Subgroups — #26122065

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#15688878	C₂	1	4	yes	maximal
3	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
8	#15688878	—	2	8	no	—
9	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄³×C₄₇	—	2	4	no	—
14	C₂²×C₄₇⋊C₈	—	2	8	no	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₂²×C₄₇⋊C₈	—	2	4	no	—
17	C₂²×C₄₇⋊C₈	—	2	8	no	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
19	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	—	2	4	no	—
25	C₂×C₄²×C₄₇	—	2	4	no	—
26	C₂×C₄²×C₄₇	—	2	8	no	—
27	C₂×C₄²×C₄₇	—	2	4	no	—
28	C₂×C₄²×C₄₇	—	2	4	no	—
29	C₂×C₄²×C₄₇	—	2	4	no	—
30	C₂×C₄²×C₄₇	C₈	1	8	yes	complemented
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	D₄	1	4	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	4	yes	—
35	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
36	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
37	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	2	4	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
53	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₄²×C₄₇	—	2	16	no	—
62	C₄²×C₄₇	—	2	16	no	—
63	C₄²×C₄₇	—	2	8	no	—
64	C₄²×C₄₇	—	2	8	no	—
65	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂×C₄₇⋊C₈	—	2	16	no	—
69	C₂×C₄₇⋊C₈	—	4	8	no	—
70	C₂×C₄₇⋊C₈	—	4	4	no	—
71	C₂×C₄₇⋊C₈	—	4	4	no	—
72	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
73	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₄₇	#117	1	2	yes	—
77	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
78	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
79	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	#95	1	4	yes	—
90	C₂×C₄×C₄₇	#95	1	4	yes	—
91	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
92	C₂×C₄×C₄₇	#97	1	4	yes	—
93	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
94	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
95	C₂×C₄×C₄₇	C₄⋊C₈	1	16	yes	complemented
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
100	C₂×C₄×C₄₇	#121	1	4	yes	—
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	#124	1	2	yes	—
103	C₂×C₄×C₄₇	#125	1	2	yes	—
104	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
107	C₄₇⋊C₈	—	4	8	no	—
108	#10832	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
110	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
111	C₂²×C₄₇	#380	1	2	yes	—
112	C₂²×C₄₇	#382	1	1	yes	characteristic
113	C₂²×C₄₇	#384	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
116	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
117	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
118	C₂²×C₄₇	#407	1	2	yes	—
119	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
120	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
121	C₂²×C₄₇	#423	1	4	yes	—
122	C₂²×C₄₇	#423	1	2	yes	—
123	C₂²×C₄₇	#431	1	4	yes	—
124	C₂²×C₄₇	#432	1	4	yes	—
125	C₄×C₄₇	—	2	16	no	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	#336	1	8	yes	complemented
128	C₄×C₄₇	#389	1	4	yes	—
129	C₄×C₄₇	#390	1	2	yes	—
130	C₄×C₄₇	#391	1	2	yes	—
131	C₂²×C₄⋊C₈	—	47	2	no	—
132	C₂×C₄³	D₄₇	1	1	yes	characteristic
133	C₂×#336	—	47	4	no	—
134	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
135	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
136	C₂×C₄₇	#1728	1	1	yes	characteristic
137	C₂×C₄₇	C₄×C₄⋊C₈	1	2	yes	—
138	C₂×C₄₇	#1767	1	2	yes	—
139	C₂×C₄₇	#1797	1	4	yes	—
140	C₂×C₄₇	#1831	1	2	yes	—
141	C₂×C₄₇	#1842	1	2	yes	—
142	C₂³×C₈	—	47	4	no	—
143	C₂²×C₄²	Dic₄₇	1	2	yes	—
144	C₂²×C₄²	Dic₄₇	1	2	yes	—
145	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
146	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	47	8	no	—
148	C₂×C₄⋊C₈	—	94	4	no	—
149	C₂×C₄⋊C₈	—	94	4	no	—
150	C₄³	—	2	4	no	—
151	#336	—	94	8	no	—
152	C₄₇	#10832	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
154	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂²×C₈	—	47	8	no	—
158	C₂²×C₈	—	94	8	no	—
159	C₂²×C₈	—	94	4	no	—
160	C₂²×C₈	—	94	8	no	—
161	C₂²×C₈	—	94	4	no	—
162	C₂×C₄²	—	2	8	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	—	2	4	no	—
