Subgroups — C₂×#27212905 #41013103

Order 27008.

488 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	C₂²×#16809946	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212905	C₂	1	4	yes	maximal, direct product factor
4	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₁₁	—	2	1	no	—
6	C₂×#16809946	—	2	1	no	—
7	C₂×#16809946	—	2	2	no	—
8	C₂×#16809946	C₄	1	4	yes	complemented
9	C₂×#16809946	C₂²	1	2	yes	—
10	C₂²×C₂₁₁⋊C₈	—	2	1	no	—
11	C₂×C₄×Dic₂₁₁	—	2	1	no	—
12	C₂×C₄×Dic₂₁₁	—	2	2	no	—
13	C₂×C₄×Dic₂₁₁	C₄	1	4	yes	complemented
14	C₂×C₄×Dic₂₁₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	#117×C₂₁₁	—	2	1	no	—
17	#117×C₂₁₁	—	2	1	no	—
18	#117×C₂₁₁	C₄	1	2	yes	complemented
19	#117×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	complemented
21	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
22	C₂×C₄²×C₂₁₁	—	2	1	no	—
23	C₂²×Dic₂₁₁	—	2	8	no	—
24	C₂²×Dic₂₁₁	—	2	4	no	—
25	C₂²×Dic₂₁₁	—	2	2	no	—
26	#16809946	—	2	4	no	—
27	#16809946	—	2	2	no	—
28	#16809946	C₂×C₄	1	4	yes	complemented
29	C₂×C₂₁₁⋊C₈	—	2	4	no	—
30	C₂×C₂₁₁⋊C₈	—	2	2	no	—
31	C₄×Dic₂₁₁	—	2	4	no	—
32	C₄×Dic₂₁₁	—	2	2	no	—
33	C₄×Dic₂₁₁	C₂×C₄	1	4	yes	complemented
34	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	—	2	2	no	—
36	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
44	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
45	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₁₁	—	2	2	no	—
47	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
48	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
49	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
50	C₄²×C₂₁₁	—	2	2	no	—
51	C₄²×C₂₁₁	—	2	2	no	—
52	C₂×Dic₂₁₁	—	2	16	no	—
53	C₂×Dic₂₁₁	—	2	8	no	—
54	C₂×Dic₂₁₁	—	2	4	no	—
55	C₂₁₁⋊C₈	—	2	4	no	—
56	C₂³×C₂₁₁	—	2	4	no	—
57	C₂³×C₂₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
61	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₁₁	—	2	4	no	—
63	C₂×C₄×C₂₁₁	—	4	4	no	—
64	C₂×C₄×C₂₁₁	C₄²	1	2	yes	—
65	C₂×C₄×C₂₁₁	C₄²	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₂₁₁	—	2	4	no	—
83	C₂²×C₂₁₁	—	2	4	no	—
84	C₂²×C₂₁₁	—	2	2	no	—
85	C₂²×C₂₁₁	—	2	2	no	—
86	C₂²×C₂₁₁	#95	1	2	yes	—
87	C₂²×C₂₁₁	#95	1	2	yes	—
88	C₂²×C₂₁₁	#104	1	4	yes	—
89	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₂₁₁	—	4	4	no	—
93	C₄×C₂₁₁	#95	1	2	yes	—
94	C₄×C₂₁₁	#95	1	2	yes	—
95	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
96	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	Dic₂₁₁	—	2	8	no	—
100	C₂×C₂₁₁	—	2	1	no	—
101	C₂×C₂₁₁	—	2	2	no	—
102	C₂×C₂₁₁	—	2	1	no	—
103	C₂×C₂₁₁	#339	1	4	yes	complemented
104	C₂×C₂₁₁	C₂×#95	1	1	yes	characteristic
105	C₂×C₂₁₁	C₂×#104	1	2	yes	—
106	C₂₁₁	C₂×#339	1	1	yes	characteristic, complemented, Sylow
107	C₂×#339	—	211	1	no	maximal, Sylow
108	C₂²×M₁₆	—	211	1	no	—
109	C₂×#117	D₂₁₁	1	1	yes	characteristic
110	C₂²×C₄²	—	211	1	no	—
111	#339	—	211	4	no	—
112	C₂³×C₄	—	422	1	no	—
113	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
114	C₂×M₁₆	—	422	1	no	—
115	C₂×M₁₆	—	422	2	no	—
116	C₂×M₁₆	—	211	4	no	—
117	C₂×M₁₆	—	211	2	no	—
118	C₂²×C₈	—	422	1	no	—
119	#117	—	2	1	no	—
120	#117	—	2	1	no	—
121	#117	Dic₂₁₁	1	2	yes	complemented
122	#117	C₂×D₂₁₁	1	2	yes	—
123	C₂×C₄⋊C₄	Dic₂₁₁	1	2	yes	complemented
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	422	1	no	—
127	C₂×C₄²	—	422	2	no	—
128	C₂×C₄²	—	211	4	no	—
129	C₂×C₄²	—	211	2	no	—
130	C₂⁴	C₂²⋊D₂₁₁	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	422	8	no	—
