Subgroups — C₂×#20244354 #26220834

Order 12032.

1246 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
3	#20244354	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	4	yes	maximal
5	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
10	#15688989	—	2	2	no	—
11	#15688989	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	—	2	2	no	—
13	#15688885	C₂²	1	16	yes	—
14	C₂×#125×C₄₇	—	2	2	no	—
15	C₂×#125×C₄₇	—	2	4	no	—
16	C₂×#125×C₄₇	C₄	1	8	yes	complemented
17	C₂×#125×C₄₇	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	8	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
29	C₄×C₄₇⋊C₈	—	2	8	no	—
30	C₄₇⋊#95	—	2	8	no	—
31	C₂³×C₄×C₄₇	—	2	2	no	—
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	#125×C₄₇	—	2	16	no	—
35	#125×C₄₇	—	2	8	no	—
36	#125×C₄₇	C₂×C₄	1	16	yes	complemented
37	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	4	8	no	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	4	4	no	—
62	C₂²×Dic₄₇	—	4	8	no	—
63	C₄⋊C₄×C₄₇	—	2	32	no	—
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
69	C₄²×C₄₇	C₂×D₄	1	4	yes	—
70	C₄⋊Dic₄₇	—	2	8	no	—
71	C₄⋊Dic₄₇	—	4	4	no	—
72	C₂×C₄₇⋊C₈	—	2	16	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
76	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂³×C₄₇	#124	1	2	yes	—
78	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
79	C₂³×C₄₇	#135	1	8	yes	—
80	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	16	no	—
83	C₂×C₄×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	4	8	no	—
85	C₂×C₄×C₄₇	—	4	16	no	—
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
87	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
89	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
90	C₂×C₄×C₄₇	#124	1	2	yes	—
91	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
96	C₂×Dic₄₇	—	4	4	no	—
97	C₂×Dic₄₇	—	4	8	no	—
98	C₂×Dic₄₇	—	8	8	no	—
99	C₄₇⋊C₈	—	2	8	no	—
100	C₂×#1991	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	#391	1	4	yes	—
102	C₂²×C₄₇	#416	1	8	yes	—
103	C₂²×C₄₇	#488	1	8	yes	—
104	C₂²×C₄₇	#499	1	8	yes	—
105	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#135	1	4	yes	—
109	C₄×C₄₇	—	2	8	no	—
110	C₄×C₄₇	—	4	8	no	—
111	C₄×C₄₇	#391	1	4	yes	—
112	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
113	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
114	Dic₄₇	—	8	4	no	—
115	C₂²×#125	D₄₇	1	1	yes	characteristic
116	C₂²×C₄×C₈	—	47	1	no	—
117	C₂×#382	—	47	1	no	—
118	#1991	—	47	8	no	—
119	C₂×#340	—	47	4	no	—
120	C₂×C₄₇	#1991	1	8	yes	complemented
121	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#416	1	2	yes	—
123	C₂×C₄₇	C₂×#488	1	2	yes	—
124	C₂×C₄₇	C₂×#499	1	2	yes	—
125	C₂³×C₈	—	47	2	no	—
126	C₂×#125	—	2	2	no	—
127	C₂×#125	—	2	4	no	—
128	C₂×#125	Dic₄₇	1	8	yes	complemented
129	C₂×#125	C₂×D₄₇	1	4	yes	—
130	C₂²×C₄⋊C₄	—	2	2	no	—
131	C₂²×C₄⋊C₄	—	47	2	no	—
132	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
133	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	C₂×C₄×C₈	—	47	4	no	—
135	C₂×C₄×C₈	—	94	4	no	—
136	#382	—	47	4	no	—
137	#382	—	94	2	no	—
138	C₂×#95	—	94	2	no	—
139	#340	—	47	16	no	—
140	C₄₇	C₂×#1991	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	—	94	2	no	—
143	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
144	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	C₂²×C₈	—	47	8	no	—
146	C₂²×C₈	—	47	4	no	—
147	C₂²×C₈	—	94	4	no	—
148	C₂²×C₈	—	94	4	no	—
149	#125	—	2	16	no	—
150	#125	—	2	8	no	—
151	#125	C₂×Dic₄₇	1	16	yes	complemented
152	C₂×C₄⋊C₄	—	2	16	no	—
153	C₂×C₄⋊C₄	—	2	8	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	47	4	no	—
157	C₂×C₄⋊C₄	—	94	8	no	—
158	C₂×C₄⋊C₄	—	94	2	no	—
159	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
160	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
161	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
162	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
164	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
165	C₄×C₈	—	94	8	no	—
166	#95	—	94	8	no	—
167	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	4	8	no	—
172	C₂²×C₄	—	94	4	no	—
173	C₂²×C₄	—	188	4	no	—
174	C₂²×C₄	—	188	8	no	—
175	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
176	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
177	C₂²×C₄	#10625354	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	C₂×C₈	—	47	8	no	—
184	C₂×C₈	—	94	16	no	—
185	C₂×C₈	—	94	4	no	—
186	C₂×C₈	—	94	4	no	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	94	8	no	—
191	C₄⋊C₄	—	188	4	no	—
192	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
193	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
194	C₄²	C₂²×Dic₄₇	1	4	yes	—
195	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
196	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
197	C₂³	#13094013	1	2	yes	—
198	C₂³	#13094022	1	2	yes	—
199	C₂³	#13094044	1	8	yes	—
200	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂×C₄	—	2	16	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	16	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	—	376	8	no	—
207	C₂×C₄	—	188	4	no	—
208	C₂×C₄	—	188	8	no	—
209	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
211	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
212	C₂×C₄	#13094000	1	8	yes	—
213	C₂×C₄	#13094013	1	2	yes	—
214	C₂×C₄	#13094022	1	2	yes	—
215	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
218	C₂×C₄	C₂×#10625354	1	2	yes	—
219	C₈	—	94	8	no	—
220	C₂²	#15689030	1	4	yes	—
221	C₂²	#15689119	1	8	yes	—
222	C₂²	#15689480	1	8	yes	—
223	C₂²	#15689522	1	8	yes	—
224	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂²	C₂×#13094013	1	1	yes	characteristic
226	C₂²	C₂×#13094022	1	1	yes	characteristic
227	C₂²	C₂×#13094044	1	4	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	—	376	4	no	—
231	C₄	#15689030	1	4	yes	—
232	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
233	C₄	C₂×#13094000	1	2	yes	—
234	C₂	#20244354	1	8	yes	direct product factor
235	C₂	C₂×#15689030	1	1	yes	characteristic
236	C₂	C₂×#15689119	1	2	yes	—
237	C₂	C₂×#15689480	1	2	yes	—
238	C₂	C₂×#15689522	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	C₂×#1991
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
3. #20244354	C₂	direct (N × H)
16. C₂×#125×C₄₇	C₄	semidirect
36. #125×C₄₇	C₂×C₄	semidirect
120. C₂×C₄₇	#1991	semidirect
128. C₂×#125	Dic₄₇	semidirect
140. C₄₇	C₂×#1991	semidirect
151. #125	C₂×Dic₄₇	semidirect
234. C₂	#20244354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688989	2
#20244354	2
C₂×#15688885	2
C₂²×#125×C₄₇	2
C₂×#1991	47

Subgroups — C2×#20244354 #26220834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244354 #26220834