Subgroups — #26220356

Order 12032.

1566 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689132	C₂	1	4	yes	maximal
3	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
4	#20244312	C₂	1	8	yes	maximal, complemented
5	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689147	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15689132	—	2	16	no	—
11	C₂×#13093948	C₂²	1	16	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
14	#15688969	—	2	1	no	—
15	C₄²×Dic₄₇	—	2	1	no	—
16	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
18	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
20	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
21	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#10625329	—	2	8	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13093983	—	2	4	no	—
28	#13093983	D₄	1	8	yes	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	#13093948	—	2	32	no	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₂²×Q₈×C₄₇	—	2	4	no	—
37	C₂²×Q₈×C₄₇	—	2	2	no	—
38	C₂²×Q₈×C₄₇	—	2	8	no	—
39	C₂²×Q₈×C₄₇	—	2	8	no	—
40	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
41	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
42	C₂³×C₄×C₄₇	—	2	2	no	—
43	C₂³×C₄×C₄₇	D₄	1	2	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₄₇	—	2	8	no	—
46	C₄⋊Q₈×C₄₇	—	2	8	no	—
47	C₄⋊Q₈×C₄₇	—	2	8	no	—
48	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	D₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₄₇	—	2	8	no	—
59	C₂×Q₈×C₄₇	—	2	4	no	—
60	C₂×Q₈×C₄₇	—	2	32	no	—
61	C₂×Q₈×C₄₇	—	2	8	no	—
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂×Q₈×C₄₇	—	4	8	no	—
64	C₂×Q₈×C₄₇	—	4	8	no	—
65	C₂×Q₈×C₄₇	—	4	8	no	—
66	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	8	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	4	8	no	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₄₇	#41	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	2	16	no	—
82	C₄⋊C₄×C₄₇	—	2	16	no	—
83	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
84	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
86	C₄²×C₄₇	C₂×D₄	1	4	yes	—
87	C₂²⋊Dic₄₇	—	2	8	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	C₂×D₄	1	8	yes	—
90	C₄×Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	4	8	no	—
92	#10625329	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	2	8	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂³×C₄₇	#121	1	4	yes	—
98	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₄₇	—	2	16	no	—
100	Q₈×C₄₇	—	4	4	no	—
101	Q₈×C₄₇	—	4	4	no	—
102	Q₈×C₄₇	—	4	16	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	16	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	—	4	16	no	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
112	C₂×C₄×C₄₇	#121	1	4	yes	—
113	C₂×C₄×C₄₇	#124	1	4	yes	—
114	C₂×C₄×C₄₇	#127	1	4	yes	—
115	C₂×C₄×C₄₇	#137	1	8	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	4	8	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	#21259	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	2	8	no	—
126	C₂²×C₄₇	#390	1	4	yes	—
127	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#121	1	2	yes	—
129	C₂²×C₄₇	C₂×#137	1	4	yes	—
130	C₄×C₄₇	—	2	4	no	—
131	C₄×C₄₇	—	4	8	no	—
132	C₄×C₄₇	#390	1	4	yes	—
133	C₄×C₄₇	#419	1	8	yes	complemented
134	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
135	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
136	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
137	Dic₄₇	—	4	8	no	—
138	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
139	C₂×#423	—	47	1	no	—
140	C₂×#419	—	47	4	no	—
141	C₄×#117	—	47	1	no	—
142	#1981	—	47	8	no	—
143	C₂×C₄₇	—	2	4	no	—
144	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#419	1	2	yes	—
146	C₂×C₄₇	#3039	1	4	yes	—
147	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
148	C₂²×M₁₆	—	47	2	no	—
149	C₂×C₄⋊Q₈	—	2	1	no	—
150	C₂×C₄⋊Q₈	—	2	1	no	—
