Subgroups — #26220813

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244346	C₂	1	8	yes	maximal, complemented
4	#20243107	C₂	1	4	yes	maximal
5	C₂²×#125×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	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
11	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
12	#15688969	—	2	1	no	—
13	C₄²×Dic₄₇	—	2	1	no	—
14	#15688885	C₂²	1	16	yes	—
15	C₂×#125×C₄₇	—	2	1	no	—
16	C₂×#125×C₄₇	—	2	1	no	—
17	C₂×#125×C₄₇	—	2	4	no	—
18	C₂×#125×C₄₇	C₄	1	8	yes	complemented
19	C₂×#125×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	2	8	no	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	#13093983	—	2	4	no	—
27	#13093983	Q₈	1	8	yes	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
29	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	4	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₂³×C₄×C₄₇	—	2	2	no	—
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#125×C₄₇	—	2	8	no	—
38	#125×C₄₇	—	2	8	no	—
39	#125×C₄₇	—	2	8	no	—
40	#125×C₄₇	C₂×C₄	1	16	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	—	4	8	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₄₇	#41	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	2	16	no	—
69	C₄⋊C₄×C₄₇	—	2	16	no	—
70	C₄⋊C₄×C₄₇	—	2	8	no	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
76	C₄²×C₄₇	C₂×D₄	1	4	yes	—
77	C₂²⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	—
80	C₄×Dic₄₇	—	2	8	no	—
81	C₄×Dic₄₇	—	4	8	no	—
82	#10625329	—	4	8	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
87	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	—	4	16	no	—
94	C₂×C₄×C₄₇	—	4	8	no	—
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
99	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
100	C₂×C₄×C₄₇	#124	1	4	yes	—
101	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×Dic₄₇	—	2	8	no	—
107	C₂×Dic₄₇	—	4	8	no	—
108	C₄₇⋊C₈	—	4	4	no	—
109	#21314	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	8	no	—
111	C₂²×C₄₇	—	2	4	no	—
112	C₂²×C₄₇	#391	1	4	yes	—
113	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
115	C₂²×C₄₇	#575	1	4	yes	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	4	8	no	—
119	C₄×C₄₇	#391	1	4	yes	—
120	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
121	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
122	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	Dic₄₇	—	4	8	no	—
124	C₂²×#125	D₄₇	1	1	yes	characteristic
125	C₂×#423	—	47	1	no	—
126	C₄×#117	—	47	1	no	—
127	#1989	—	47	8	no	—
128	#1736	—	47	4	no	—
129	C₂×C₄₇	—	2	4	no	—
130	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
131	C₂×C₄₇	#2896	1	2	yes	—
132	C₂×C₄₇	#3080	1	4	yes	—
133	C₂²×M₁₆	—	47	2	no	—
134	C₂×#125	—	2	1	no	—
135	C₂×#125	—	2	1	no	—
136	C₂×#125	—	2	4	no	—
137	C₂×#125	Dic₄₇	1	8	yes	complemented
138	C₂×#125	C₂×D₄₇	1	4	yes	—
139	C₂×#117	—	47	2	no	—
140	C₂²×C₄⋊C₄	—	2	2	no	—
141	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
142	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
143	#423	—	94	4	no	—
144	C₂×C₄⋊C₈	—	47	4	no	—
145	C₄×C₄⋊C₄	—	47	4	no	—
146	C₄×C₂²⋊C₄	—	94	2	no	—
147	#376	—	94	1	no	—
148	C₄³	—	94	1	no	—
149	#340	—	47	16	no	—
150	C₄₇	#21314	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
153	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
154	C₂×M₁₆	—	94	8	no	—
155	C₂×M₁₆	—	94	4	no	—
156	C₂²×C₈	—	47	4	no	—
157	#125	—	2	8	no	—
158	#125	—	2	8	no	—
159	#125	—	2	8	no	—
160	#125	C₂×Dic₄₇	1	16	yes	—
161	#117	—	47	8	no	—
162	#117	—	94	4	no	—
163	C₂×C₄⋊C₄	—	2	16	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	47	4	no	—
169	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
170	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
171	C₂×C₂²⋊C₄	—	94	2	no	—
172	C₂×C₄²	—	94	2	no	—
173	C₂×C₄²	—	94	4	no	—
174	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
175	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
178	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
179	C₄⋊C₈	—	94	8	no	—
180	#95	—	94	4	no	—
181	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	—	94	4	no	—
187	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
188	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
189	C₂²×C₄	#10625354	1	4	yes	—
190	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	—	188	8	no	—
197	C₂×C₈	—	94	4	no	—
198	C₂×C₈	—	94	8	no	—
199	C₄⋊C₄	—	2	16	no	—
200	C₄⋊C₄	—	2	16	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	47	8	no	—
204	C₄⋊C₄	—	94	4	no	—
205	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
206	C₂²⋊C₄	—	94	8	no	—
207	C₄²	—	94	8	no	—
208	C₄²	—	188	8	no	—
209	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
210	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
211	C₄²	C₂²×Dic₄₇	1	4	yes	—
212	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
215	C₂³	#13094000	1	4	yes	—
216	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	16	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	94	8	no	—
224	C₂×C₄	—	188	8	no	—
225	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
228	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
229	C₂×C₄	#13094000	1	4	yes	—
230	C₂×C₄	#13094013	1	4	yes	—
231	C₂×C₄	#13094022	1	4	yes	—
232	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
235	C₂×C₄	C₂×#10625354	1	2	yes	—
236	C₈	—	188	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	8	no	—
239	C₂²	#15689030	1	4	yes	—
240	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
241	C₂²	C₂×#13094000	1	2	yes	—
242	C₂²	#15689921	1	4	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	—	188	8	no	—
247	C₄	#15689030	1	4	yes	—
248	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
249	C₄	C₂×#13094013	1	1	yes	characteristic
250	C₄	C₂×#13094022	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	C₂×#15689030	1	1	yes	characteristic
253	C₂	#20251907	1	2	yes	—
254	C₂	#20253206	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#21314
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
3. #20244346	C₂	semidirect
18. C₂×#125×C₄₇	C₄	semidirect
137. C₂×#125	Dic₄₇	semidirect
150. C₄₇	#21314	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
C₄×#13093983	2
#20244346	2
#20243107	2
C₂²×#125×C₄₇	2
#21314	47