Subgroups — C₂×#20253912 #27085894

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689680	C₂	1	2	yes	maximal
2	#20253912	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689532	C₂	1	8	yes	maximal, complemented
4	C₂×#15689161	C₂	1	4	yes	maximal
5	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689680	—	2	8	no	—
7	#15689680	C₂²	1	16	yes	—
8	C₂×#13094024	C₂²	1	4	yes	—
9	C₂×#13094016	C₂²	1	4	yes	—
10	C₂×#13094003	—	2	4	no	—
11	C₂×C₄×Dic₉₄	—	2	4	no	—
12	C₂×#13093983	C₂²	1	4	yes	—
13	#15689532	C₂²	1	64	yes	complemented
14	#15689161	C₂²	1	32	yes	—
15	C₂×#13093958	C₂²	1	8	yes	—
16	C₂×#13093956	C₂²	1	8	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₄₇	C₂²	1	4	yes	—
20	C₂×#97×C₄₇	C₂²	1	8	yes	—
21	C₂²×Dic₉₄	—	2	4	no	—
22	#13094024	—	2	8	no	—
23	#13094024	C₂³	1	16	yes	—
24	#13094016	—	2	8	no	—
25	#13094016	C₂³	1	16	yes	—
26	#13094003	—	2	32	no	—
27	C₄×Dic₉₄	—	2	32	no	—
28	#13093983	—	2	8	no	—
29	#13093983	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
32	C₂×Dic₄₇⋊C₄	—	2	8	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	#13093958	C₂³	1	32	yes	—
36	#13093956	C₂³	1	32	yes	—
37	C₂³×C₄×C₄₇	D₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	Q₈	1	16	yes	complemented
40	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	8	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
46	#97×C₄₇	C₂³	1	16	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
56	C₂²×Dic₄₇	—	2	8	no	—
57	C₂×Dic₉₄	—	2	16	no	—
58	C₂×Dic₉₄	—	4	16	no	—
59	C₂×C₈×C₄₇	C₂×Q₈	1	16	yes	—
60	C₂×C₈×C₄₇	C₂×Q₈	1	32	yes	complemented
61	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
62	C₄²×C₄₇	C₂×D₄	1	8	yes	—
63	C₄²×C₄₇	C₂×D₄	1	4	yes	—
64	C₄²×C₄₇	C₂⁴	1	4	yes	—
65	C₂²⋊Dic₄₇	—	2	16	no	—
66	C₄⋊Dic₄₇	—	2	16	no	—
67	C₄⋊Dic₄₇	—	2	8	no	—
68	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
69	Dic₄₇⋊C₄	—	2	32	no	—
70	C₄×Dic₄₇	—	2	16	no	—
71	C₂³×C₄₇	—	2	4	no	—
72	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
74	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	8	no	—
76	C₂×C₄×C₄₇	—	4	8	no	—
77	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
78	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
79	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
80	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
84	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₄₇	—	2	16	no	—
86	C₂×Dic₄₇	—	4	16	no	—
87	Dic₉₄	—	4	32	no	—
88	C₈×C₄₇	C₂²×Q₈	1	16	yes	—
89	C₂×#3162	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	8	no	—
91	C₂²×C₄₇	—	2	4	no	—
92	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
93	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
94	C₂²×C₄₇	#575	1	4	yes	—
95	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
96	C₄×C₄₇	—	2	8	no	—
97	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
98	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
99	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
100	Dic₄₇	—	4	16	no	—
101	C₂×#533	—	47	2	no	—
102	#3162	—	47	16	no	—
103	C₂×#501	—	47	8	no	—
104	C₂×#427	—	47	4	no	—
105	C₂²×#97	D₄₇	1	1	yes	characteristic
106	C₂×C₄₇	—	2	4	no	—
107	C₂×C₄₇	#3162	1	4	yes	complemented
108	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#575	1	2	yes	—
110	C₂³×C₈	C₂×D₄₇	1	2	yes	—
111	#533	—	47	16	no	—
112	#533	—	94	8	no	—
113	C₂×C₄⋊Q₈	—	47	4	no	—
114	C₂×#125	—	47	4	no	—
115	C₂×C₂²⋊Q₈	—	94	4	no	—
116	C₂×C₄×Q₈	—	94	4	no	—
117	C₂×#117	—	47	4	no	—
118	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
119	#501	—	47	64	no	—
