Subgroups — #26147650

Order 12032.

2078 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	C₂×#15689030	C₂	1	4	yes	maximal
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241617	C₂	1	8	yes	maximal
5	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
10	#15689030	—	2	16	no	—
11	#15688993	—	2	4	no	—
12	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
13	C₄²×Dic₄₇	—	2	4	no	—
14	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
17	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×Dic₄₇	C₂³	1	4	yes	—
21	C₂³×Dic₄₇	C₂³	1	4	yes	—
22	C₂×C₄⋊Dic₄₇	—	2	8	no	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	8	no	—
25	C₂×C₄⋊Dic₄₇	—	2	16	no	—
26	C₂×C₄×Dic₄₇	—	2	4	no	—
27	C₂×C₄×Dic₄₇	—	2	8	no	—
28	C₂×C₄×Dic₄₇	—	2	4	no	—
29	C₂×C₄×Dic₄₇	—	2	8	no	—
30	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
31	C₂×C₄×Dic₄₇	Q₈	1	16	yes	complemented
32	C₄₇⋊#95	—	2	32	no	—
33	C₂²×Q₈×C₄₇	—	2	8	no	—
34	C₂²×Q₈×C₄₇	—	2	4	no	—
35	C₂²×Q₈×C₄₇	—	2	4	no	—
36	C₂²×Q₈×C₄₇	—	2	8	no	—
37	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₄₇	—	2	32	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	D₄	1	4	yes	—
47	C₂×C₄²×C₄₇	D₄	1	4	yes	—
48	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂×Q₈×C₄₇	—	2	16	no	—
50	C₂×Q₈×C₄₇	—	2	32	no	—
51	C₂×Q₈×C₄₇	—	4	16	no	—
52	C₂×Q₈×C₄₇	—	4	8	no	—
53	C₂×Q₈×C₄₇	—	4	8	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
67	C₂²×C₄×C₄₇	#41	1	4	yes	—
68	C₂²×C₄×C₄₇	#41	1	8	yes	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	8	no	—
72	C₂²×Dic₄₇	—	2	8	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	16	no	—
75	C₂²×Dic₄₇	C₂×D₄	1	8	yes	—
76	C₂²×Dic₄₇	C₂×Q₈	1	16	yes	—
77	C₄⋊C₄×C₄₇	—	2	32	no	—
78	C₄²×C₄₇	C₂²⋊C₄	1	16	yes	—
79	C₄⋊Dic₄₇	—	4	16	no	—
80	C₄⋊Dic₄₇	—	4	8	no	—
81	C₄×Dic₄₇	—	2	16	no	—
82	C₄×Dic₄₇	—	2	32	no	—
83	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
85	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
88	C₂³×C₄₇	C₂×#41	1	2	yes	—
89	C₂³×C₄₇	C₂×#41	1	4	yes	—
90	Q₈×C₄₇	—	4	32	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	16	no	—
93	C₂×C₄×C₄₇	—	4	8	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
99	C₂×C₄×C₄₇	C₄×Q₈	1	16	yes	—
100	C₂×C₄×C₄₇	#121	1	8	yes	—
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	16	yes	—
102	C₂×C₄×C₄₇	#124	1	4	yes	—
103	C₂×C₄×C₄₇	#124	1	8	yes	—
104	C₂×C₄×C₄₇	#127	1	4	yes	—
105	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
106	C₂×Dic₄₇	—	2	8	no	—
107	C₂×Dic₄₇	—	2	16	no	—
108	C₂×Dic₄₇	—	2	8	no	—
109	C₂×Dic₄₇	—	2	16	no	—
110	C₂×Dic₄₇	—	4	8	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	16	no	—
113	C₂×Dic₄₇	C₄⋊Q₈	1	16	yes	complemented
114	#14714	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
117	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
118	C₂²×C₄₇	C₂×#121	1	2	yes	—
119	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	4	yes	—
120	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#124	1	2	yes	—
122	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
124	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
125	C₂²×C₄₇	#532	1	4	yes	—
126	C₂²×C₄₇	#533	1	8	yes	—
127	C₄×C₄₇	—	4	8	no	—
128	C₄×C₄₇	#390	1	8	yes	—
129	C₄×C₄₇	#391	1	16	yes	complemented
130	Dic₄₇	—	2	16	no	—
131	Dic₄₇	—	8	4	no	—
132	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
133	C₂×#391	—	47	4	no	—
134	C₂×#384	—	47	1	no	—
135	C₂×C₄³	—	47	1	no	—
136	C₄⋊#95	—	47	8	no	—
137	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#391	1	2	yes	—
139	C₂×C₄₇	C₄×C₄⋊Q₈	1	4	yes	—
140	C₂×C₄₇	#2857	1	4	yes	—
141	C₂×C₄₇	#2858	1	1	yes	characteristic
142	C₂×C₄₇	#2862	1	2	yes	—
143	C₂×C₄₇	#2868	1	1	yes	characteristic
144	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
145	C₂×C₄⋊Q₈	—	2	4	no	—
146	C₂×C₄⋊Q₈	—	2	4	no	—
147	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
148	C₂²×C₄⋊C₄	—	47	4	no	—
149	C₂²×C₄⋊C₄	—	47	2	no	—
