Subgroups — C₂×#20244357 #26220844

Order 12032.

1294 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094016	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20244357	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688886	C₂	1	4	yes	maximal
5	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	—	2	2	no	—
7	C₂×#13094016	—	2	4	no	—
8	C₂×#13094016	C₄	1	8	yes	complemented
9	C₂×#13094016	C₂²	1	4	yes	—
10	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
14	#15688993	—	2	2	no	—
15	#15688993	C₂²	1	4	yes	—
16	#15688886	C₂²	1	16	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₄₇	—	2	4	no	—
20	C₂×#97×C₄₇	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂³×Dic₄₇	—	2	2	no	—
23	#13094016	—	2	16	no	—
24	#13094016	—	2	8	no	—
25	#13094016	C₂×C₄	1	16	yes	complemented
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	8	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	8	no	—
31	C₂×C₄⋊Dic₄₇	—	4	8	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	16	no	—
36	C₂×Dic₄₇⋊C₄	—	2	8	no	—
37	C₂³×C₄×C₄₇	D₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	2	4	no	—
40	C₂²×C₈×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
42	C₂²×C₈×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	#97×C₄₇	—	2	8	no	—
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₄₇	#41	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	8	no	—
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	C₂²×Dic₄₇	—	4	4	no	—
65	C₂²×Dic₄₇	—	4	8	no	—
66	C₂×C₈×C₄₇	—	2	16	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
70	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
72	C₄²×C₄₇	C₂×D₄	1	4	yes	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄⋊Dic₄₇	—	2	4	no	—
75	C₄⋊Dic₄₇	—	2	8	no	—
76	C₄⋊Dic₄₇	—	4	16	no	—
77	C₄⋊Dic₄₇	—	4	4	no	—
78	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
79	Dic₄₇⋊C₄	—	2	32	no	—
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
81	C₂³×C₄₇	#124	1	2	yes	—
82	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
83	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
84	C₂³×C₄₇	#137	1	4	yes	—
85	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	16	no	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
92	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
93	C₂×C₄×C₄₇	#124	1	2	yes	—
94	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	4	4	no	—
101	C₂×Dic₄₇	—	4	4	no	—
102	C₂×Dic₄₇	—	4	16	no	—
103	C₂×Dic₄₇	—	4	4	no	—
104	C₂×Dic₄₇	—	4	8	no	—
105	C₂×Dic₄₇	—	8	8	no	—
106	C₈×C₄₇	—	2	8	no	—
107	C₂×#1992	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	#391	1	4	yes	—
109	C₂²×C₄₇	#417	1	8	yes	—
110	C₂²×C₄₇	#490	1	4	yes	—
111	C₂²×C₄₇	#491	1	4	yes	—
112	C₂²×C₄₇	#501	1	8	yes	—
113	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
117	C₂²×C₄₇	C₂×#137	1	2	yes	—
118	C₄×C₄₇	—	2	8	no	—
119	C₄×C₄₇	#391	1	4	yes	—
120	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
121	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
122	Dic₄₇	—	4	8	no	—
123	Dic₄₇	—	8	4	no	—
124	C₂²×#125	—	47	1	no	—
125	C₂²×#97	D₄₇	1	1	yes	characteristic
126	C₂×#384	—	47	1	no	—
127	#1992	—	47	8	no	—
128	C₂×#340	—	47	4	no	—
129	C₂×C₄₇	#1992	1	8	yes	complemented
130	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#417	1	2	yes	—
132	C₂×C₄₇	C₂×#490	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#501	1	2	yes	—
135	C₂³×C₈	C₂×D₄₇	1	2	yes	—
136	C₂×#125	—	47	8	no	—
137	C₂×#125	—	47	4	no	—
138	C₂×#125	—	94	2	no	—
139	C₂×#125	—	94	4	no	—
140	C₂²×C₄⋊C₄	—	47	2	no	—
141	C₂²×C₄⋊C₄	—	47	2	no	—
142	C₂²×C₄⋊C₄	—	94	2	no	—
143	C₂²×C₄⋊C₄	—	94	2	no	—
144	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
145	C₂×#97	—	2	4	no	—
146	C₂×#97	C₂×D₄₇	1	4	yes	—
147	#384	—	47	4	no	—
148	#384	—	94	2	no	—
149	#340	—	47	16	no	—
