Subgroups — #26115253

Order 12032.

720 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093935	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	D₄⋊(C₄₇⋊C₈)	—	2	8	no	—
12	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	—	2	1	no	—
15	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#390	—	2	1	no	—
17	C₄₇×#390	—	2	1	no	—
18	C₄₇×#390	C₄	1	2	yes	—
19	C₄₇×#384	C₄	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	complemented
26	#13093935	—	4	2	no	—
27	C₄×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×D₄×C₄₇	—	2	1	no	—
29	C₂²×D₄×C₄₇	—	2	1	no	—
30	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	—	2	2	no	—
32	#127×C₄₇	—	2	4	no	—
33	C₄×D₄×C₄₇	—	2	4	no	—
34	C₄×D₄×C₄₇	—	2	4	no	—
35	C₄×D₄×C₄₇	—	4	4	no	—
36	C₄×D₄×C₄₇	—	4	2	no	—
37	C₄×D₄×C₄₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂⁴×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	—	2	4	no	—
53	C₂×D₄×C₄₇	—	2	4	no	—
54	C₂×D₄×C₄₇	—	2	4	no	—
55	C₂×D₄×C₄₇	—	2	2	no	—
56	C₂×D₄×C₄₇	—	2	4	no	—
57	C₂×D₄×C₄₇	—	4	2	no	—
58	C₂×D₄×C₄₇	—	4	2	no	—
59	C₂×D₄×C₄₇	—	4	2	no	—
60	C₂×D₄×C₄₇	—	4	2	no	—
61	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	4	4	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	4	2	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₂²⋊C₄×C₄₇	—	4	4	no	—
78	C₂²⋊C₄×C₄₇	—	4	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	2	no	—
80	C₂²⋊C₄×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	4	1	no	—
86	C₄²×C₄₇	D₈	1	2	yes	—
87	C₄²×C₄₇	SD₁₆	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	#41	1	2	yes	—
90	C₂×C₄₇⋊C₈	—	2	4	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	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	D₄×C₄₇	—	2	8	no	—
105	D₄×C₄₇	—	4	8	no	—
106	D₄×C₄₇	—	4	8	no	—
107	D₄×C₄₇	—	4	4	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	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	8	4	no	—
123	C₂×C₄×C₄₇	—	8	4	no	—
124	C₂×C₄×C₄₇	#102	1	2	yes	—
125	C₂×C₄×C₄₇	#102	1	2	yes	—
126	C₂×C₄×C₄₇	#104	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₇	#121	1	2	yes	—
129	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₄₇	#131	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
136	C₄₇⋊C₈	—	4	4	no	—
137	C₄₇⋊C₈	—	4	4	no	—
138	#9678	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	—	4	4	no	—
141	C₂²×C₄₇	—	4	4	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	—	8	2	no	—
144	C₂²×C₄₇	—	8	2	no	—
145	C₂²×C₄₇	—	8	2	no	—
146	C₂²×C₄₇	#390	1	1	yes	characteristic
147	C₂²×C₄₇	#407	1	1	yes	characteristic
148	C₂²×C₄₇	#408	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
150	C₂²×C₄₇	#418	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
152	C₂²×C₄₇	#421	1	1	yes	characteristic
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	4	1	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	#435	1	4	yes	—
161	C₄×C₄₇	#459	1	2	yes	—
162	C₄×C₄₇	#460	1	2	yes	—
163	C₄×#127	D₄₇	1	1	yes	characteristic
164	#1767	—	47	1	no	—
165	C₄×C₄⋊C₈	—	47	1	no	—
166	C₂×D₄⋊C₈	—	47	4	no	—
167	C₂×C₄₇	—	8	4	no	—
168	C₂×C₄₇	#1486	1	1	yes	characteristic
169	C₂×C₄₇	#1490	1	1	yes	characteristic
170	C₂×C₄₇	#1492	1	1	yes	characteristic
171	C₂×C₄₇	#1496	1	1	yes	characteristic
172	C₂×C₄₇	#1955	1	1	yes	characteristic