133	C₂²×C₄	—	422	4	no	—
134	C₂²×C₄	—	422	2	no	—
135	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
136	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
137	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
142	C₂²×C₄	D₈₄₄	1	2	yes	—
143	C₂²×C₄	Dic₄₂₂	1	2	yes	—
144	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
145	M₁₆	—	422	4	no	—
146	M₁₆	—	422	2	no	—
147	M₁₆	—	211	4	no	—
148	C₂×C₈	—	422	4	no	—
149	C₂×C₈	—	422	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	complemented
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	422	4	no	—
157	C₄²	—	422	2	no	—
158	C₄²	—	211	4	no	—
159	C₂³	—	2	4	no	—
160	C₂³	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
161	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
162	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
163	C₂³	Dic₂₁₁⋊C₄	1	2	yes	—
164	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	422	8	no	—
168	C₂×C₄	—	422	16	no	—
169	C₂×C₄	—	422	4	no	—
170	C₂×C₄	C₄×Dic₂₁₁	1	2	yes	—
171	C₂×C₄	C₄×Dic₂₁₁	1	2	yes	—
172	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
173	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
174	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
175	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
177	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
178	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
179	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
180	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
181	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
182	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
186	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
187	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
188	C₈	—	422	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂₁₁⋊#95	1	2	yes	—
194	C₂²	C₂₁₁⋊#95	1	2	yes	—
195	C₂²	#20269904	1	4	yes	—
196	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
197	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
198	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
199	C₄	—	4	4	no	—
200	C₄	—	422	8	no	—
201	C₄	C₂₁₁⋊#95	1	2	yes	—
202	C₄	C₂₁₁⋊#95	1	2	yes	—
203	C₄	C₂×C₄×Dic₂₁₁	1	1	yes	characteristic
204	C₄	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
205	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
206	C₄	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#27212905	1	4	yes	direct product factor
211	C₂	C₂×C₂₁₁⋊#95	1	1	yes	characteristic
212	C₂	C₂×#20269904	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#117×C₂₁₁	C₂
socle	C₂³×C₂₁₁	C₂²⋊C₄
prime radical	C₂⁴×C₂₁₁	D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#117	D₂₁₁
hypocenter	C₂₁₁	C₂×#339
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #27212905	C₂	direct (N × H)
8. C₂×#16809946	C₄	semidirect
13. C₂×C₄×Dic₂₁₁	C₄	semidirect
18. #117×C₂₁₁	C₄	semidirect
20. C₂×C₄⋊C₄×C₂₁₁	C₄	semidirect
28. #16809946	C₂×C₄	semidirect
33. C₄×Dic₂₁₁	C₂×C₄	semidirect
47. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
103. C₂×C₂₁₁	#339	semidirect
106. C₂₁₁	C₂×#339	semidirect
121. #117	Dic₂₁₁	semidirect
123. C₂×C₄⋊C₄	Dic₂₁₁	semidirect
151. C₄⋊C₄	C₂×Dic₂₁₁	semidirect
210. C₂	#27212905	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16809946	2
C₂²×C₄×Dic₂₁₁	2
#27212905	2
C₂×#117×C₂₁₁	2
C₂×#339	211

Subgroups — C2×#27212905 #41013103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212905 #41013103