₄₇⋊C₈	—	4	2	no	—
115	C₂³×C₄₇	—	4	1	no	—
116	C₂³×C₄₇	—	4	1	no	—
117	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂³×C₄₇	#120	1	2	yes	—
119	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
120	C₂³×C₄₇	#124	1	1	yes	characteristic
121	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
122	Q₈×C₄₇	—	8	8	no	—
123	C₂×C₄×C₄₇	—	2	2	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	4	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	8	4	no	—
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₄₇	#120	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₄₇	#124	1	2	yes	—
145	C₂×C₄×C₄₇	#124	1	2	yes	—
146	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
148	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
150	C₂×Dic₄₇	—	4	4	no	—
151	C₂×Dic₄₇	—	8	2	no	—
152	C₂×Dic₄₇	—	8	2	no	—
153	C₄₇⋊C₈	—	4	4	no	—
154	#11388	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	1	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	4	2	no	—
160	C₂²×C₄₇	—	4	1	no	—
161	C₂²×C₄₇	—	4	1	no	—
162	C₂²×C₄₇	#379	1	1	yes	characteristic
163	C₂²×C₄₇	#391	1	1	yes	characteristic
164	C₂²×C₄₇	#391	1	1	yes	characteristic
165	C₂²×C₄₇	#393	1	2	yes	—
166	C₂²×C₄₇	#395	1	2	yes	—
167	C₄×C₄₇	—	4	4	no	—
168	C₄×C₄₇	—	4	2	no	—
169	C₄×C₄₇	—	4	2	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	#391	1	2	yes	—
173	C₄×C₄₇	#391	1	2	yes	—
174	C₄×C₄₇	#393	1	2	yes	—
175	C₄×C₄₇	#395	1	2	yes	—
176	Dic₄₇	—	8	4	no	—
177	C₂×#544	D₄₇	1	1	yes	characteristic
178	#1742	—	47	1	no	—
179	#1740	—	47	1	no	—
180	#1736	—	47	2	no	—
181	#1733	—	47	2	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	—	2	2	no	—
186	C₂×C₄₇	#1446	1	1	yes	characteristic
187	C₂×C₄₇	#1795	1	1	yes	characteristic
188	C₂×C₄₇	#1796	1	1	yes	characteristic
189	C₂²×M₁₆	—	47	2	no	—
190	#544	—	2	4	no	—
191	#544	—	2	4	no	—
192	C₂×C₄⋊Q₈	—	2	1	no	—
193	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	complemented
194	C₂×#125	—	2	1	no	—
195	C₂×#125	Dic₄₇	1	2	yes	complemented
196	C₂×C₂²⋊Q₈	—	2	2	no	—
197	C₂×#117	—	47	2	no	—
198	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
200	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
201	#343	—	94	1	no	—
202	#343	—	94	1	no	—
203	#342	—	94	2	no	—
204	#340	—	94	4	no	—
205	#339	—	94	4	no	—
206	C₄₇	#11388	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	—	2	2	no	—
208	C₂³×C₄	—	2	1	no	—
209	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
210	C₂×M₁₆	—	47	4	no	—
211	C₂×M₁₆	—	94	2	no	—
212	C₂×M₁₆	—	94	2	no	—
213	C₂×M₁₆	—	94	2	no	—
214	C₂×M₁₆	—	94	2	no	—
215	C₂×M₁₆	—	94	2	no	—
216	C₂²×C₈	—	94	2	no	—
217	C₄⋊Q₈	—	2	8	no	—
218	C₄⋊Q₈	—	2	4	no	—
219	C₄⋊Q₈	—	2	4	no	—
220	#125	—	2	8	no	—
221	#125	—	2	4	no	—
222	#125	—	2	4	no	—
223	C₂²⋊Q₈	—	2	16	no	—
224	#117	—	2	2	no	—
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	47	4	no	—
228	#117	—	94	2	no	—
229	#117	—	94	2	no	—
230	#117	—	94	2	no	—
231	#117	C₂²⋊D₄₇	1	4	yes	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	94	2	no	—
238	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
239	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
240	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
243	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
244	C₂×C₂²⋊C₄	—	2	1	no	—
245	C₂×C₂²⋊C₄	—	94	2	no	—
246	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	—	94	2	no	—
249	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
250	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂×Q₈	—	2	8	no	—
