Subgroups — C₂×#27212914 #41013133

Order 27008.

432 conjugacy classes of subgroups.

The subgroup lattice has 222 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	#27212914	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₁₁	C₄	1	2	yes	—
6	C₂×#16809946	—	2	2	no	—
7	C₂×#16809946	—	2	1	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	1	no	—
13	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
14	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
15	C₂×C₄⋊Dic₂₁₁	C₄	1	2	yes	—
16	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
17	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₁₁	—	2	2	no	—
19	C₂×M₁₆×C₂₁₁	—	2	1	no	—
20	C₂×M₁₆×C₂₁₁	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
22	C₂²×C₈×C₂₁₁	—	2	1	no	—
23	C₂²×Dic₂₁₁	—	2	2	no	—
24	C₂²×Dic₂₁₁	—	2	2	no	—
25	C₂²×Dic₂₁₁	—	4	4	no	—
26	C₂²×Dic₂₁₁	C₂×C₄	1	4	yes	—
27	#16809946	—	2	4	no	—
28	#16809946	—	2	2	no	—
29	#16809946	C₂×C₄	1	4	yes	complemented
30	C₂×C₂₁₁⋊C₈	—	2	4	no	—
31	C₂×C₂₁₁⋊C₈	—	2	2	no	—
32	C₄⋊Dic₂₁₁	—	2	2	no	—
33	C₄⋊Dic₂₁₁	—	2	2	no	—
34	C₄⋊Dic₂₁₁	—	2	2	no	—
35	C₄⋊Dic₂₁₁	—	2	2	no	—
36	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
41	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
46	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₁₁	—	2	4	no	—
48	M₁₆×C₂₁₁	—	2	2	no	—
49	M₁₆×C₂₁₁	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₂₁₁	—	2	4	no	—
51	C₂×C₈×C₂₁₁	—	2	2	no	—
52	C₂×Dic₂₁₁	—	2	4	no	—
53	C₂×Dic₂₁₁	—	4	4	no	—
54	C₂×Dic₂₁₁	—	4	4	no	—
55	C₂×Dic₂₁₁	—	4	4	no	—
56	C₂₁₁⋊C₈	—	2	4	no	—
57	C₂³×C₂₁₁	—	2	2	no	—
58	C₂³×C₂₁₁	—	2	2	no	—
59	C₂³×C₂₁₁	C₄²	1	2	yes	—
60	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
63	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₁₁	C₄²	1	2	yes	—
65	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
70	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
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	C₂×D₄	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	C₈×C₂₁₁	—	2	4	no	—
83	C₂²×C₂₁₁	—	2	4	no	—
84	C₂²×C₂₁₁	—	2	4	no	—
85	C₂²×C₂₁₁	—	2	1	no	—
86	C₂²×C₂₁₁	—	2	1	no	—
87	C₂²×C₂₁₁	—	2	1	no	—
88	C₂²×C₂₁₁	—	2	1	no	—
89	C₂²×C₂₁₁	#95	1	2	yes	—
90	C₂²×C₂₁₁	#95	1	2	yes	—
91	C₂²×C₂₁₁	#100	1	2	yes	—
92	C₂²×C₂₁₁	#101	1	2	yes	—
93	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
94	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₂₁₁	#95	1	2	yes	—
97	C₄×C₂₁₁	#95	1	2	yes	—
98	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	Dic₂₁₁	—	4	4	no	—
103	C₂×C₂₁₁	—	2	2	no	—
104	C₂×C₂₁₁	—	2	1	no	—
105	C₂×C₂₁₁	—	2	1	no	—
106	C₂×C₂₁₁	#343	1	4	yes	complemented
107	C₂×C₂₁₁	C₂×#95	1	1	yes	characteristic
108	C₂×C₂₁₁	C₂×#100	1	1	yes	characteristic
109	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
110	C₂₁₁	C₂×#343	1	1	yes	characteristic, complemented, Sylow
111	C₂×#343	—	211	1	no	maximal, Sylow
112	C₂²×M₁₆	—	211	1	no	—
113	C₂²×M₁₆	D₂₁₁	1	1	yes	characteristic
114	C₂²×C₄⋊C₄	—	211	1	no	—
115	#343	—	211	4	no	—
116	C₂³×C₄	—	211	2	no	—
117	C₂³×C₄	C₂×D₂₁₁	1	1	yes	characteristic
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₂×M₁₆	—	422	2	no	—
121	C₂×M₁₆	—	422	1	no	—
122	C₂×M₁₆	—	211	4	no	—
123	C₂×M₁₆	—	211	2	no	—
124	C₂×M₁₆	Dic₂₁₁	1	4	yes	complemented
125	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	422	1	no	—
128	C₂×C₄⋊C₄	—	422	1	no	—
129	C₂×C₄⋊C₄	—	422	1	no	—
130	C₂×C₄⋊C₄	—	422	1	no	—
131	C₂×C₄⋊C₄	—	422	1	no	—
132	C₂×C₄⋊C₄	—	211	2	no	—
133	C₂×C₄⋊C₄	—	211	2	no	—
134	C₂⁴	C₄×D₂₁₁	1	1	yes	characteristic
135	C₂²×C₄	—	844	4	no	—
136	C₂²×C₄	—	422	2	no	—
137	C₂²×C₄	—	422	2	no	—
138	C₂²×C₄	—	211	4	no	—
139	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
