Subgroups — #26122173

Order 12032.

982 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093955	C₂	1	2	yes	maximal
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688885	C₂	1	2	yes	maximal
4	#20241618	C₂	1	1	yes	maximal, characteristic
5	C₄₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×#13093955	—	2	4	no	—
10	C₂×#13093955	—	2	4	no	—
11	C₂×#13093955	C₄	1	8	yes	complemented
12	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₄₇⋊#95	—	2	2	no	—
15	#15688885	—	2	4	no	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#95×C₄₇	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
28	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
29	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
36	#13093955	—	2	16	no	—
37	C₄×C₄₇⋊C₈	—	2	8	no	—
38	C₄₇⋊#95	—	4	4	no	—
39	C₂³×C₄×C₄₇	—	2	2	no	—
40	C₂³×C₄×C₄₇	D₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
52	#95×C₄₇	—	4	4	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
63	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	4	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
95	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
96	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	8	yes	—
97	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
98	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	C₄×D₄	1	2	yes	—
102	C₂³×C₄₇	#121	1	2	yes	—
103	C₂³×C₄₇	#127	1	1	yes	characteristic
104	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
106	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
107	C₂³×C₄₇	#135	1	4	yes	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	8	no	—
114	C₂×C₄×C₄₇	—	8	4	no	—
115	C₂×C₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	—	8	2	no	—
117	C₂×C₄×C₄₇	#106	1	8	yes	complemented
118	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	#121	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	#124	1	2	yes	—
133	C₂×C₄×C₄₇	#125	1	2	yes	—
134	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	C₂×Dic₄₇	—	8	4	no	—
139	C₂×Dic₄₇	—	8	2	no	—
140	C₂×Dic₄₇	—	8	2	no	—
141	C₄₇⋊C₈	—	2	4	no	—
142	C₄₇⋊C₈	#95	1	8	yes	complemented
143	#10861	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₇	#380	1	1	yes	characteristic
146	C₂²×C₄₇	#384	1	1	yes	characteristic
147	C₂²×C₄₇	#389	1	2	yes	—
148	C₂²×C₄₇	#390	1	2	yes	—
149	C₂²×C₄₇	C₂×#106	1	2	yes	—
150	C₂²×C₄₇	#427	1	2	yes	—
151	C₂²×C₄₇	C₄×SD₁₆	1	4	yes	—
152	C₂²×C₄₇	C₄×SD₁₆	1	4	yes	—
153	C₂²×C₄₇	#465	1	4	yes	—
154	C₂²×C₄₇	#465	1	4	yes	—
155	C₂²×C₄₇	#492	1	1	yes	characteristic
156	C₂²×C₄₇	#492	1	1	yes	characteristic
157	C₂²×C₄₇	#495	1	2	yes	—
158	C₂²×C₄₇	#498	1	2	yes	—
159	C₂²×C₄₇	#499	1	2	yes	—
160	C₄×C₄₇	—	2	8	no	—
161	C₄×C₄₇	—	8	4	no	—
162	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
163	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₄₇	#380	1	1	yes	characteristic
165	C₄×C₄₇	#382	1	1	yes	characteristic
166	C₄×C₄₇	#389	1	2	yes	—
167	C₄×C₄₇	#391	1	2	yes	—
168	Dic₄₇	—	8	4	no	—
169	C₂²×#106	—	47	2	no	—
170	C₂²×C₄×C₈	—	47	1	no	—
171	C₂×#340	—	47	2	no	—
172	C₄⋊#95	—	47	1	no	—
173	C₄⋊#95	D₄₇	1	1	yes	characteristic
174	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
175	C₂×C₄₇	C₄×#106	1	2	yes	—
176	C₂×C₄₇	#1807	1	2	yes	—
177	C₂×C₄₇	#1844	1	1	yes	characteristic
178	C₂×C₄₇	#1846	1	1	yes	characteristic
179	C₂×C₄₇	#1941	1	4	yes	—
180	C₂×C₄₇	#1968	1	2	yes	—
181	C₂×C₄₇	#1968	1	2	yes	—
182	C₂³×C₈	—	47	1	no	—
183	C₂³×C₈	—	47	1	no	—
184	C₂²×C₄⋊C₄	—	47	2	no	—
185	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
186	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
187	C₂×#106	—	47	8	no	—
188	C₂×#106	—	94	4	no	—
189	C₂×#106	—	94	4	no	—
190	C₂×C₄×C₈	—	47	4	no	—
191	C₂×C₄×C₈	—	94	4	no	—
192	C₂×#95	—	2	2	no	—
193	C₂×#95	—	94	2	no	—
194	#340	—	94	4	no	—
195	C₄₇	#10861	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	2	2	no	—
197	C₂³×C₄	—	94	2	no	—
198	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
199	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	47	4	no	—
202	C₂²×C₈	—	47	4	no	—
203	C₂²×C₈	—	47	2	no	—
204	C₂²×C₈	—	47	2	no	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	—	47	2	no	—
207	C₂²×C₈	—	47	2	no	—
208	C₂²×C₈	—	94	2	no	—
