Subgroups — #26116264

Order 12032.

528 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×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×C₄×C₄₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
13	C₄₇×#391	—	2	1	no	—
14	C₄₇×#389	—	2	1	no	—
15	C₄₇×#389	C₄	1	2	yes	—
16	C₄₇×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
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	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	Q₈	1	4	yes	complemented
25	C₄×C₄₇⋊C₈	—	2	4	no	—
26	C₄×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₄₇	—	2	1	no	—
28	C₄⋊Q₈×C₄₇	—	2	2	no	—
29	C₄⋊Q₈×C₄₇	C₈	1	4	yes	complemented
30	C₄×Q₈×C₄₇	—	4	2	no	—
31	C₄×Q₈×C₄₇	—	4	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×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	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	2	no	—
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂×Q₈×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₄₇	—	2	4	no	—
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₄⋊C₄×C₄₇	—	4	4	no	—
65	C₄⋊C₄×C₄₇	—	4	2	no	—
66	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
67	C₄²×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	4	1	no	—
71	C₄²×C₄₇	—	4	2	no	—
72	C₄²×C₄₇	—	4	2	no	—
73	C₄²×C₄₇	C₂×C₈	1	2	yes	—
74	C₄²×C₄₇	M₁₆	1	2	yes	—
75	C₄²×C₄₇	D₈	1	2	yes	—
76	C₄²×C₄₇	SD₁₆	1	2	yes	—
77	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
78	C₄²×C₄₇	#41	1	2	yes	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₄₇	—	4	8	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₄₇	#102	1	2	yes	—
105	C₂×C₄×C₄₇	#102	1	2	yes	—
106	C₂×C₄×C₄₇	#104	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#131	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#137	1	2	yes	—
117	C₄₇⋊C₈	—	4	4	no	—
118	C₄₇⋊C₈	—	4	4	no	—
119	#9865	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	#391	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₄₇	#408	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
124	C₂²×C₄₇	#419	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
126	C₂²×C₄₇	#421	1	1	yes	characteristic
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	4	4	no	—
132	C₄×C₄₇	—	4	1	no	—
133	C₄×C₄₇	—	8	2	no	—
134	C₄×C₄₇	D₄⋊C₈	1	4	yes	—
135	C₄×C₄₇	C₈×Q₈	1	2	yes	—
136	C₄×C₄₇	#446	1	2	yes	—
137	C₄×C₄₇	#474	1	2	yes	—
138	C₄×C₄₇	#475	1	2	yes	—
139	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
140	C₄×C₄⋊C₈	—	47	1	no	—
141	C₄×C₄⋊C₈	—	47	1	no	—
142	#1279	—	47	4	no	—
143	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
144	C₂×C₄₇	#1479	1	1	yes	characteristic
145	C₂×C₄₇	#1480	1	1	yes	characteristic
146	C₂×C₄₇	#1485	1	1	yes	characteristic
147	C₂×C₄₇	#1984	1	1	yes	characteristic
148	C₂×C₄₇	#1986	1	1	yes	characteristic
149	C₂×C₄₇	#1988	1	1	yes	characteristic
150	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
151	C₂×C₄×Q₈	—	2	1	no	—
152	C₂×C₄⋊C₈	—	47	2	no	—
153	C₂×C₄⋊C₈	—	94	1	no	—
154	C₂×C₄×C₈	—	47	2	no	—
155	C₂×C₄×C₈	—	94	1	no	—
156	#391	—	2	1	no	—
157	#389	—	2	1	no	—
158	#389	Dic₄₇	1	2	yes	—
159	#384	Dic₄₇	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	2	1	no	—
161	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
162	C₄³	C₂×D₄₇	1	1	yes	characteristic
163	#336	—	94	1	no	—
164	#336	—	94	1	no	—
165	C₄₇	#9865	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×C₈	—	94	2	no	—
168	C₂²×C₈	—	94	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	C₄₇⋊C₈	1	4	yes	complemented
171	C₄×Q₈	—	4	2	no	—
172	C₄×Q₈	—	4	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄⋊C₈	—	47	4	no	—
188	C₄⋊C₈	—	94	4	no	—
189	C₄⋊C₈	—	94	2	no	—
190	C₄×C₈	—	94	4	no	—
191	C₄×C₈	—	188	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂×C₈	—	94	4	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	188	2	no	—
209	C₂×C₈	—	188	2	no	—
210	C₂×C₈	—	188	2	no	—
211	C₂×C₈	—	188	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
226	C₄²	#10625329	1	2	yes	—
227	C₄²	#10625334	1	2	yes	—
228	C₄²	#10625335	1	2	yes	—
229	C₄²	Q₈×D₄₇	1	2	yes	—
230	C₄²	#10625354	1	2	yes	—
231	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
232	Q₈	—	4	8	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
249	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
250	C₂×C₄	#13093945	1	2	yes	—
251	C₂×C₄	#13093945	1	2	yes	—
252	C₂×C₄	#13093951	1	2	yes	—
253	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
254	C₂×C₄	#13094000	1	2	yes	—
255	C₂×C₄	#13094013	1	1	yes	characteristic
256	C₂×C₄	#13094022	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
259	C₂×C₄	#13094031	1	2	yes	—
260	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
261	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
262	C₂×C₄	#13094055	1	2	yes	—
263	C₈	—	188	4	no	—
264	C₈	—	188	4	no	—
265	C₂²	#15689030	1	1	yes	characteristic
266	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
267	C₂²	#15689090	1	1	yes	characteristic
268	C₂²	C₂×#13093945	1	1	yes	characteristic
269	C₂²	#15689132	1	1	yes	characteristic
270	C₂²	C₂×#13093951	1	1	yes	characteristic
271	C₂²	#15689140	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	1	no	—
278	C₄	—	8	2	no	—
279	C₄	D₄⋊(C₄₇⋊C₈)	1	4	yes	—
280	C₄	Q₈×C₄₇⋊C₈	1	2	yes	—
281	C₄	#15689254	1	2	yes	—
282	C₄	#15689400	1	2	yes	—
283	C₄	#15689405	1	2	yes	—
284	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
285	C₂	#20241783	1	1	yes	characteristic
286	C₂	#20241787	1	1	yes	characteristic
287	C₂	#20241809	1	1	yes	characteristic
288	C₂	#20244325	1	1	yes	characteristic
289	C₂	#20244333	1	1	yes	characteristic
290	C₂	#20244341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#9865
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊(C₄₇⋊C₈)	Q₈	semidirect
29. C₄⋊Q₈×C₄₇	C₈	semidirect
165. C₄₇	#9865	semidirect
170. C₄⋊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
C₄×C₄⋊(C₄₇⋊C₈)	2
#20241229	2
C₄×C₄⋊Q₈×C₄₇	2
#9865	47