Subgroups — C₂×#20241446 #26127560

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×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20241446	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
9	#15688876	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
12	C₄₇×#500	—	2	1	no	—
13	C₄₇×#500	—	2	1	no	—
14	C₄₇×#500	C₄	1	2	yes	—
15	C₄₇×#500	C₂²	1	2	yes	—
16	C₂×#107×C₄₇	—	2	1	no	—
17	C₂×#107×C₄₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₈	—	4	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
24	C₂²×Q₈×C₄₇	—	4	1	no	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₇	—	2	2	no	—
28	C₄⋊Q₈×C₄₇	—	2	2	no	—
29	C₄⋊Q₈×C₄₇	—	4	1	no	—
30	C₄⋊Q₈×C₄₇	—	4	1	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	#107×C₄₇	—	2	2	no	—
35	#107×C₄₇	—	2	2	no	—
36	#107×C₄₇	—	2	2	no	—
37	#107×C₄₇	C₂×C₄	1	4	yes	—
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	C₂×Q₈×C₄₇	—	4	2	no	—
43	C₂×Q₈×C₄₇	—	4	2	no	—
44	C₂×Q₈×C₄₇	—	8	4	no	—
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
53	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	4	2	no	—
58	C₄⋊C₄×C₄₇	—	4	2	no	—
59	C₄⋊C₄×C₄₇	—	4	2	no	—
60	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₄₇	D₈	1	1	yes	characteristic
62	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
63	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	8	2	no	—
66	C₄₇⋊C₁₆	—	4	2	no	—
67	C₂³×C₄₇	#102	1	1	yes	characteristic
68	Q₈×C₄₇	—	8	4	no	—
69	Q₈×C₄₇	—	8	4	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	4	no	—
72	C₂×C₄×C₄₇	—	8	4	no	—
73	C₂×C₄×C₄₇	#101	1	2	yes	—
74	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#103	1	2	yes	—
78	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
79	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	2	1	no	—
85	C₈×C₄₇	#103	1	2	yes	—
86	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₄₇⋊C₈	—	8	2	no	—
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	—	8	4	no	—
97	C₄×C₄₇	#332	1	2	yes	—
98	C₄×C₄₇	#358	1	2	yes	—
99	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
100	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
101	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
102	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
103	C₂×#500	D₄₇	1	1	yes	characteristic
104	C₂×C₄⋊C₁₆	—	47	1	no	—
105	C₂×#335	—	47	1	no	—
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	—	2	1	no	—
112	C₂²×C₁₆	—	94	1	no	—
113	#500	—	2	1	no	—
114	#500	—	2	1	no	—
115	#500	Dic₄₇	1	2	yes	—
116	#500	C₂×D₄₇	1	2	yes	—
117	C₂×#107	—	2	1	no	—
118	C₂×#107	Dic₄₇	1	2	yes	—
119	C₂×C₄⋊C₈	—	94	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	—	47	2	no	—
124	#335	—	94	1	no	—
125	C₄₇	C₂×#1365	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	4	1	no	—
127	C₂²×C₈	—	188	1	no	—
128	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	4	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	—	2	2	no	—
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#107	C₂×Dic₄₇	1	4	yes	—
143	C₄⋊C₈	—	94	2	no	—
144	C₄⋊C₈	—	188	1	no	—
145	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₄₇	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	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	8	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	#10625334	1	1	yes	characteristic
154	C₂²×C₄	#10625335	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	376	2	no	—
158	C₂×C₈	—	188	2	no	—
159	C₂×C₈	#10625336	1	2	yes	—
160	C₂×C₈	#10625337	1	2	yes	—
161	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
162	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
163	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	#10625334	1	1	yes	characteristic
170	C₄²	#10625335	1	1	yes	characteristic
171	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
173	C₁₆	—	188	2	no	—
174	C₂³	#13093945	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#13093943	1	2	yes	—
181	C₂×C₄	#13093945	1	1	yes	characteristic
182	C₂×C₄	#13093945	1	1	yes	characteristic
183	C₂×C₄	#13093945	1	1	yes	characteristic
184	C₂×C₄	#13093948	1	2	yes	—
185	C₂×C₄	#13093967	1	2	yes	—
186	C₂×C₄	#13093970	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
188	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
189	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	376	2	no	—
193	C₈	#13093948	1	2	yes	—
194	C₈	C₂×#10625336	1	1	yes	characteristic
195	C₈	C₂×#10625337	1	1	yes	characteristic
196	C₂²	#15688869	1	2	yes	—
197	C₂²	#15688927	1	2	yes	—
198	C₂²	#15688936	1	2	yes	—
199	C₂²	C₂×#13093945	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	#15688869	1	2	yes	—
204	C₄	#15688927	1	2	yes	—
205	C₄	C₂×#13093943	1	1	yes	characteristic
206	C₄	C₂×#13093948	1	1	yes	characteristic
207	C₄	C₂×#13093967	1	1	yes	characteristic
208	C₄	C₂×#13093970	1	1	yes	characteristic
209	C₂	#20241446	1	4	yes	direct product factor
210	C₂	C₂×#15688869	1	1	yes	characteristic
211	C₂	C₂×#15688927	1	1	yes	characteristic
212	C₂	C₂×#15688936	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#500	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×#500

Complements

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

N	H ≅ G/N	type
3. #20241446	C₂	direct (N × H)
107. C₂×C₄₇	#1365	semidirect
125. C₄₇	C₂×#1365	semidirect
209. C₂	#20241446	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×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688876	2
#20241446	2
C₂×C₄₇×#500	2
C₂×#1365	47

Subgroups — C2×#20241446 #26127560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241446 #26127560