209	C₂²×C₈	—	94	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	47	4	no	—
216	C₂×C₄⋊C₄	—	94	2	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄⋊C₄	—	94	2	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	94	2	no	—
221	C₂×C₄⋊C₄	C₄×D₄₇	1	4	yes	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	Dic₉₄	1	2	yes	—
225	C₂×C₄²	C₄×D₄₇	1	2	yes	—
226	#106	—	94	16	no	—
227	C₄×C₈	—	94	8	no	—
228	#95	—	4	4	no	—
229	#95	—	188	4	no	—
230	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	—	188	4	no	—
242	C₂²×C₄	—	188	4	no	—
243	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
244	C₂²×C₄	#10625335	1	4	yes	—
245	C₂²×C₄	#10625336	1	4	yes	—
246	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
249	C₂²×C₄	#10625350	1	2	yes	—
250	C₂²×C₄	D₄×D₄₇	1	2	yes	—
251	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	#10625352	1	2	yes	—
254	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
256	C₂×C₈	—	47	4	no	—
257	C₂×C₈	—	47	4	no	—
258	C₂×C₈	—	47	8	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	47	8	no	—
261	C₂×C₈	—	47	4	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	94	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	94	4	no	—
271	C₄⋊C₄	—	94	4	no	—
272	C₄⋊C₄	—	188	2	no	—
273	C₄⋊C₄	—	188	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	4	no	—
277	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
278	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
279	C₂³	#13093996	1	2	yes	—
280	C₂³	#13094020	1	1	yes	characteristic
281	C₂³	#13094022	1	1	yes	characteristic
282	C₂³	C₂×#10625336	1	2	yes	—
283	C₂³	C₂×#10625335	1	2	yes	—
284	C₂³	#13094044	1	4	yes	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	376	2	no	—
295	C₂×C₄	—	376	2	no	—
296	C₂×C₄	—	376	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	—	188	8	no	—
299	C₂×C₄	#13093955	1	8	yes	complemented
300	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
303	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
305	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	#13093980	1	1	yes	characteristic
307	C₂×C₄	#13093981	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
309	C₂×C₄	#13093989	1	2	yes	—
310	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
311	C₂×C₄	#13093997	1	2	yes	—
312	C₂×C₄	#13094001	1	2	yes	—
313	C₂×C₄	#13094002	1	2	yes	—
314	C₂×C₄	#13094010	1	2	yes	—
315	C₂×C₄	#13094014	1	2	yes	—
316	C₂×C₄	#13094023	1	1	yes	characteristic
317	C₂×C₄	#13094023	1	1	yes	characteristic
318	C₈	—	47	8	no	—
319	C₈	—	94	4	no	—
320	C₂²	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
321	C₂²	#15688981	1	1	yes	characteristic
322	C₂²	#15688992	1	1	yes	characteristic
323	C₂²	#15689018	1	2	yes	—
324	C₂²	#15689023	1	2	yes	—
325	C₂²	C₂×#13093955	1	2	yes	—
326	C₂²	#15689159	1	2	yes	—
327	C₂²	C₄×#10625335	1	4	yes	—
328	C₂²	C₄×#10625336	1	4	yes	—
329	C₂²	#15689350	1	4	yes	—
330	C₂²	#15689354	1	4	yes	—
331	C₂²	#15689495	1	1	yes	characteristic
332	C₂²	#15689497	1	1	yes	characteristic
333	C₂²	#15689505	1	2	yes	—
334	C₂²	#15689519	1	2	yes	—
335	C₂²	#15689522	1	2	yes	—
336	C₄	—	2	8	no	—
337	C₄	—	8	4	no	—
338	C₄	—	376	4	no	—
339	C₄	#95×D₄₇	1	1	yes	characteristic
340	C₄	#15688974	1	1	yes	characteristic
341	C₄	#15688982	1	1	yes	characteristic
342	C₄	#15688987	1	1	yes	characteristic
343	C₄	#15689020	1	2	yes	—
344	C₄	#15689027	1	2	yes	—
345	C₂	Dic₄₇⋊#95	1	1	yes	characteristic
346	C₂	C₄×#13093955	1	2	yes	—
347	C₂	#20243411	1	2	yes	—
348	C₂	#20243544	1	1	yes	characteristic
349	C₂	#20243550	1	1	yes	characteristic
350	C₂	#20244113	1	4	yes	—
351	C₂	#20244241	1	2	yes	—
352	C₂	#20244245	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×C₄⋊#95	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄⋊#95	D₄₇
hypocenter	C₄₇	#10861
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₄⋊#95

Complements

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

N	H ≅ G/N	type
11. C₂×#13093955	C₄	semidirect
13. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
117. C₂×C₄×C₄₇	#106	semidirect
142. C₄₇⋊C₈	#95	semidirect
195. C₄₇	#10861	semidirect
299. C₂×C₄	#13093955	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093955	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688885	2
#20241618	2
C₄₇×C₄⋊#95	2
#10861	47