150	C₄₇	C₂×#1992	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	94	2	no	—
152	C₂³×C₄	—	94	2	no	—
153	C₂³×C₄	D₁₈₈	1	2	yes	—
154	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	Dic₉₄	1	8	yes	—
158	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
159	#125	—	47	16	no	—
160	#125	—	94	16	no	—
161	#125	—	94	8	no	—
162	C₂×C₄⋊C₄	—	47	8	no	—
163	C₂×C₄⋊C₄	—	47	4	no	—
164	C₂×C₄⋊C₄	—	47	4	no	—
165	C₂×C₄⋊C₄	—	94	16	no	—
166	C₂×C₄⋊C₄	—	94	2	no	—
167	C₂×C₄⋊C₄	—	94	8	no	—
168	C₂×C₄⋊C₄	—	94	4	no	—
169	C₂×C₄⋊C₄	—	94	8	no	—
170	C₂×C₄⋊C₄	—	94	2	no	—
171	C₂×C₄⋊C₄	—	94	8	no	—
172	C₂×C₄⋊C₄	—	188	8	no	—
173	C₂×C₄²	C₄×D₄₇	1	4	yes	—
174	C₂×C₄²	D₁₈₈	1	2	yes	—
175	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
176	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
177	#97	—	2	8	no	—
178	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	94	8	no	—
182	C₂²×C₄	—	94	4	no	—
183	C₂²×C₄	—	94	4	no	—
184	C₂²×C₄	—	188	4	no	—
185	C₂²×C₄	—	188	8	no	—
186	C₂²×C₄	—	188	4	no	—
187	C₂²×C₄	—	188	4	no	—
188	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
189	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
190	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
191	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	#10625350	1	4	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	C₂×C₈	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
200	C₄⋊C₄	—	47	8	no	—
201	C₄⋊C₄	—	94	32	no	—
202	C₄⋊C₄	—	94	8	no	—
203	C₄⋊C₄	—	94	4	no	—
204	C₄⋊C₄	—	94	8	no	—
205	C₄⋊C₄	—	188	16	no	—
206	C₄⋊C₄	—	188	4	no	—
207	C₄²	(C₂×D₄₇)⋊C₄	1	8	yes	—
208	C₄²	C₂×C₄×D₄₇	1	4	yes	—
209	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
210	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
211	C₂³	#13094012	1	2	yes	—
212	C₂³	#13094024	1	2	yes	—
213	C₂³	#13094052	1	4	yes	—
214	C₂³	#13094057	1	4	yes	—
215	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	94	8	no	—
220	C₂×C₄	—	376	8	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	—	188	8	no	—
223	C₂×C₄	—	188	16	no	—
224	C₂×C₄	—	188	4	no	—
225	C₂×C₄	—	188	4	no	—
226	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	8	yes	—
227	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
228	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
229	C₂×C₄	#13094003	1	8	yes	—
230	C₂×C₄	#13094012	1	2	yes	—
231	C₂×C₄	#13094024	1	2	yes	—
232	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
235	C₂×C₄	C₂×#10625350	1	2	yes	—
236	C₈	—	2	8	no	—
237	C₂²	#15689029	1	4	yes	—
238	C₂²	#15689125	1	8	yes	—
239	C₂²	#15689488	1	4	yes	—
240	C₂²	#15689492	1	4	yes	—
241	C₂²	#15689532	1	8	yes	—
242	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂²	C₂×#13094012	1	1	yes	characteristic
244	C₂²	C₂×#13094024	1	1	yes	characteristic
245	C₂²	C₂×#13094052	1	2	yes	—
246	C₂²	C₂×#13094057	1	2	yes	—
247	C₄	—	2	8	no	—
248	C₄	—	376	4	no	—
249	C₄	—	188	8	no	—
250	C₄	#15689029	1	4	yes	—
251	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
252	C₄	C₂×#13094003	1	2	yes	—
253	C₂	#20244357	1	8	yes	direct product factor
254	C₂	C₂×#15689029	1	1	yes	characteristic
255	C₂	C₂×#15689125	1	2	yes	—
256	C₂	C₂×#15689488	1	1	yes	characteristic
257	C₂	C₂×#15689492	1	1	yes	characteristic
258	C₂	C₂×#15689532	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂×#1992
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #20244357	C₂	direct (N × H)
8. C₂×#13094016	C₄	semidirect
25. #13094016	C₂×C₄	semidirect
129. C₂×C₄₇	#1992	semidirect
150. C₄₇	C₂×#1992	semidirect
253. C₂	#20244357	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×#13094016	2
C₂×#15688993	2
#20244357	2
C₂×#15688886	2
C₂²×#97×C₄₇	2
C₂×#1992	47

Subgroups — C2×#20244357 #26220844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244357 #26220844