Subgroups — C₂×#20245617 #26371568

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	#20245617	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688926	C₂	1	4	yes	maximal
4	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	—	2	1	no	—
7	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
8	C₂×#13094024	—	2	1	no	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15689503	—	2	2	no	—
11	#15689503	C₂²	1	4	yes	—
12	#15688926	C₂²	1	16	yes	—
13	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
15	C₄₇×#500	—	2	2	no	—
16	C₄₇×#500	C₂²	1	4	yes	—
17	C₂×#107×C₄₇	—	2	1	no	—
18	C₂×#107×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₉₄	—	2	2	no	—
21	C₂×Dic₁₈₈	—	2	8	no	—
22	C₂×Dic₁₈₈	—	2	2	no	—
23	C₂×Dic₁₈₈	—	2	4	no	—
24	C₂×Dic₁₈₈	—	4	4	no	—
25	C₂×Dic₁₈₈	C₂³	1	4	yes	—
26	#13094024	—	2	4	no	—
27	#13094024	—	4	2	no	—
28	C₂×C₄⋊Dic₄₇	—	4	1	no	—
29	C₂×C₄⋊Dic₄₇	—	4	1	no	—
30	C₂×C₄₇⋊C₁₆	—	2	4	no	—
31	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
32	C₂²×Q₈×C₄₇	—	4	1	no	—
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₄₇	—	2	4	no	—
36	C₄⋊Q₈×C₄₇	—	4	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#107×C₄₇	—	2	2	no	—
41	#107×C₄₇	—	2	4	no	—
42	#107×C₄₇	C₂³	1	4	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	C₄×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×Q₈×C₄₇	—	4	4	no	—
46	C₂×Q₈×C₄₇	—	8	4	no	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	4	2	no	—
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×Dic₉₄	—	4	4	no	—
54	C₂×Dic₉₄	—	4	4	no	—
55	C₂×Dic₉₄	—	4	2	no	—
56	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
57	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₄₇	#41	1	4	yes	—
60	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	—	2	4	no	—
62	C₄⋊C₄×C₄₇	—	4	2	no	—
63	C₄⋊C₄×C₄₇	—	4	4	no	—
64	C₄²×C₄₇	D₈	1	2	yes	—
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₄₇	—	4	4	no	—
67	C₄⋊Dic₄₇	—	4	4	no	—
68	Dic₁₈₈	—	2	8	no	—
69	Dic₁₈₈	—	4	4	no	—
70	Dic₁₈₈	—	4	8	no	—
71	C₄₇⋊C₁₆	—	2	4	no	—
72	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₄₇	—	8	8	no	—
74	C₂×C₄×C₄₇	—	4	4	no	—
75	C₂×C₄×C₄₇	—	8	4	no	—
76	C₂×C₄×C₄₇	SD₃₂	1	4	yes	—
77	C₂×C₄×C₄₇	Q₃₂	1	4	yes	—
78	C₂×C₄×C₄₇	#124	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₇	—	4	4	no	—
85	C₂×Dic₄₇	—	8	4	no	—
86	Dic₉₄	—	4	4	no	—
87	Dic₉₄	—	4	4	no	—
88	Dic₉₄	—	8	4	no	—
89	C₈×C₄₇	#124	1	2	yes	—
90	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
91	C₈×C₄₇	C₂×#41	1	2	yes	—
92	C₂×#2241	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	#486	1	2	yes	—
94	C₂²×C₄₇	C₂×SD₃₂	1	2	yes	—
95	C₂²×C₄₇	C₂×Q₃₂	1	2	yes	—
96	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₄₇	—	8	4	no	—
98	C₄×C₄₇	#486	1	2	yes	—
99	C₄×C₄₇	C₂×SD₃₂	1	4	yes	—
100	C₄×C₄₇	C₂×Q₃₂	1	4	yes	—
101	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
102	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
103	Dic₄₇	—	8	4	no	—
104	C₂×#500	D₄₇	1	1	yes	characteristic
105	C₂×#494	—	47	1	no	—
106	#2241	—	47	8	no	—
107	C₂×#358	—	47	4	no	—
108	C₂×C₄×C₁₆	—	47	1	no	—
109	C₂×C₄₇	#2241	1	4	yes	complemented
110	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
111	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
112	C₂×C₄₇	C₂²×Q₃₂	1	1	yes	characteristic
113	C₂²×Q₁₆	—	47	2	no	—
114	C₂²×Q₁₆	—	94	1	no	—