140	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
141	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
142	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
143	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
144	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
145	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
146	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
147	C₂²×C₄	Dic₄₂₂	1	2	yes	—
148	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
149	M₁₆	—	2	4	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	422	4	no	—
152	M₁₆	—	422	2	no	—
153	M₁₆	—	211	4	no	—
154	M₁₆	C₂×Dic₂₁₁	1	4	yes	complemented
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	422	4	no	—
158	C₂×C₈	—	422	2	no	—
159	C₄⋊C₄	—	422	2	no	—
160	C₄⋊C₄	—	422	2	no	—
161	C₄⋊C₄	—	422	2	no	—
162	C₄⋊C₄	—	422	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₄×Dic₂₁₁	1	2	yes	—
166	C₂³	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
167	C₂³	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
168	C₂³	Dic₂₁₁⋊C₄	1	2	yes	—
169	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
170	C₂×C₄	—	844	4	no	—
171	C₂×C₄	—	844	4	no	—
172	C₂×C₄	—	844	4	no	—
173	C₂×C₄	—	422	4	no	—
174	C₂×C₄	C₄×Dic₂₁₁	1	2	yes	—
175	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
177	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
178	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
179	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
180	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
181	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
182	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
183	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
184	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
185	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
186	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
190	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	422	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	C₂₁₁⋊#95	1	2	yes	—
201	C₂²	C₂₁₁⋊#95	1	2	yes	—
202	C₂²	#20269893	1	2	yes	—
203	C₂²	#20269895	1	2	yes	—
204	C₂²	C₂×C₄×Dic₂₁₁	1	1	yes	characteristic
205	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
206	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
207	C₄	—	844	4	no	—
208	C₄	C₂₁₁⋊#95	1	2	yes	—
209	C₄	C₂₁₁⋊#95	1	2	yes	—
210	C₄	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
211	C₄	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
212	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
213	C₄	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#27212914	1	4	yes	direct product factor
218	C₂	C₂×C₂₁₁⋊#95	1	1	yes	characteristic
219	C₂	C₂×#20269893	1	1	yes	characteristic
220	C₂	C₂×#20269895	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₂₁₁)⋊C₄
commutator	C₂²×C₂₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂²×M₁₆×C₂₁₁	C₂
socle	C₂³×C₂₁₁	C₂²⋊C₄
prime radical	C₂⁴×C₂₁₁	C₂×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂²×M₁₆	D₂₁₁
hypocenter	C₂₁₁	C₂×#343
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27212914	C₂	direct (N × H)
8. C₂×#16809946	C₄	semidirect
20. C₂×M₁₆×C₂₁₁	C₄	semidirect
29. #16809946	C₂×C₄	semidirect
49. M₁₆×C₂₁₁	C₂×C₄	semidirect
106. C₂×C₂₁₁	#343	semidirect
110. C₂₁₁	C₂×#343	semidirect
124. C₂×M₁₆	Dic₂₁₁	semidirect
154. M₁₆	C₂×Dic₂₁₁	semidirect
217. C₂	#27212914	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
#27212914	2
C₂²×M₁₆×C₂₁₁	2
C₂×#343	211

Subgroups — C2×#27212914 #41013133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212914 #41013133