Subgroups — C₂×#20298572 #28399031

Order 14464.

788 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813581	C₂	1	2	yes	maximal, complemented
2	#20298572	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16813537	C₂	1	4	yes	maximal, complemented
4	C₂²×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	#16813581	—	2	4	no	—
6	#16813581	C₂²	1	8	yes	complemented
7	C₂×C₄⋊Dic₁₁₃	—	2	2	no	—
8	C₂×C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
9	C₂×C₄×Dic₁₁₃	—	2	2	no	—
10	#16813537	C₂²	1	16	yes	complemented
11	C₂×#41×C₁₁₃	—	2	2	no	—
12	C₂×#41×C₁₁₃	—	2	2	no	—
13	C₂×#41×C₁₁₃	—	2	4	no	—
14	C₂×#41×C₁₁₃	C₄	1	4	yes	complemented
15	C₂×#41×C₁₁₃	C₂²	1	4	yes	—
16	C₂²×Q₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×D₄×C₁₁₃	C₄	1	1	yes	characteristic, complemented
18	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
19	C₂³×C₄×C₁₁₃	C₄	1	2	yes	complemented
20	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₁₃	—	2	4	no	—
22	C₄⋊Dic₁₁₃	—	2	8	no	—
23	C₂²⋊Dic₁₁₃	—	2	4	no	—
24	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
25	C₄×Dic₁₁₃	—	2	8	no	—
26	C₂⁴×C₁₁₃	C₂×C₄	1	2	yes	—
27	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
28	#41×C₁₁₃	—	2	8	no	—
29	#41×C₁₁₃	—	2	8	no	—
30	#41×C₁₁₃	—	2	8	no	—
31	#41×C₁₁₃	—	2	8	no	—
32	C₂×Q₈×C₁₁₃	—	2	2	no	—
33	C₂×Q₈×C₁₁₃	—	2	2	no	—
34	C₂×Q₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
35	C₂×D₄×C₁₁₃	—	2	4	no	—
36	C₂×D₄×C₁₁₃	—	2	4	no	—
37	C₂×D₄×C₁₁₃	—	2	2	no	—
38	C₂×D₄×C₁₁₃	—	2	2	no	—
39	C₂×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
40	C₂×D₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
41	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
42	C₂²×C₄×C₁₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₁₁₃	C₂×C₄	1	8	yes	complemented
50	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
51	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
52	C₂²×C₄×C₁₁₃	D₄	1	4	yes	—
53	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
54	C₂³×C₁₁₃	—	2	2	no	—
55	C₂³×C₁₁₃	—	2	2	no	—
56	C₂³×C₁₁₃	—	4	2	no	—
57	C₂³×C₁₁₃	—	4	4	no	—
58	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂³×C₁₁₃	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₁₁₃	C₂×D₄	1	2	yes	—
62	C₂³×C₁₁₃	C₂×D₄	1	2	yes	—
63	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
64	Q₈×C₁₁₃	—	2	8	no	—
65	D₄×C₁₁₃	—	2	8	no	—
66	D₄×C₁₁₃	—	2	8	no	—
67	D₄×C₁₁₃	—	2	8	no	—
68	C₂×C₄×C₁₁₃	—	2	2	no	—
69	C₂×C₄×C₁₁₃	—	2	2	no	—
70	C₂×C₄×C₁₁₃	—	2	8	no	—
71	C₂×C₄×C₁₁₃	—	2	8	no	—
72	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
84	C₂×Dic₁₁₃	—	2	8	no	—
85	C₂×Dic₁₁₃	—	4	8	no	—
86	C₂²×C₁₁₃	—	2	4	no	—
87	C₂²×C₁₁₃	—	2	2	no	—
88	C₂²×C₁₁₃	—	2	4	no	—
89	C₂²×C₁₁₃	—	4	1	no	—
90	C₂²×C₁₁₃	—	4	2	no	—
91	C₂²×C₁₁₃	—	4	1	no	—
92	C₂²×C₁₁₃	—	4	4	no	—
93	C₂²×C₁₁₃	—	4	4	no	—
94	C₂²×C₁₁₃	—	4	4	no	—
95	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
100	C₄×C₁₁₃	—	2	4	no	—
101	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
102	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
104	Dic₁₁₃	—	4	8	no	—
105	C₂×C₁₁₃	—	2	2	no	—
106	C₂×C₁₁₃	—	2	1	no	—
107	C₂×C₁₁₃	—	2	1	no	—
108	C₂×C₁₁₃	—	2	2	no	—
109	C₂×C₁₁₃	—	4	4	no	—
110	C₂×C₁₁₃	#410	1	2	yes	complemented