150	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
151	C₂²×C₄²	—	47	4	no	—
152	C₂²×C₄²	—	47	2	no	—
153	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
154	#391	—	94	16	no	—
155	#384	—	94	4	no	—
156	C₂×#95	—	47	16	no	—
157	C₄³	—	94	4	no	—
158	C₄₇	#14714	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₈	—	2	8	no	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	—	2	4	no	—
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
164	C₂³×C₄	—	47	4	no	—
165	C₂³×C₄	—	47	4	no	—
166	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
169	C₄⋊Q₈	—	2	32	no	—
170	C₂×C₄⋊C₄	—	2	8	no	—
171	C₂×C₄⋊C₄	—	2	8	no	—
172	C₂×C₄⋊C₄	—	94	8	no	—
173	C₂×C₄⋊C₄	—	94	16	no	—
174	C₂×C₄⋊C₄	—	94	8	no	—
175	C₂×C₄⋊C₄	—	94	4	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
177	C₂×C₄²	—	47	8	no	—
178	C₂×C₄²	—	47	16	no	—
179	C₂×C₄²	—	94	4	no	—
180	C₂×C₄²	—	94	8	no	—
181	C₂×C₄²	—	94	4	no	—
182	C₂×C₄²	—	94	8	no	—
183	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
184	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
185	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
186	#95	—	94	32	no	—
187	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
188	C₂×Q₈	—	2	16	no	—
189	C₂×Q₈	—	2	32	no	—
190	C₂×Q₈	—	4	16	no	—
191	C₂×Q₈	—	4	8	no	—
192	C₂×Q₈	—	4	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	47	8	no	—
198	C₂²×C₄	—	47	16	no	—
199	C₂²×C₄	—	94	4	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	—	94	8	no	—
202	C₂²×C₄	—	94	8	no	—
203	C₂²×C₄	—	94	4	no	—
204	C₂²×C₄	—	94	16	no	—
205	C₂²×C₄	D₄×D₄₇	1	4	yes	—
206	C₂²×C₄	#10625352	1	4	yes	—
207	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
208	C₂²×C₄	#10625354	1	8	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
210	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
216	C₄⋊C₄	—	2	32	no	—
217	C₄⋊C₄	—	188	16	no	—
218	C₄⋊C₄	—	188	8	no	—
219	C₄²	—	94	16	no	—
220	C₄²	—	94	32	no	—
221	C₄²	C₂²⋊Dic₄₇	1	16	yes	—
222	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
223	C₂³	C₂×D₄×D₄₇	1	2	yes	—
224	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂³	C₂×Q₈×D₄₇	1	4	yes	—
227	C₂³	C₂×#10625352	1	2	yes	—
228	C₂³	C₂×#10625354	1	4	yes	—
229	Q₈	—	4	32	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	47	16	no	—
236	C₂×C₄	—	94	8	no	—
237	C₂×C₄	—	94	16	no	—
238	C₂×C₄	—	94	8	no	—
239	C₂×C₄	—	94	16	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	16	no	—
243	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
245	C₂×C₄	D₄×Dic₄₇	1	8	yes	—
246	C₂×C₄	Q₈×Dic₄₇	1	16	yes	—
247	C₂×C₄	#13093995	1	8	yes	—
248	C₂×C₄	#13094000	1	16	yes	—
249	C₂×C₄	#13094011	1	4	yes	—
250	C₂×C₄	#13094013	1	8	yes	—
251	C₂×C₄	#13094020	1	4	yes	—
252	C₂×C₄	#13094022	1	8	yes	—
253	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
254	C₂²	C₂×D₄×Dic₄₇	1	2	yes	—
255	C₂²	C₂×Q₈×Dic₄₇	1	4	yes	—
256	C₂²	C₂×#13093995	1	2	yes	—
257	C₂²	C₂×#13094000	1	4	yes	—
258	C₂²	C₂×#13094011	1	1	yes	characteristic
259	C₂²	C₂×#13094013	1	2	yes	—
260	C₂²	C₂×#13094020	1	1	yes	characteristic
261	C₂²	C₄⋊Q₈×D₄₇	1	4	yes	—
262	C₂²	C₂×#13094022	1	2	yes	—
263	C₂²	#15689674	1	4	yes	—
264	C₂²	#15689678	1	8	yes	—
265	C₄	—	4	8	no	—
266	C₄	—	94	16	no	—
267	C₄	—	376	4	no	—
268	C₄	#15689026	1	8	yes	—
269	C₄	#15689030	1	16	yes	complemented
270	C₂	C₂×#15689026	1	1	yes	characteristic
271	C₂	C₂×#15689030	1	2	yes	—
272	C₂	C₄⋊Q₈×Dic₄₇	1	4	yes	—
273	C₂	#20251735	1	4	yes	—
274	C₂	#20251746	1	1	yes	characteristic
275	C₂	#20251765	1	2	yes	—
276	C₂	#20251784	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#14714
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
31. C₂×C₄×Dic₄₇	Q₈	semidirect
113. C₂×Dic₄₇	C₄⋊Q₈	semidirect
129. C₄×C₄₇	#391	semidirect
147. C₂×C₄⋊Q₈	Dic₄₇	semidirect
158. C₄₇	#14714	semidirect
269. C₄	#15689030	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689030	2
C₂×#15688993	2
C₂×C₄²×Dic₄₇	2
#20241617	2
C₂²×C₄⋊Q₈×C₄₇	2
#14714	47