Subgroups — C₂×#20241442 #26127555

Order 12032.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 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×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20241442	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	—	2	1	no	—
6	#15689528	—	2	1	no	—
7	#15689528	—	2	1	no	—
8	#15689528	C₄	1	2	yes	—
9	#15689528	C₂²	1	2	yes	—
10	C₂×#13093958	—	2	1	no	—
11	C₂×#13093958	C₄	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
13	#15688876	—	2	1	no	—
14	#15688876	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₄₇	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×C₄⋊C₁₆	—	2	1	no	—
18	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×Dic₉₄	—	4	1	no	—
20	C₂²×C₄₇⋊C₈	—	4	1	no	—
21	#13094024	—	2	2	no	—
22	#13094024	—	2	2	no	—
23	#13094024	—	4	1	no	—
24	#13094024	—	4	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	4	1	no	—
27	#13093958	—	2	2	no	—
28	#13093958	—	2	2	no	—
29	#13093958	—	2	2	no	—
30	#13093958	C₂×C₄	1	4	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	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	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₂×C₁₆×C₄₇	—	4	2	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	2	no	—
49	C₂×Dic₉₄	—	8	2	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
54	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₄₇	D₈	1	1	yes	characteristic
59	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₄₇	—	2	4	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	—	8	2	no	—
67	C₁₆×C₄₇	—	4	2	no	—
68	C₂³×C₄₇	#102	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	#101	1	2	yes	—
71	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#103	1	2	yes	—
75	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
76	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×Dic₄₇	—	4	4	no	—
81	C₂×Dic₄₇	—	8	4	no	—
82	Dic₉₄	—	8	4	no	—
83	Dic₉₄	—	8	2	no	—
84	Dic₉₄	—	8	2	no	—
85	C₈×C₄₇	—	2	1	no	—
86	C₈×C₄₇	—	2	1	no	—
87	C₈×C₄₇	#103	1	2	yes	—
88	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₄₇⋊C₈	—	8	2	no	—
91	C₂×#1364	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	#332	1	2	yes	—
93	C₂²×C₄₇	#357	1	2	yes	—
94	C₂²×C₄₇	#362	1	2	yes	—
95	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	#332	1	2	yes	—
99	C₄×C₄₇	#357	1	2	yes	—
100	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
101	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
102	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
103	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
104	Dic₄₇	—	8	4	no	—
105	C₂×#500	—	47	1	no	—
106	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
107	C₂×#335	—	47	1	no	—
108	#1364	—	47	4	no	—
109	C₂×C₄₇	#1364	1	4	yes	complemented
110	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
111	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
112	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	94	1	no	—
114	C₂²×C₁₆	—	2	1	no	—
115	#500	—	47	2	no	—
116	#500	—	47	2	no	—
117	#500	—	94	1	no	—
118	#500	—	94	1	no	—
119	C₂×#107	—	47	2	no	—
120	C₂×#107	—	94	1	no	—
121	C₂×C₄⋊C₈	—	94	1	no	—
122	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
123	C₄⋊C₁₆	—	2	1	no	—
124	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
125	#335	—	47	2	no	—
126	#335	—	94	1	no	—
127	C₄₇	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	188	1	no	—
129	C₂²×C₈	—	188	1	no	—
130	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
132	C₄⋊Q₈	—	94	2	no	—
133	C₄⋊Q₈	—	94	2	no	—
134	C₄⋊Q₈	—	188	1	no	—
135	C₄⋊Q₈	—	188	1	no	—
136	C₂×C₄⋊C₄	—	94	2	no	—
137	C₂×C₄⋊C₄	—	188	1	no	—
138	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	4	2	no	—
141	#107	—	47	4	no	—
142	#107	—	94	2	no	—
143	#107	—	94	2	no	—
144	#107	—	94	2	no	—
145	C₄⋊C₈	—	94	2	no	—
146	C₄⋊C₈	—	188	1	no	—
147	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
148	C₄×C₈	D₁₈₈	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
151	C₂×Q₈	—	376	2	no	—
152	C₂×Q₈	—	376	2	no	—
153	C₂×Q₈	—	188	2	no	—
154	C₂×Q₈	—	188	2	no	—
155	C₂²×C₄	—	188	2	no	—
156	C₂²×C₄	#10625325	1	1	yes	characteristic
157	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
158	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	376	2	no	—
161	C₂×C₈	—	188	2	no	—
162	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
163	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
164	C₂×C₈	#10625335	1	2	yes	—
165	C₂×C₈	#10625337	1	2	yes	—
166	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₄⋊C₄	—	94	4	no	—
169	C₄⋊C₄	—	188	2	no	—
170	C₄⋊C₄	—	188	2	no	—
171	C₄⋊C₄	—	188	2	no	—
172	C₄²	#10625325	1	1	yes	characteristic
173	C₄²	D₃₇₆	1	1	yes	characteristic
174	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
175	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
176	C₁₆	—	4	2	no	—
177	C₂³	#13093946	1	1	yes	characteristic
178	Q₈	—	376	2	no	—
179	Q₈	—	376	2	no	—
180	Q₈	—	376	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	376	4	no	—
183	C₂×C₄	—	188	4	no	—
184	C₂×C₄	#13093942	1	2	yes	—
185	C₂×C₄	#13093946	1	1	yes	characteristic
186	C₂×C₄	#13093946	1	1	yes	characteristic
187	C₂×C₄	#13093946	1	1	yes	characteristic
188	C₂×C₄	#13093947	1	2	yes	—
189	C₂×C₄	D₇₅₂	1	2	yes	—
190	C₂×C₄	#13093969	1	2	yes	—
191	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
193	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	376	2	no	—
197	C₈	#13093947	1	2	yes	—
198	C₈	C₂×#10625335	1	1	yes	characteristic
199	C₈	C₂×#10625337	1	1	yes	characteristic
200	C₂²	#15688868	1	2	yes	—
201	C₂²	#15688925	1	2	yes	—
202	C₂²	#15688940	1	2	yes	—
203	C₂²	C₂×#13093946	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	376	4	no	—
207	C₄	#15688868	1	2	yes	—
208	C₄	#15688925	1	2	yes	—
209	C₄	C₂×#13093942	1	1	yes	characteristic
210	C₄	C₂×#13093947	1	1	yes	characteristic
211	C₄	C₂×D₇₅₂	1	1	yes	characteristic
212	C₄	C₂×#13093969	1	1	yes	characteristic
213	C₂	#20241442	1	4	yes	direct product factor
214	C₂	C₂×#15688868	1	1	yes	characteristic
215	C₂	C₂×#15688925	1	1	yes	characteristic
216	C₂	C₂×#15688940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093946
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#1364
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20241442	C₂	direct (N × H)
109. C₂×C₄₇	#1364	semidirect
127. C₄₇	C₂×#1364	semidirect
213. C₂	#20241442	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×#15688876	2
#20241442	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×#1364	47

Subgroups — C2×#20241442 #26127555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241442 #26127555