Subgroups — #26216671

Order 12032.

1114 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689150	C₂	1	4	yes	maximal
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#20243542	C₂	1	4	yes	maximal, complemented
5	#20243540	C₂	1	4	yes	maximal, complemented
6	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15689150	—	2	16	no	—
10	#15689147	—	2	4	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	16	yes	—
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
23	C₄₇×#376	C₂²	1	2	yes	—
24	C₄³×C₄₇	C₂²	1	2	yes	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	8	no	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	8	no	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	32	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	#117×C₄₇	D₄	1	8	yes	—
40	#117×C₄₇	Q₈	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	4	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
52	#95×C₄₇	C₂×C₄	1	4	yes	—
53	#95×C₄₇	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
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	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₄₇	C₄⋊C₄	1	16	yes	—
71	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	16	yes	—
72	C₄²×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	—	4	4	no	—
74	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
75	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
76	C₄²×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	C₂×D₄	1	2	yes	—
78	C₄²×C₄₇	C₂×D₄	1	8	yes	—
79	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
80	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
81	C₄²×C₄₇	C₂×Q₈	1	8	yes	—
82	#10625329	—	4	8	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	8	no	—
87	C₂×C₄₇⋊C₈	—	4	8	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	#127	1	4	yes	—
108	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
109	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
110	C₂×C₄×C₄₇	#131	1	4	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
115	C₄₇⋊C₈	—	4	4	no	—
116	C₄₇⋊C₈	—	4	4	no	—
117	#20793	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	8	no	—
121	C₂²×C₄₇	#384	1	4	yes	—
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
124	C₂²×C₄₇	C₂×#131	1	2	yes	—
125	C₂²×C₄₇	#568	1	2	yes	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	16	no	—
130	C₄×C₄₇	#384	1	4	yes	—
131	C₄×C₄₇	#424	1	4	yes	complemented
132	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
133	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
135	C₂×#424	—	47	4	no	—
136	C₂×#423	—	47	1	no	—
137	C₂²×C₄⋊C₈	—	47	1	no	—
138	C₄×#117	D₄₇	1	1	yes	characteristic
139	#1843	—	47	4	no	—
140	#1842	—	47	4	no	—
141	C₂×C₄₇	—	2	4	no	—
142	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
144	C₂×C₄₇	#2905	1	1	yes	characteristic
145	C₂×C₄₇	#2906	1	4	yes	—
146	C₂²×M₁₆	—	47	2	no	—
147	C₂³×C₈	—	47	2	no	—
148	C₂×#117	C₂×D₄₇	1	2	yes	—
149	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
150	#424	—	94	16	no	—
151	#423	—	94	4	no	—
152	C₂×C₄⋊C₈	—	47	2	no	—
153	C₂×C₄⋊C₈	—	47	2	no	—
154	C₂×C₄⋊C₈	—	47	16	no	—
155	C₂×C₄⋊C₈	—	47	2	no	—
156	C₂×C₄⋊C₈	—	47	2	no	—
157	C₂×C₄⋊C₈	—	94	4	no	—
158	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
159	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
160	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
161	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
162	#376	C₂×D₄₇	1	2	yes	—
163	C₄³	C₂×D₄₇	1	2	yes	—
164	C₄₇	#20793	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	94	4	no	—
169	C₂×M₁₆	—	94	8	no	—
170	C₂²×C₈	—	47	4	no	—
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	—	94	4	no	—
173	C₂²×C₈	—	94	8	no	—
174	#117	Dic₉₄	1	8	yes	—
175	#117	D₁₈₈	1	8	yes	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
178	C₂×C₄²	Dic₉₄	1	4	yes	—
179	C₂×C₄²	D₁₈₈	1	4	yes	—
180	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
181	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
182	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄⋊C₈	—	94	4	no	—
188	C₄⋊C₈	—	94	4	no	—
189	C₄⋊C₈	—	94	4	no	—
190	C₄⋊C₈	—	94	4	no	—
191	C₄⋊C₈	—	94	32	no	—
192	#95	C₂×Dic₄₇	1	4	yes	—
193	#95	C₂×Dic₄₇	1	4	yes	—
194	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
198	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
199	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
200	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
201	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
202	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
203	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
210	M₁₆	—	188	8	no	—
211	C₂×C₈	—	94	4	no	—
212	C₂×C₈	—	94	8	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	8	no	—
215	C₂×C₈	—	188	8	no	—
216	C₄⋊C₄	C₄⋊Dic₄₇	1	16	yes	—
217	C₂²⋊C₄	C₄⋊Dic₄₇	1	16	yes	—
218	C₄²	—	2	8	no	—
219	C₄²	—	4	4	no	—
220	C₄²	C₄⋊Dic₄₇	1	4	yes	—
221	C₄²	C₄⋊Dic₄₇	1	4	yes	—
222	C₄²	C₂×Dic₉₄	1	2	yes	—
223	C₄²	C₂×Dic₉₄	1	2	yes	—
224	C₄²	C₂×Dic₉₄	1	8	yes	—
225	C₄²	C₂×D₁₈₈	1	2	yes	—
226	C₄²	C₂×D₁₈₈	1	2	yes	—
227	C₄²	C₂×D₁₈₈	1	8	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
231	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
232	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₄₇	1	16	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
245	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
246	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
247	C₂×C₄	#13094021	1	4	yes	—
248	C₂×C₄	#13094024	1	8	yes	—
249	C₂×C₄	#13094024	1	4	yes	—
250	C₂×C₄	#13094031	1	4	yes	—
251	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
253	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
254	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
255	C₈	—	188	4	no	—
256	C₈	—	188	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	#15688993	1	4	yes	—
261	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
262	C₂²	C₂²×C₄⋊Dic₄₇	1	2	yes	—
263	C₂²	C₂×#13094031	1	2	yes	—
264	C₂²	#15689890	1	2	yes	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	16	no	—
269	C₄	#15688993	1	4	yes	—
270	C₄	#15689150	1	4	yes	complemented
271	C₄	C₂×#13094021	1	1	yes	characteristic
272	C₄	C₂×#13094024	1	2	yes	—
273	C₄	C₂×#13094024	1	1	yes	characteristic
274	C₂	—	2	4	no	—
275	C₂	C₂×#15688993	1	1	yes	characteristic
276	C₂	C₂×#15689150	1	1	yes	characteristic
277	C₂	#20251952	1	1	yes	characteristic
278	C₂	#20251956	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20793
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. #20243542	C₂	semidirect
5. #20243540	C₂	semidirect
131. C₄×C₄₇	#424	semidirect
164. C₄₇	#20793	semidirect
270. C₄	#15689150	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689150	2
C₂×#15689147	2
C₂²×C₄⋊(C₄₇⋊C₈)	2
#20243542	2
#20243540	2
C₄×#117×C₄₇	2
#20793	47