Subgroups — #26220862

Order 12032.

1270 conjugacy classes of subgroups.

The subgroup lattice has 302 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×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20244358	C₂	1	4	yes	maximal, complemented
4	#20244350	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	—	2	2	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₄×#10625329	—	2	4	no	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	#15688993	—	2	2	no	—
14	#15688993	C₂²	1	2	yes	—
15	#15688993	C₂²	1	2	yes	—
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	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	4	8	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	#13093935	—	2	4	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₂²×Q₈×C₄₇	—	2	2	no	—
39	C₂³×C₄×C₄₇	D₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₄₇	—	2	4	no	—
42	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
43	#125×C₄₇	—	2	4	no	—
44	#125×C₄₇	C₂×C₄	1	8	yes	complemented
45	C₂²⋊Q₈×C₄₇	—	2	16	no	—
46	C₄×Q₈×C₄₇	—	2	16	no	—
47	#117×C₄₇	—	2	4	no	—
48	#117×C₄₇	C₂×C₄	1	8	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₄₇	D₄	1	2	yes	—
56	C₂×C₄²×C₄₇	D₄	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₄₇	—	2	8	no	—
61	C₂×Q₈×C₄₇	—	4	4	no	—
62	C₂×Q₈×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	—	4	8	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	2	16	no	—
79	C₄⋊C₄×C₄₇	—	2	8	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
82	C₂²⋊C₄×C₄₇	—	2	8	no	—
83	C₄²×C₄₇	—	2	8	no	—
84	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	4	8	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	4	8	no	—
93	#10625329	—	2	16	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₄₇	#124	1	2	yes	—
101	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₄₇	—	4	16	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	16	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
115	C₂×C₄×C₄₇	#124	1	2	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
118	C₂×C₄×C₄₇	#137	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	8	no	—
125	C₂×Dic₄₇	—	8	8	no	—
126	C₄₇⋊C₈	—	2	8	no	—
127	#21322	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	#391	1	4	yes	—
133	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₂²×C₄₇	C₂×#137	1	2	yes	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	4	8	no	—
140	C₄×C₄₇	#391	1	4	yes	—
141	C₄×C₄₇	#417	1	8	yes	—
142	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
143	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	Dic₄₇	—	8	4	no	—
145	C₂×#533	D₄₇	1	1	yes	characteristic
146	C₂×C₄×M₁₆	—	47	1	no	—
147	C₂×#384	—	47	1	no	—
148	#1992	—	47	4	no	—
149	#1990	—	47	4	no	—
150	#1736	—	47	4	no	—
151	C₂×C₄₇	—	2	4	no	—
152	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#417	1	2	yes	—
154	C₂×C₄₇	#3138	1	1	yes	characteristic
155	C₂×C₄₇	#3139	1	1	yes	characteristic
156	C₂×C₄₇	#3165	1	2	yes	—
157	C₂²×M₁₆	—	47	2	no	—
158	#533	—	2	4	no	—
159	#533	Dic₄₇	1	8	yes	complemented
160	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
161	C₂×#125	C₂×D₄₇	1	2	yes	—
162	C₂×C₂²⋊Q₈	—	2	2	no	—
163	C₂×C₄×Q₈	—	2	2	no	—
164	C₂×#117	C₂×D₄₇	1	2	yes	—
165	C₂²×C₄⋊C₄	—	47	2	no	—
166	C₂²×C₄⋊C₄	—	94	2	no	—
167	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
168	C₄×M₁₆	—	94	4	no	—
169	C₂×#97	—	47	2	no	—
170	C₂×C₄×C₈	—	47	2	no	—
171	#384	—	47	2	no	—
172	#384	—	47	2	no	—
173	#384	—	94	2	no	—
174	#340	—	47	16	no	—
175	C₄₇	#21322	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	2	2	no	—
177	C₂³×C₄	—	94	2	no	—
178	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×M₁₆	—	47	8	no	—
181	C₂×M₁₆	—	94	4	no	—
182	C₂×M₁₆	—	94	4	no	—
183	C₂²×C₈	—	47	4	no	—
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	complemented
186	#125	—	2	4	no	—
187	#125	C₂×Dic₄₇	1	8	yes	complemented
188	C₂²⋊Q₈	—	2	16	no	—
189	C₄×Q₈	—	2	16	no	—
190	#117	—	2	4	no	—
191	#117	C₂×Dic₄₇	1	8	yes	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	47	4	no	—
195	C₂×C₄⋊C₄	—	94	8	no	—
196	C₂×C₄⋊C₄	—	94	2	no	—
197	C₂×C₄⋊C₄	—	94	4	no	—
198	C₂×C₄⋊C₄	—	94	4	no	—
199	C₂×C₄⋊C₄	—	188	8	no	—
200	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
208	#97	—	94	4	no	—
209	C₄×C₈	—	94	4	no	—
210	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	188	8	no	—
219	C₂²×C₄	—	188	4	no	—
220	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
221	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
222	C₂²×C₄	#10625354	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	M₁₆	—	94	16	no	—
230	C₂×C₈	—	47	8	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	94	4	no	—
233	C₄⋊C₄	—	2	16	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	94	8	no	—
237	C₄⋊C₄	—	188	8	no	—
238	C₄⋊C₄	—	188	4	no	—
239	C₄⋊C₄	—	188	8	no	—
240	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₄²	—	2	8	no	—
243	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
244	C₄²	C₂²×Dic₄₇	1	2	yes	—
245	C₄²	C₂²×Dic₄₇	1	2	yes	—
246	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
251	C₂³	#13094013	1	2	yes	—
252	C₂³	#13094022	1	2	yes	—
253	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
254	Q₈	—	4	16	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	376	8	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	—	188	8	no	—
264	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
265	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
267	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
268	C₂×C₄	#13094000	1	8	yes	—
269	C₂×C₄	#13094013	1	2	yes	—
270	C₂×C₄	#13094022	1	2	yes	—
271	C₂×C₄	#13094049	1	4	yes	—
272	C₂×C₄	#13094054	1	4	yes	—
273	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
276	C₂×C₄	C₂×#10625354	1	2	yes	—
277	C₈	—	94	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	#15689030	1	4	yes	—
283	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
284	C₂²	C₂×#13094013	1	1	yes	characteristic
285	C₂²	C₂×#13094022	1	1	yes	characteristic
286	C₂²	C₂×#13094049	1	2	yes	—
287	C₂²	C₂×#13094054	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	8	no	—
290	C₄	—	376	4	no	—
291	C₄	#15689030	1	4	yes	—
292	C₄	#15689124	1	8	yes	—
293	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
294	C₄	C₂×#13094000	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	C₂×#15689030	1	1	yes	characteristic
297	C₂	C₂×#15689124	1	2	yes	—
298	C₂	#20253764	1	1	yes	characteristic
299	C₂	#20253770	1	1	yes	characteristic
300	C₂	#20253935	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	#21322
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. #20244358	C₂	semidirect
4. #20244350	C₂	semidirect
18. C₄₇×#533	C₄	semidirect
28. C₂×#10625329	Q₈	semidirect
42. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
44. #125×C₄₇	C₂×C₄	semidirect
159. #533	Dic₄₇	semidirect
175. C₄₇	#21322	semidirect
185. C₄⋊Q₈	C₂×Dic₄₇	semidirect
187. #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×#15688993	2
#20244358	2
#20244350	2
#20243109	2
C₂×C₄₇×#533	2
#21322	47