Subgroups — #26116318

Order 12032.

680 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241231	C₂	1	4	yes	maximal
4	C₂×C₄₇×#391	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#391	—	2	1	no	—
15	C₄₇×#391	—	2	1	no	—
16	C₄₇×#391	—	2	1	no	—
17	C₄₇×#391	—	2	1	no	—
18	C₂×#95×C₄₇	C₄	1	2	yes	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	Q₈	1	4	yes	complemented
26	C₄×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₄₇	—	2	1	no	—
28	C₂²×Q₈×C₄₇	—	2	1	no	—
29	C₂²×Q₈×C₄₇	—	2	2	no	—
30	C₂²×Q₈×C₄₇	—	4	2	no	—
31	C₂²×Q₈×C₄₇	—	4	2	no	—
32	C₂²×Q₈×C₄₇	C₈	1	4	yes	complemented
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	2	no	—
48	#95×C₄₇	—	2	2	no	—
49	#95×C₄₇	—	2	2	no	—
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	2	4	no	—
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	4	2	no	—
54	C₂×Q₈×C₄₇	—	4	2	no	—
55	C₂×Q₈×C₄₇	—	4	8	no	—
56	C₂×Q₈×C₄₇	—	4	8	no	—
57	C₂×Q₈×C₄₇	—	4	2	no	—
58	C₂×Q₈×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	—	4	1	no	—
70	C₂²×C₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
80	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	4	1	no	—
90	C₄²×C₄₇	—	4	1	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
103	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂³×C₄₇	#131	1	2	yes	—
106	Q₈×C₄₇	—	4	8	no	—
107	Q₈×C₄₇	—	4	8	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	8	4	no	—
133	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
135	C₂×C₄×C₄₇	#99	1	2	yes	—
136	C₂×C₄×C₄₇	#100	1	2	yes	—
137	C₂×C₄×C₄₇	#101	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
141	C₄₇⋊C₈	—	4	4	no	—
142	C₄₇⋊C₈	—	8	2	no	—
143	#9878	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	#391	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₂²×C₄₇	#408	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
156	C₂²×C₄₇	#410	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
159	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
160	C₂²×C₄₇	#446	1	2	yes	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	4	4	no	—
167	C₄×C₄₇	—	4	1	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₄×C₄₇	#323	1	4	yes	—
170	C₂×#391	D₄₇	1	1	yes	characteristic
171	#1797	—	47	1	no	—
172	C₄×C₂²⋊C₈	—	47	1	no	—
173	#1280	—	47	4	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	—	2	2	no	—
176	C₂×C₄₇	C₂×#323	1	1	yes	characteristic
177	C₂×C₄₇	#1461	1	1	yes	characteristic
178	C₂×C₄₇	#1469	1	1	yes	characteristic
179	C₂×C₄₇	#1472	1	1	yes	characteristic
180	C₂×C₄₇	#1984	1	1	yes	characteristic
181	C₂×C₄₇	#1996	1	1	yes	characteristic
182	C₂×C₄₇	#1997	1	1	yes	characteristic
183	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	94	1	no	—
187	C₂×C₂²⋊C₈	—	47	2	no	—
188	C₂×C₂²⋊C₈	—	47	2	no	—
189	C₂×C₄×C₈	—	94	1	no	—
190	#391	—	2	1	no	—
191	#391	—	2	1	no	—
192	#391	—	2	1	no	—
193	#391	—	2	1	no	—
194	C₂×#95	Dic₄₇	1	2	yes	—
195	C₂×#95	C₂×D₄₇	1	2	yes	—
196	#336	—	94	1	no	—
197	#336	—	94	1	no	—
198	C₄₇	#9878	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×Q₈	—	4	2	no	—
203	C₂²×Q₈	—	4	2	no	—
204	C₂²×Q₈	C₄₇⋊C₈	1	4	yes	complemented
205	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
207	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
208	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
209	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
210	C₂²×C₈	—	94	2	no	—
211	C₂²×C₈	—	94	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	4	2	no	—
215	C₂×C₄²	—	4	2	no	—
216	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₄⋊C₈	—	188	2	no	—
221	C₂²⋊C₈	—	47	4	no	—
222	C₂²⋊C₈	—	94	2	no	—
223	C₂²⋊C₈	—	94	4	no	—
224	C₄×C₈	—	188	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
254	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
255	C₂²×C₄	#10625329	1	2	yes	—
256	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
260	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
261	C₂²×C₄	#10625354	1	2	yes	—
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	188	2	no	—
266	C₂×C₈	—	188	2	no	—
267	C₂×C₈	—	188	2	no	—
268	C₂×C₈	—	188	2	no	—
269	C₂×C₈	—	188	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
281	C₂³	Q₈×Dic₄₇	1	2	yes	—
282	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
283	C₂³	C₂×#10625329	1	1	yes	characteristic
284	C₂³	#13094031	1	2	yes	—
285	Q₈	—	4	8	no	—
286	Q₈	—	4	8	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
313	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
314	C₂×C₄	#13093939	1	2	yes	—
315	C₂×C₄	#13093941	1	2	yes	—
316	C₂×C₄	#13093943	1	2	yes	—
317	C₂×C₄	#13094000	1	2	yes	—
318	C₂×C₄	#13094013	1	1	yes	characteristic
319	C₂×C₄	#13094022	1	1	yes	characteristic
320	C₈	—	376	2	no	—
321	C₈	—	188	4	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	#15689030	1	1	yes	characteristic
331	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
332	C₂²	#15689090	1	1	yes	characteristic
333	C₂²	C₂×#13093939	1	1	yes	characteristic
334	C₂²	#15689097	1	1	yes	characteristic
335	C₂²	C₂×#13093941	1	1	yes	characteristic
336	C₂²	C₂×#13093943	1	1	yes	characteristic
337	C₂²	Q₈×C₄₇⋊C₈	1	2	yes	—
338	C₂²	#15689254	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	4	no	—
345	C₄	—	4	1	no	—
346	C₄	—	8	2	no	—
347	C₄	#15688847	1	4	yes	—
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	C₂×#15688847	1	1	yes	characteristic
351	C₂	#20241712	1	1	yes	characteristic
352	C₂	#20241740	1	1	yes	characteristic
353	C₂	#20241752	1	1	yes	characteristic
354	C₂	#20244325	1	1	yes	characteristic
355	C₂	#20244374	1	1	yes	characteristic
356	C₂	#20244378	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#391	D₄₇
hypocenter	C₄₇	#9878
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#391

Complements

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

N	H ≅ G/N	type
25. C₂²⋊(C₄₇⋊C₈)	Q₈	semidirect
32. C₂²×Q₈×C₄₇	C₈	semidirect
198. C₄₇	#9878	semidirect
204. C₂²×Q₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20241231	2
C₂×C₄₇×#391	2
#9878	47