252	C₂×Q₈	—	8	4	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	8	2	no	—
258	C₂²×C₄	—	188	2	no	—
259	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
260	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
261	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
262	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
263	C₂²×C₄	#10625354	1	2	yes	—
264	C₂²×C₄	#10625354	1	2	yes	—
265	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
268	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	M₁₆	—	94	4	no	—
270	M₁₆	—	94	4	no	—
271	M₁₆	—	188	2	no	—
272	M₁₆	—	188	2	no	—
273	C₂×C₈	—	94	4	no	—
274	C₂×C₈	—	188	2	no	—
275	C₂×C₈	—	188	2	no	—
276	C₄⋊C₄	—	2	8	no	—
277	C₄⋊C₄	—	2	8	no	—
278	C₄⋊C₄	—	2	8	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	94	4	no	—
286	C₄⋊C₄	—	188	2	no	—
287	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
288	C₄⋊C₄	C₂²⋊Dic₄₇	1	2	yes	—
289	C₄⋊C₄	C₂²⋊Dic₄₇	1	2	yes	—
290	C₂²⋊C₄	—	2	4	no	—
291	C₂²⋊C₄	—	2	4	no	—
292	C₂²⋊C₄	—	94	4	no	—
293	C₂²⋊C₄	—	188	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	4	no	—
297	C₄²	—	94	4	no	—
298	C₄²	—	188	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	1	no	—
301	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
302	C₂³	#13093993	1	2	yes	—
303	C₂³	#13094000	1	2	yes	—
304	C₂³	#13094013	1	1	yes	characteristic
305	C₂³	#13094022	1	1	yes	characteristic
306	Q₈	—	8	8	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	376	2	no	—
319	C₂×C₄	—	376	2	no	—
320	C₂×C₄	—	188	4	no	—
321	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
322	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
323	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
324	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
325	C₂×C₄	#13093993	1	2	yes	—
326	C₂×C₄	#13094000	1	2	yes	—
327	C₂×C₄	#13094000	1	2	yes	—
328	C₂×C₄	#13094000	1	2	yes	—
329	C₂×C₄	#13094000	1	2	yes	—
330	C₂×C₄	#13094000	1	2	yes	—
331	C₂×C₄	#13094013	1	2	yes	—
332	C₂×C₄	#13094013	1	2	yes	—
333	C₂×C₄	#13094013	1	1	yes	characteristic
334	C₂×C₄	#13094022	1	2	yes	—
335	C₂×C₄	#13094022	1	2	yes	—
336	C₂×C₄	#13094022	1	1	yes	characteristic
337	C₈	—	188	4	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	#15688979	1	1	yes	characteristic
346	C₂²	#15689030	1	1	yes	characteristic
347	C₂²	#15689030	1	1	yes	characteristic
348	C₂²	#15689034	1	2	yes	—
349	C₂²	#15689042	1	2	yes	—
350	C₄	—	4	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	376	4	no	—
356	C₄	#15689030	1	2	yes	—
357	C₄	#15689030	1	2	yes	—
358	C₄	#15689034	1	2	yes	—
359	C₄	#15689042	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	2	no	—
364	C₂	#20241659	1	1	yes	characteristic
365	C₂	#20243358	1	1	yes	characteristic
366	C₂	#20243363	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688979
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#544	C₂
socle	C₂²×C₄₇	#379
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#544	D₄₇
hypocenter	C₄₇	#11388
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#544

Complements

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

N	H ≅ G/N	type
16. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
18. C₂×#125×C₄₇	C₄	semidirect
28. C₂×#10625329	Q₈	semidirect
33. #13093983	Q₈	semidirect
193. C₂×C₄⋊Q₈	Dic₄₇	semidirect
195. C₂×#125	Dic₄₇	semidirect
206. C₄₇	#11388	semidirect

Maximal subgroups

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

subgroup	index
#20243127	2
#20243121	2
#20243107	2
#20243097	2
C₂×C₄₇×#544	2
#11388	47