165	C₂×C₄²	—	2	4	no	—
166	C₂×C₄²	—	2	4	no	—
167	C₂×C₄²	—	2	4	no	—
168	C₂×C₄²	C₄₇⋊C₈	1	8	yes	complemented
169	C₂×C₄²	Dic₉₄	1	4	yes	—
170	C₂×C₄²	Dic₉₄	1	2	yes	—
171	C₂×C₄²	C₄×D₄₇	1	4	yes	—
172	C₂×C₄²	D₁₈₈	1	4	yes	—
173	C₂×C₄²	D₁₈₈	1	2	yes	—
174	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
175	C₄⋊C₈	—	94	16	no	—
176	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₂×C₄₇⋊C₈	1	8	yes	—
183	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
184	C₂²×C₄	#10625329	1	8	yes	—
185	C₂²×C₄	#10625329	1	4	yes	—
186	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
187	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
189	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
190	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
191	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
192	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
193	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
196	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
198	C₂²×C₄	#10625350	1	4	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
200	C₂×C₈	—	94	8	no	—
201	C₂×C₈	—	94	16	no	—
202	C₂×C₈	—	188	8	no	—
203	C₂×C₈	—	188	4	no	—
204	C₂×C₈	—	188	4	no	—
205	C₄²	—	2	16	no	—
206	C₄²	—	2	16	no	—
207	C₄²	—	2	8	no	—
208	C₄²	—	2	8	no	—
209	C₄²	Dic₄₇⋊C₄	1	8	yes	—
210	C₄²	(C₂×D₄₇)⋊C₄	1	8	yes	—
211	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
212	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
214	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
215	C₂³	#13093983	1	2	yes	—
216	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
217	C₂³	C₂×#10625329	1	2	yes	—
218	C₂³	C₂×#10625329	1	4	yes	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	C₄₇⋊#95	1	4	yes	—
229	C₂×C₄	C₄₇⋊#95	1	4	yes	—
230	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
231	C₂×C₄	#13093935	1	4	yes	—
232	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
233	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
234	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	16	yes	complemented
235	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
237	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
238	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
239	C₂×C₄	#13093998	1	4	yes	—
240	C₂×C₄	#13094003	1	4	yes	—
241	C₂×C₄	#13094012	1	2	yes	—
242	C₂×C₄	#13094016	1	2	yes	—
243	C₂×C₄	#13094021	1	1	yes	characteristic
244	C₂×C₄	#13094024	1	2	yes	—
245	C₂×C₄	#13094024	1	1	yes	characteristic
246	C₈	—	188	8	no	—
247	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
248	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
249	C₂²	#15688983	1	2	yes	—
250	C₂²	#15688989	1	1	yes	characteristic
251	C₂²	#15688993	1	1	yes	characteristic
252	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
253	C₂²	C₂×#13093935	1	1	yes	characteristic
254	C₂²	C₄×#10625329	1	2	yes	—
255	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
256	C₂²	#15689087	1	2	yes	—
257	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
258	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
259	C₂²	#15689147	1	4	yes	—
260	C₂²	#15689147	1	2	yes	—
261	C₂²	#15689176	1	4	yes	—
262	C₂²	#15689180	1	4	yes	—
263	C₄	—	2	4	no	—
264	C₄	—	2	16	no	—
265	C₄	#15688878	1	8	yes	complemented
266	C₄	#15689021	1	4	yes	—
267	C₄	#15689025	1	2	yes	—
268	C₄	#15689029	1	2	yes	—
269	C₂	#20241618	1	1	yes	characteristic
270	C₂	C₂×#15688878	1	1	yes	characteristic
271	C₂	#20243082	1	1	yes	characteristic
272	C₂	C₄×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
273	C₂	#20243241	1	2	yes	—
274	C₂	#20243368	1	4	yes	—
275	C₂	#20243505	1	2	yes	—
276	C₂	#20243540	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#10832
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
30. C₂×C₄²×C₄₇	C₈	semidirect
95. C₂×C₄×C₄₇	C₄⋊C₈	semidirect
127. C₄×C₄₇	#336	semidirect
152. C₄₇	#10832	semidirect
168. C₂×C₄²	C₄₇⋊C₈	semidirect
234. C₂×C₄	C₄⋊(C₄₇⋊C₈)	semidirect
265. C₄	#15688878	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688878	2
C₂×C₄³×C₄₇	2
#10832	47