115	C₂×C₄⋊Q₈	—	2	1	no	—
116	C₂×C₄⋊Q₈	—	94	1	no	—
117	C₂²×C₁₆	—	47	2	no	—
118	#500	—	2	2	no	—
119	#500	C₂×D₄₇	1	4	yes	—
120	#494	—	47	4	no	—
121	#494	—	94	2	no	—
122	C₂×#107	—	2	1	no	—
123	C₂×#107	C₂×D₄₇	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
125	#358	—	47	16	no	—
126	C₄×C₁₆	—	47	4	no	—
127	C₄₇	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×Q₈	—	94	2	no	—
130	C₂×Q₁₆	—	47	4	no	—
131	C₂×Q₁₆	—	94	8	no	—
132	C₂×Q₁₆	—	94	2	no	—
133	C₂×Q₁₆	—	94	4	no	—
134	C₂×Q₁₆	—	188	4	no	—
135	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	4	2	no	—
139	C₄⋊Q₈	—	94	4	no	—
140	C₄⋊Q₈	—	188	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	188	1	no	—
144	C₂×C₄⋊C₄	—	188	1	no	—
145	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
146	C₂×C₁₆	—	47	4	no	—
147	C₂×C₁₆	—	94	4	no	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#107	C₂²×D₄₇	1	4	yes	—
151	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
152	C₄×C₈	C₂²×D₄₇	1	2	yes	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂×Q₈	—	94	4	no	—
156	C₂×Q₈	—	188	4	no	—
157	C₂×Q₈	—	188	4	no	—
158	C₂×Q₈	—	188	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	188	2	no	—
161	C₂²×C₄	#10625334	1	1	yes	characteristic
162	C₂²×C₄	#10625334	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
164	Q₁₆	—	94	8	no	—
165	Q₁₆	—	188	4	no	—
166	Q₁₆	—	188	8	no	—
167	C₂×C₈	#10625336	1	4	yes	—
168	C₂×C₈	#10625354	1	4	yes	—
169	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
171	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	188	4	no	—
176	C₄⋊C₄	—	188	4	no	—
177	C₄²	#10625334	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
179	C₁₆	—	94	4	no	—
180	C₂³	C₂×#10625334	1	1	yes	characteristic
181	Q₈	—	8	8	no	—
182	Q₈	—	376	4	no	—
183	Q₈	—	188	4	no	—
184	Q₈	—	188	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	376	4	no	—
188	C₂×C₄	—	188	4	no	—
189	C₂×C₄	#13093968	1	4	yes	—
190	C₂×C₄	#13093970	1	4	yes	—
191	C₂×C₄	#13094013	1	2	yes	—
192	C₂×C₄	C₂×#10625334	1	2	yes	—
193	C₂×C₄	C₂×#10625334	1	2	yes	—
194	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
195	C₂×C₄	C₂×#10625336	1	2	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₈	#13094013	1	2	yes	—
198	C₈	C₂×#10625336	1	4	yes	—
199	C₈	C₂×#10625354	1	2	yes	—
200	C₂²	#15689471	1	2	yes	—
201	C₂²	C₂×#13093968	1	2	yes	—
202	C₂²	C₂×#13093970	1	2	yes	—
203	C₂²	C₂²×#10625334	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	376	4	no	—
206	C₄	#15689471	1	2	yes	—
207	C₄	C₂×#13093968	1	4	yes	—
208	C₄	C₂×#13093970	1	4	yes	—
209	C₄	C₂×#13094013	1	1	yes	characteristic
210	C₄	C₂²×#10625336	1	1	yes	characteristic
211	C₂	#20245617	1	4	yes	direct product factor
212	C₂	C₂×#15689471	1	1	yes	characteristic
213	C₂	C₂²×#13093968	1	1	yes	characteristic
214	C₂	C₂²×#13093970	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	C₂×#2241
LCS	G ▷ C₂×C₈×C₄₇ ▷ 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
2. #20245617	C₂	direct (N × H)
109. C₂×C₄₇	#2241	semidirect
127. C₄₇	C₂×#2241	semidirect
211. C₂	#20245617	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×#15689503	2
#20245617	2
C₂×#15688926	2
C₂×C₄×C₄₇⋊C₁₆	2
C₂×C₄₇×#500	2
C₂×#2241	47

Subgroups — C2×#20245617 #26371568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245617 #26371568