173	C₂×C₄₇	#1980	1	1	yes	characteristic
174	C₂×C₄₇	#1982	1	1	yes	characteristic
175	C₂×#127	Dic₄₇	1	1	yes	characteristic
176	C₂×C₄×D₄	—	2	1	no	—
177	C₂×C₄×D₄	—	2	1	no	—
178	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
179	C₂×C₄⋊C₈	—	47	2	no	—
180	C₂×C₄⋊C₈	—	94	1	no	—
181	C₂×#97	—	94	1	no	—
182	C₂×C₄×C₈	—	47	2	no	—
183	#390	—	2	1	no	—
184	#390	—	2	1	no	—
185	#390	Dic₄₇	1	2	yes	—
186	#384	Dic₄₇	1	1	yes	characteristic
187	C₄³	C₂×D₄₇	1	1	yes	characteristic
188	#336	—	94	1	no	—
189	#336	—	94	1	no	—
190	D₄⋊C₈	—	94	8	no	—
191	C₄₇	#9678	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
195	C₂³×C₄	—	2	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	#127	—	2	4	no	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	—	4	4	no	—
202	C₄×D₄	—	4	2	no	—
203	C₄×D₄	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
217	C₄⋊C₈	—	47	4	no	—
218	C₄⋊C₈	—	94	4	no	—
219	C₄⋊C₈	—	94	2	no	—
220	#97	—	188	2	no	—
221	C₄×C₈	—	94	4	no	—
222	C₂⁴	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	#10625329	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	188	2	no	—
248	C₂×C₈	—	188	2	no	—
249	C₂×C₈	—	188	2	no	—
250	C₂×C₈	—	188	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	#10625334	1	2	yes	—
264	C₄²	#10625335	1	2	yes	—
265	C₄²	D₄×D₄₇	1	2	yes	—
266	C₄²	#10625352	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
275	D₄	—	2	8	no	—
276	D₄	—	4	8	no	—
277	D₄	—	4	8	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	#13093945	1	2	yes	—
296	C₂×C₄	#13093945	1	2	yes	—
297	C₂×C₄	#13093951	1	2	yes	—
298	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
299	C₂×C₄	#13093995	1	2	yes	—
300	C₂×C₄	#13094011	1	1	yes	characteristic
301	C₂×C₄	#13094020	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625329	1	2	yes	—
303	C₂×C₄	#13094031	1	2	yes	—
304	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
305	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
306	C₂×C₄	#13094050	1	2	yes	—
307	C₈	—	188	4	no	—
308	C₈	—	188	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	#15689026	1	1	yes	characteristic
317	C₂²	#15689087	1	1	yes	characteristic
318	C₂²	#15689090	1	1	yes	characteristic
319	C₂²	C₂×#13093945	1	1	yes	characteristic
320	C₂²	#15689128	1	1	yes	characteristic
321	C₂²	C₂×#13093951	1	1	yes	characteristic
322	C₂²	#15689140	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	8	2	no	—
330	C₄	#15689195	1	4	yes	—
331	C₄	#15689315	1	2	yes	—
332	C₄	#15689320	1	2	yes	—
333	C₂	—	8	4	no	—
334	C₂	#20241813	1	1	yes	characteristic
335	C₂	#20241831	1	1	yes	characteristic
336	C₂	#20241839	1	1	yes	characteristic
337	C₂	#20241857	1	1	yes	characteristic
338	C₂	#20244187	1	1	yes	characteristic
339	C₂	#20244308	1	1	yes	characteristic
340	C₂	#20244316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#9678
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20243241	C₂	semidirect
2. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. C₂×D₄⋊(C₄₇⋊C₈)	C₂	semidirect
25. C₄⋊(C₄₇⋊C₈)	D₄	semidirect
191. C₄₇	#9678	semidirect

Maximal subgroups

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

subgroup	index
#20243241	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×D₄⋊(C₄₇⋊C₈)	2
C₄×#127×C₄₇	2
#9678	47