151	C₂×C₄⋊Q₈	—	2	4	no	—
152	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
153	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
154	C₂×#117	—	47	2	no	—
155	C₂²×C₄⋊C₄	—	2	2	no	—
156	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
157	#423	—	94	4	no	—
158	C₂×C₄⋊C₈	—	47	4	no	—
159	#419	—	94	16	no	—
160	C₂×#103	—	47	16	no	—
161	C₄×C₄⋊C₄	—	47	4	no	—
162	C₄×C₂²⋊C₄	—	94	2	no	—
163	#376	—	94	1	no	—
164	C₄³	—	94	1	no	—
165	C₄₇	#21259	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	4	no	—
167	C₂²×Q₈	—	2	2	no	—
168	C₂²×Q₈	—	2	8	no	—
169	C₂²×Q₈	—	2	8	no	—
170	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
171	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	94	8	no	—
176	C₂×M₁₆	—	94	4	no	—
177	C₂²×C₈	—	47	4	no	—
178	C₄⋊Q₈	—	2	8	no	—
179	C₄⋊Q₈	—	2	8	no	—
180	C₄⋊Q₈	—	2	8	no	—
181	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
182	#117	—	47	8	no	—
183	#117	—	94	4	no	—
184	C₂×C₄⋊C₄	—	2	16	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₂²⋊C₄	—	94	2	no	—
189	C₂×C₄²	—	94	2	no	—
190	C₂×C₄²	—	94	4	no	—
191	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
192	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
195	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
196	C₄⋊C₈	—	94	8	no	—
197	#103	—	94	32	no	—
198	#95	—	94	4	no	—
199	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	32	no	—
203	C₂×Q₈	—	2	8	no	—
204	C₂×Q₈	—	2	8	no	—
205	C₂×Q₈	—	4	8	no	—
206	C₂×Q₈	—	4	8	no	—
207	C₂×Q₈	—	4	8	no	—
208	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	8	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
215	C₂²×C₄	D₄×D₄₇	1	4	yes	—
216	C₂²×C₄	#10625352	1	4	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
222	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
223	M₁₆	—	188	8	no	—
224	C₂×C₈	—	94	8	no	—
225	C₂×C₈	—	94	4	no	—
226	C₄⋊C₄	—	2	16	no	—
227	C₄⋊C₄	—	2	16	no	—
228	C₄⋊C₄	—	47	8	no	—
229	C₄⋊C₄	—	94	4	no	—
230	C₂²⋊C₄	—	94	8	no	—
231	C₄²	—	94	8	no	—
232	C₄²	—	188	8	no	—
233	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
234	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
235	C₄²	C₂²×Dic₄₇	1	4	yes	—
236	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
239	C₂³	#13093995	1	4	yes	—
240	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
241	Q₈	—	2	16	no	—
242	Q₈	—	4	16	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	16	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	94	8	no	—
251	C₂×C₄	—	188	8	no	—
252	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
255	C₂×C₄	D₄×Dic₄₇	1	8	yes	—
256	C₂×C₄	#13093995	1	4	yes	—
257	C₂×C₄	#13094011	1	4	yes	—
258	C₂×C₄	#13094020	1	4	yes	—
259	C₂×C₄	#13094055	1	8	yes	—
260	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
262	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₄	C₂×#10625352	1	2	yes	—
264	C₈	—	188	4	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	#15689026	1	4	yes	—
268	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
269	C₂²	C₂×#13093995	1	2	yes	—
270	C₂²	C₂×#13094055	1	4	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	4	8	no	—
273	C₄	—	188	8	no	—
274	C₄	#15689026	1	4	yes	—
275	C₄	#15689132	1	8	yes	complemented
276	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
277	C₄	C₂×#13094011	1	1	yes	characteristic
278	C₄	C₂×#13094020	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	C₂×#15689026	1	1	yes	characteristic
281	C₂	C₂×#15689132	1	2	yes	—
282	C₂	#20252871	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#21259
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #20244312	C₂	semidirect
20. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
133. C₄×C₄₇	#419	semidirect
152. C₂×C₄⋊Q₈	Dic₄₇	semidirect
165. C₄₇	#21259	semidirect
275. C₄	#15689132	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×#15689132	2
C₄×#13093983	2
#20244312	2
C₂²×C₄⋊Q₈×C₄₇	2
#21259	47