120	#427	—	47	32	no	—
121	C₂×#107	—	47	8	no	—
122	C₂×#106	—	47	8	no	—
123	C₂×#97	C₂×D₄₇	1	4	yes	—
124	C₂×#97	C₂×D₄₇	1	8	yes	—
125	C₄₇	C₂×#3162	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	94	4	no	—
127	C₂³×C₄	D₁₈₈	1	2	yes	—
128	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₂²×C₈	Dic₉₄	1	16	yes	complemented
130	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
131	C₂²×C₈	C₂²×D₄₇	1	8	yes	—
132	C₄⋊Q₈	—	47	16	no	—
133	C₄⋊Q₈	—	94	8	no	—
134	#125	—	47	16	no	—
135	#125	—	94	8	no	—
136	C₂²⋊Q₈	—	94	32	no	—
137	C₄×Q₈	—	94	32	no	—
138	#117	—	47	16	no	—
139	#117	—	94	8	no	—
140	C₂×C₄⋊C₄	—	47	8	no	—
141	C₂×C₄⋊C₄	—	94	8	no	—
142	C₂×C₄⋊C₄	—	94	4	no	—
143	C₂×C₂²⋊C₄	—	94	4	no	—
144	C₂×C₄²	—	94	4	no	—
145	C₂×C₄²	D₁₈₈	1	4	yes	—
146	C₂×C₄²	D₁₈₈	1	2	yes	—
147	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
148	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
149	#107	—	47	32	no	—
150	#106	—	47	32	no	—
151	#97	C₂²×D₄₇	1	16	yes	—
152	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
153	C₂×Q₈	—	94	16	no	—
154	C₂×Q₈	—	188	16	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	94	8	no	—
157	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
158	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
159	C₂²×C₄	C₂×D₁₈₈	1	8	yes	—
160	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
161	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
162	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
164	C₂×C₈	C₂×Dic₉₄	1	16	yes	—
165	C₂×C₈	C₂×Dic₉₄	1	32	yes	complemented
166	C₂×C₈	C₂³×D₄₇	1	8	yes	—
167	C₄⋊C₄	—	47	16	no	—
168	C₄⋊C₄	—	94	32	no	—
169	C₄⋊C₄	—	94	8	no	—
170	C₄⋊C₄	—	94	16	no	—
171	C₂²⋊C₄	—	94	16	no	—
172	C₄²	—	94	16	no	—
173	C₄²	C₂×D₁₈₈	1	8	yes	—
174	C₄²	C₂×D₁₈₈	1	4	yes	—
175	C₄²	C₂³×D₄₇	1	4	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	#13094024	1	4	yes	—
178	C₂³	C₂²×D₁₈₈	1	2	yes	—
179	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
180	Q₈	—	188	32	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	94	16	no	—
184	C₂×C₄	—	188	16	no	—
185	C₂×C₄	#13094024	1	8	yes	—
186	C₂×C₄	#13094024	1	4	yes	—
187	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
188	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
189	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
191	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
192	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
193	C₈	C₂²×Dic₉₄	1	16	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×#13094024	1	4	yes	—
197	C₂²	C₂×#13094024	1	2	yes	—
198	C₂²	#15689923	1	4	yes	—
199	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	188	16	no	—
202	C₄	C₂×#13094024	1	2	yes	—
203	C₄	C₂×#13094024	1	4	yes	—
204	C₄	C₂³×Dic₉₄	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	#20253912	1	4	yes	direct product factor
207	C₂	C₂²×#13094024	1	1	yes	characteristic
208	C₂	C₂×#15689923	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂×#3162
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #20253912	C₂	direct (N × H)
3. C₂×#15689532	C₂	semidirect
13. #15689532	C₂²	semidirect
39. C₂²×C₈×C₄₇	Q₈	semidirect
60. C₂×C₈×C₄₇	C₂×Q₈	semidirect
107. C₂×C₄₇	#3162	semidirect
125. C₄₇	C₂×#3162	semidirect
129. C₂²×C₈	Dic₉₄	semidirect
165. C₂×C₈	C₂×Dic₉₄	semidirect
206. C₂	#20253912	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×#15689680	2
#20253912	2
C₂×#15689532	2
C₂×#15689161	2
C₂²×#97×C₄₇	2
C₂×#3162	47

Subgroups — C2×#20253912 #27085894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253912 #27085894