Subgroups — #26104480

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243531	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241209	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×#13093935	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₄₇	—	2	1	no	—
13	C₂×C₄×D₄×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
15	C₄₇×#390	—	2	1	no	—
16	C₄₇×#390	—	2	1	no	—
17	C₄₇×#390	C₂²	1	2	yes	—
18	C₄₇×#384	C₄	1	1	yes	characteristic
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
23	#13093935	—	4	2	no	—
24	C₂²×D₄×C₄₇	—	2	1	no	—
25	C₂²×D₄×C₄₇	—	2	1	no	—
26	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	—	2	2	no	—
28	#127×C₄₇	—	2	4	no	—
29	C₄×D₄×C₄₇	—	4	4	no	—
30	C₄×D₄×C₄₇	—	4	4	no	—
31	C₄×D₄×C₄₇	—	4	2	no	—
32	C₄×D₄×C₄₇	—	4	2	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
46	C₂⁴×C₄₇	—	2	2	no	—
47	C₂×D₄×C₄₇	—	2	4	no	—
48	C₂×D₄×C₄₇	—	2	4	no	—
49	C₂×D₄×C₄₇	—	2	2	no	—
50	C₂×D₄×C₄₇	—	4	2	no	—
51	C₂×D₄×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	4	2	no	—
53	C₂×D₄×C₄₇	—	4	2	no	—
54	C₂×D₄×C₄₇	—	4	2	no	—
55	C₂×D₄×C₄₇	—	4	2	no	—
56	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	4	2	no	—
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₄⋊C₄×C₄₇	—	4	2	no	—
71	C₂²⋊C₄×C₄₇	—	4	4	no	—
72	C₂²⋊C₄×C₄₇	—	4	2	no	—
73	C₂²⋊C₄×C₄₇	—	4	2	no	—
74	C₂²⋊C₄×C₄₇	—	4	4	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₄²×C₄₇	D₈	1	2	yes	—
82	C₄²×C₄₇	SD₁₆	1	2	yes	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	D₄×C₄₇	—	4	4	no	—
98	D₄×C₄₇	—	4	4	no	—
99	D₄×C₄₇	—	4	8	no	—
100	D₄×C₄₇	—	4	8	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	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	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	8	8	no	—
119	C₂×C₄×C₄₇	#100	1	2	yes	—
120	C₂×C₄×C₄₇	#102	1	2	yes	—
121	C₂×C₄×C₄₇	#102	1	2	yes	—
122	C₂×C₄×C₄₇	#104	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
127	C₄₇⋊C₈	—	4	4	no	—
128	C₄₇⋊C₈	—	8	2	no	—
129	#7621	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	4	4	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	—	4	4	no	—
133	C₂²×C₄₇	—	8	2	no	—
134	C₂²×C₄₇	—	8	2	no	—
135	C₂²×C₄₇	—	8	2	no	—
136	C₂²×C₄₇	—	8	2	no	—
137	C₂²×C₄₇	#407	1	1	yes	characteristic
138	C₂²×C₄₇	#410	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
141	C₂²×C₄₇	#418	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
143	C₂²×C₄₇	#421	1	1	yes	characteristic
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	8	2	no	—
153	C₄×#127	D₄₇	1	1	yes	characteristic
154	#1839	—	47	1	no	—
155	#1767	—	47	1	no	—
156	#1270	—	47	4	no	—
157	C₂×C₄₇	—	8	4	no	—
158	C₂×C₄₇	#1466	1	1	yes	characteristic
159	C₂×C₄₇	#1486	1	1	yes	characteristic
160	C₂×C₄₇	#1490	1	1	yes	characteristic
161	C₂×C₄₇	#1502	1	1	yes	characteristic
162	C₂×C₄₇	#1514	1	1	yes	characteristic
163	C₂×C₄₇	#1530	1	1	yes	characteristic
164	C₂×C₄₇	#1545	1	1	yes	characteristic
165	C₂×#127	Dic₄₇	1	1	yes	characteristic
166	C₂×C₄×D₄	—	2	1	no	—
167	C₂×C₄×D₄	—	2	1	no	—
168	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
169	C₂×C₄⋊C₈	—	94	1	no	—
170	C₂×#97	—	94	1	no	—
171	#390	—	2	1	no	—
172	#390	—	2	1	no	—
173	#390	C₂×D₄₇	1	2	yes	—
174	#384	Dic₄₇	1	1	yes	characteristic
175	C₄³	C₂×D₄₇	1	1	yes	characteristic
176	#336	—	47	2	no	—
177	#336	—	47	2	no	—
178	#336	—	94	1	no	—
179	#336	—	94	1	no	—
180	C₄₇	#7621	1	1	yes	characteristic, complemented, Sylow
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂²×C₈	—	94	2	no	—
186	C₂²×C₈	—	94	2	no	—
187	#127	—	2	4	no	—
188	C₄×D₄	—	4	4	no	—
189	C₄×D₄	—	4	4	no	—
190	C₄×D₄	—	4	2	no	—
191	C₄×D₄	—	4	2	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊C₈	—	94	4	no	—
206	#97	—	188	2	no	—
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	#10625329	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	188	2	no	—
231	C₂×C₈	—	188	2	no	—
232	C₂×C₈	—	188	2	no	—
233	C₂×C₈	—	188	2	no	—
234	C₂×C₈	—	188	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	#10625334	1	2	yes	—
249	C₄²	#10625335	1	2	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
258	D₄	—	4	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	8	no	—
261	D₄	—	4	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	#13093941	1	2	yes	—
281	C₂×C₄	#13093945	1	2	yes	—
282	C₂×C₄	#13093945	1	2	yes	—
283	C₂×C₄	#13093951	1	2	yes	—
284	C₂×C₄	C₂×#10625329	1	2	yes	—
285	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
287	C₂×C₄	#13094050	1	2	yes	—
288	C₈	—	376	2	no	—
289	C₈	—	188	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	#15689087	1	1	yes	characteristic
298	C₂²	#15689097	1	1	yes	characteristic
299	C₂²	C₂×#13093941	1	1	yes	characteristic
300	C₂²	C₂×#13093945	1	1	yes	characteristic
301	C₂²	#15689128	1	1	yes	characteristic
302	C₂²	C₂×#13093951	1	1	yes	characteristic
303	C₂²	#15689140	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	2	no	—
313	C₂	—	8	4	no	—
314	C₂	#20241729	1	1	yes	characteristic
315	C₂	#20241813	1	1	yes	characteristic
316	C₂	#20241831	1	1	yes	characteristic
317	C₂	#20241883	1	1	yes	characteristic
318	C₂	#20241935	1	1	yes	characteristic
319	C₂	#20241999	1	1	yes	characteristic
320	C₂	#20242056	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
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	#7621
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20243531	C₂	semidirect
2. #20243241	C₂	semidirect
3. #20241209	C₂	semidirect
180. C₄₇	#7621	semidirect

Maximal subgroups

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

subgroup	index
#20243531	2
#20243241	2
#20241209	2
C₄×#127×C₄₇	2
#7621	47