Subgroups — #26220868

Order 12032.

1182 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×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
3	#20244362	C₂	1	4	yes	maximal, complemented
4	#20244354	C₂	1	4	yes	maximal, complemented
5	#20243109	C₂	1	4	yes	maximal
6	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₄×#10625329	—	2	4	no	—
10	C₂×#13093935	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15688989	—	2	2	no	—
13	#15688989	C₂²	1	2	yes	—
14	#15688989	C₂²	1	2	yes	—
15	C₂×C₄₇⋊#95	—	2	2	no	—
16	#15688885	C₂²	1	16	yes	—
17	C₄₇×#533	—	2	4	no	—
18	C₄₇×#533	C₄	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂×#125×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
23	C₂×#117×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	Q₈	1	8	yes	complemented
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	8	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	#13093935	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₂²×Q₈×C₄₇	—	2	2	no	—
38	C₂³×C₄×C₄₇	D₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₇	—	2	4	no	—
41	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
42	#125×C₄₇	—	2	4	no	—
43	#125×C₄₇	C₂×C₄	1	8	yes	complemented
44	C₂²⋊Q₈×C₄₇	—	2	16	no	—
45	C₄×Q₈×C₄₇	—	2	16	no	—
46	#117×C₄₇	—	2	4	no	—
47	#117×C₄₇	C₂×C₄	1	8	yes	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	D₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₄₇	—	2	8	no	—
60	C₂×Q₈×C₄₇	—	4	8	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₄₇	#41	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	4	8	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₄⋊C₄×C₄₇	—	2	16	no	—
76	C₄⋊C₄×C₄₇	—	2	8	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
79	C₂²⋊C₄×C₄₇	—	2	8	no	—
80	C₄²×C₄₇	—	2	8	no	—
81	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	C₄²×C₄₇	C₂×D₄	1	2	yes	—
86	C₄⋊Dic₄₇	—	2	8	no	—
87	C₄⋊Dic₄₇	—	4	4	no	—
88	#10625329	—	2	16	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₄₇	#124	1	2	yes	—
95	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₄₇	—	4	16	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	4	8	no	—
102	C₂×C₄×C₄₇	—	4	8	no	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
108	C₂×C₄×C₄₇	#124	1	2	yes	—
109	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	8	no	—
116	C₂×Dic₄₇	—	8	8	no	—
117	C₄₇⋊C₈	—	2	8	no	—
118	#21323	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	#391	1	4	yes	—
123	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₄₇	#575	1	4	yes	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	—	4	8	no	—
130	C₄×C₄₇	#391	1	4	yes	—
131	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
132	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
133	Dic₄₇	—	8	4	no	—
134	C₂×#533	D₄₇	1	1	yes	characteristic
135	C₂×C₄×M₁₆	—	47	1	no	—
136	C₂×#382	—	47	1	no	—
137	#1993	—	47	4	no	—
138	#1991	—	47	4	no	—
139	#1736	—	47	4	no	—
140	C₂×C₄₇	—	2	4	no	—
141	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
142	C₂×C₄₇	#2896	1	2	yes	—
143	C₂×C₄₇	#3140	1	2	yes	—
144	C₂×C₄₇	#3166	1	2	yes	—
145	C₂²×M₁₆	—	47	2	no	—
146	#533	—	2	4	no	—
147	#533	Dic₄₇	1	8	yes	complemented
148	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
149	C₂×#125	C₂×D₄₇	1	2	yes	—
150	C₂×C₂²⋊Q₈	—	2	2	no	—
151	C₂×C₄×Q₈	—	2	2	no	—
152	C₂×#117	C₂×D₄₇	1	2	yes	—
153	C₂²×C₄⋊C₄	—	47	2	no	—
154	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
155	C₄×M₁₆	—	94	4	no	—
156	C₂×#97	—	47	2	no	—
157	C₂×C₄×C₈	—	47	2	no	—
158	#382	—	47	2	no	—
159	#382	—	47	2	no	—
160	#382	—	94	2	no	—
161	C₂×#95	—	94	2	no	—
162	#340	—	47	16	no	—
163	C₄₇	#21323	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	2	2	no	—
165	C₂³×C₄	—	94	2	no	—
166	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	47	8	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂×M₁₆	—	94	4	no	—
171	C₂²×C₈	—	47	4	no	—
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	complemented
174	#125	—	2	4	no	—
175	#125	C₂×Dic₄₇	1	8	yes	complemented
176	C₂²⋊Q₈	—	2	16	no	—
177	C₄×Q₈	—	2	16	no	—
178	#117	—	2	4	no	—
179	#117	C₂×Dic₄₇	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	47	4	no	—
183	C₂×C₄⋊C₄	—	94	8	no	—
184	C₂×C₄⋊C₄	—	94	2	no	—
185	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
189	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
193	#97	—	94	4	no	—
194	C₄×C₈	—	94	4	no	—
195	#95	—	94	4	no	—
196	#95	—	94	4	no	—
197	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	94	4	no	—
203	C₂²×C₄	—	188	8	no	—
204	C₂²×C₄	—	188	4	no	—
205	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
206	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
207	C₂²×C₄	#10625354	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
214	M₁₆	—	94	16	no	—
215	C₂×C₈	—	47	8	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	4	no	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	94	8	no	—
222	C₄⋊C₄	—	188	4	no	—
223	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
224	C₂²⋊C₄	—	2	8	no	—
225	C₄²	—	2	8	no	—
226	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
227	C₄²	C₂²×Dic₄₇	1	2	yes	—
228	C₄²	C₂²×Dic₄₇	1	2	yes	—
229	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
230	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
233	C₂³	#13094013	1	2	yes	—
234	C₂³	#13094022	1	2	yes	—
235	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
236	Q₈	—	4	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	376	8	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	8	no	—
245	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
246	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
247	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
248	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
249	C₂×C₄	#13094000	1	8	yes	—
250	C₂×C₄	#13094013	1	2	yes	—
251	C₂×C₄	#13094022	1	2	yes	—
252	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
255	C₂×C₄	C₂×#10625354	1	2	yes	—
256	C₈	—	94	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	#15689030	1	4	yes	—
261	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
262	C₂²	C₂×#13094013	1	1	yes	characteristic
263	C₂²	C₂×#13094022	1	1	yes	characteristic
264	C₂²	#15689921	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	376	4	no	—
269	C₄	#15689030	1	4	yes	—
270	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
271	C₄	C₂×#13094000	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₂×#15689030	1	1	yes	characteristic
274	C₂	#20251907	1	2	yes	—
275	C₂	#20253777	1	2	yes	—
276	C₂	#20253941	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#21323
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #20244362	C₂	semidirect
4. #20244354	C₂	semidirect
18. C₄₇×#533	C₄	semidirect
28. C₂×#10625329	Q₈	semidirect
41. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
43. #125×C₄₇	C₂×C₄	semidirect
147. #533	Dic₄₇	semidirect
163. C₄₇	#21323	semidirect
173. C₄⋊Q₈	C₂×Dic₄₇	semidirect
175. #125	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
C₂×#15688989	2
#20244362	2
#20244354	2
#20243109	2
C₂×C₄₇×#533	2
#21323	47