Subgroups — C₂×#20241447 #26127561

Order 12032.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241447	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	—	2	1	no	—
6	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
7	#15689528	—	2	1	no	—
8	#15689528	—	2	1	no	—
9	#15689528	C₄	1	2	yes	—
10	#15689528	C₂²	1	2	yes	—
11	C₂×#13093958	—	2	1	no	—
12	C₂×#13093958	C₄	1	2	yes	—
13	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
14	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#335	—	2	1	no	—
18	C₄₇×#335	C₂²	1	2	yes	—
19	C₂²×Dic₉₄	—	4	1	no	—
20	#13094024	—	2	2	no	—
21	#13094024	—	2	2	no	—
22	#13094024	—	4	1	no	—
23	#13094024	—	4	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	4	1	no	—
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	C₂×C₄₇⋊C₁₆	—	4	2	no	—
28	#13093958	—	2	2	no	—
29	#13093958	—	2	2	no	—
30	#13093958	—	2	2	no	—
31	#13093958	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₄₇	—	4	1	no	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	4	1	no	—
38	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	4	2	no	—
46	C₂×Dic₉₄	—	4	2	no	—
47	C₂×Dic₉₄	—	4	2	no	—
48	C₂×Dic₉₄	—	8	4	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	4	2	no	—
51	C₂×C₈×C₄₇	—	8	2	no	—
52	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
55	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
56	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₄₇	D₈	1	1	yes	characteristic
60	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
61	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Dic₄₇	—	2	4	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₄⋊Dic₄₇	—	4	2	no	—
66	C₄₇⋊C₁₆	—	4	2	no	—
67	C₂³×C₄₇	#102	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	#101	1	2	yes	—
70	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#103	1	2	yes	—
74	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
75	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×Dic₄₇	—	4	4	no	—
80	C₂×Dic₄₇	—	8	4	no	—
81	Dic₉₄	—	8	4	no	—
82	Dic₉₄	—	8	4	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	2	1	no	—
85	C₈×C₄₇	—	8	2	no	—
86	C₈×C₄₇	#103	1	2	yes	—
87	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×#1365	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	#332	1	2	yes	—
91	C₂²×C₄₇	#358	1	2	yes	—
92	C₂²×C₄₇	#361	1	2	yes	—
93	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	#332	1	2	yes	—
97	C₄×C₄₇	#358	1	2	yes	—
98	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
99	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
100	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
101	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
102	Dic₄₇	—	8	4	no	—
103	C₂×#500	—	47	1	no	—
104	C₂×C₄⋊C₁₆	—	47	1	no	—
105	C₂×#335	D₄₇	1	1	yes	characteristic
106	#1365	—	47	4	no	—
107	C₂×C₄₇	#1365	1	4	yes	complemented
108	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
110	C₂×C₄₇	C₂×#361	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	94	1	no	—
112	C₂²×C₁₆	—	94	1	no	—
113	#500	—	47	2	no	—
114	#500	—	47	2	no	—
115	#500	—	94	1	no	—
116	#500	—	94	1	no	—
117	C₂×#107	—	47	2	no	—
118	C₂×#107	—	94	1	no	—
119	C₂×C₄⋊C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	C₄⋊C₁₆	—	47	2	no	—
122	C₄⋊C₁₆	—	94	1	no	—
123	#335	—	2	1	no	—
124	#335	C₂×D₄₇	1	2	yes	—
125	C₄₇	C₂×#1365	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	188	1	no	—
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
129	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
130	C₄⋊Q₈	—	94	2	no	—
131	C₄⋊Q₈	—	94	2	no	—
132	C₄⋊Q₈	—	188	1	no	—
133	C₄⋊Q₈	—	188	1	no	—
134	C₂×C₄⋊C₄	—	94	2	no	—
135	C₂×C₄⋊C₄	—	188	1	no	—
136	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂×C₁₆	—	94	2	no	—
138	C₂×C₁₆	—	188	2	no	—
139	#107	—	47	4	no	—
140	#107	—	94	2	no	—
141	#107	—	94	2	no	—
142	#107	—	94	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	4	1	no	—
145	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
146	C₄×C₈	D₁₈₈	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
149	C₂×Q₈	—	376	4	no	—
150	C₂×Q₈	—	188	2	no	—
151	C₂×Q₈	—	188	2	no	—
152	C₂²×C₄	—	188	2	no	—
153	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	#10625334	1	1	yes	characteristic
155	C₂²×C₄	#10625336	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	8	2	no	—
159	C₂×C₈	#10625325	1	2	yes	—
160	C₂×C₈	Dic₁₈₈	1	2	yes	—
161	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
162	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
165	C₄⋊C₄	—	94	4	no	—
166	C₄⋊C₄	—	188	2	no	—
167	C₄⋊C₄	—	188	2	no	—
168	C₄⋊C₄	—	188	2	no	—
169	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
170	C₄²	#10625334	1	1	yes	characteristic
171	C₄²	#10625336	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
173	C₁₆	—	188	2	no	—
174	C₂³	#13093944	1	1	yes	characteristic
175	Q₈	—	376	4	no	—
176	Q₈	—	376	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	376	4	no	—
179	C₂×C₄	—	188	4	no	—
180	C₂×C₄	#13093942	1	2	yes	—
181	C₂×C₄	#13093944	1	1	yes	characteristic
182	C₂×C₄	#13093944	1	1	yes	characteristic
183	C₂×C₄	#13093944	1	1	yes	characteristic
184	C₂×C₄	#13093949	1	2	yes	—
185	C₂×C₄	#13093968	1	2	yes	—
186	C₂×C₄	#13093970	1	2	yes	—
187	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
189	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	8	2	no	—
193	C₈	#13093949	1	2	yes	—
194	C₈	C₂×#10625325	1	1	yes	characteristic
195	C₈	C₂×Dic₁₈₈	1	1	yes	characteristic
196	C₂²	#15688870	1	2	yes	—
197	C₂²	#15688926	1	2	yes	—
198	C₂²	#15688935	1	2	yes	—
199	C₂²	C₂×#13093944	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	376	4	no	—
203	C₄	#15688870	1	2	yes	—
204	C₄	#15688926	1	2	yes	—
205	C₄	C₂×#13093942	1	1	yes	characteristic
206	C₄	C₂×#13093949	1	1	yes	characteristic
207	C₄	C₂×#13093968	1	1	yes	characteristic
208	C₄	C₂×#13093970	1	1	yes	characteristic
209	C₂	#20241447	1	4	yes	direct product factor
210	C₂	C₂×#15688870	1	1	yes	characteristic
211	C₂	C₂×#15688926	1	1	yes	characteristic
212	C₂	C₂×#15688935	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	C₂×#1365
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20241447	C₂	direct (N × H)
107. C₂×C₄₇	#1365	semidirect
125. C₄₇	C₂×#1365	semidirect
209. C₂	#20241447	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689528	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20241447	2
C₂×C₄₇×#335	2
C₂×#1365	47

Subgroups — C2×#20241447 #26127561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241447 #26127561