111	C₂×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#410	—	113	1	no	maximal, Sylow
113	C₁₁₃	C₂×#410	1	1	yes	characteristic, complemented, Sylow
114	C₂²×#41	D₁₁₃	1	1	yes	characteristic
115	C₂×#117	—	113	2	no	—
116	#410	—	113	8	no	—
117	C₂×#99	—	113	4	no	—
118	C₂×#41	—	2	2	no	—
119	C₂×#41	—	2	4	no	—
120	C₂×#41	—	2	2	no	—
121	C₂×#41	Dic₁₁₃	1	4	yes	complemented
122	C₂×#41	C₂×D₁₁₃	1	4	yes	—
123	C₂²×Q₈	Dic₁₁₃	1	1	yes	characteristic, complemented
124	C₂²×D₄	Dic₁₁₃	1	1	yes	characteristic, complemented
125	C₂²×D₄	C₂×D₁₁₃	1	2	yes	—
126	C₂³×C₄	Dic₁₁₃	1	2	yes	complemented
127	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
128	#117	—	113	8	no	—
129	#117	—	226	4	no	—
130	C₂×C₄⋊C₄	—	226	2	no	—
131	C₂×C₂²⋊C₄	—	113	4	no	—
132	C₂×C₄²	—	226	2	no	—
133	#99	—	113	16	no	—
134	C₂⁴	C₂×Dic₁₁₃	1	2	yes	—
135	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
136	#41	—	2	8	no	—
137	#41	—	2	8	no	—
138	#41	—	2	8	no	—
139	#41	—	2	8	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₁₁₃	1	4	yes	complemented
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	C₂×Dic₁₁₃	1	4	yes	—
148	C₂×D₄	C₂×Dic₁₁₃	1	4	yes	complemented
149	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	226	4	no	—
154	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
155	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
157	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×Dic₁₁₃	1	8	yes	complemented
159	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
160	C₂²×C₄	C₂²⋊D₁₁₃	1	4	yes	—
161	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
162	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
163	C₄⋊C₄	—	226	8	no	—
164	C₂²⋊C₄	—	113	8	no	—
165	C₂²⋊C₄	—	226	4	no	—
166	C₄²	—	226	8	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	4	2	no	—
170	C₂³	—	4	4	no	—
171	C₂³	C₂²×Dic₁₁₃	1	1	yes	characteristic
172	C₂³	C₂²×Dic₁₁₃	1	4	yes	—
173	C₂³	C₂²×Dic₁₁₃	1	1	yes	characteristic
174	C₂³	C₂×C₂²⋊D₁₁₃	1	2	yes	—
175	C₂³	C₂×C₂²⋊D₁₁₃	1	2	yes	—
176	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	D₄	—	2	8	no	—
179	D₄	—	2	8	no	—
180	D₄	—	2	8	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	452	8	no	—
186	C₂×C₄	—	226	8	no	—
187	C₂×C₄	C₂²⋊Dic₁₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊Dic₁₁₃	1	2	yes	—
189	C₂×C₄	C₂²⋊Dic₁₁₃	1	2	yes	—
190	C₂×C₄	C₂²⋊Dic₁₁₃	1	2	yes	—
191	C₂×C₄	C₂²⋊Dic₁₁₃	1	4	yes	—
192	C₂×C₄	C₂²⋊Dic₁₁₃	1	4	yes	—
193	C₂×C₄	C₂²⋊Dic₁₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₁₁₃	1	4	yes	—
195	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
196	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	4	yes	—
198	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	4	yes	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	4	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	4	no	—
208	C₂²	C₂×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
209	C₂²	C₂×C₂²⋊Dic₁₁₃	1	2	yes	—
210	C₂²	C₂×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
211	C₂²	C₂³×Dic₁₁₃	1	1	yes	characteristic
212	C₂²	C₂²×C₂²⋊D₁₁₃	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	452	8	no	—
215	C₄	C₂×C₂²⋊Dic₁₁₃	1	2	yes	—
216	C₄	C₂×C₂²⋊Dic₁₁₃	1	2	yes	—
217	C₄	C₂×C₂²⋊Dic₁₁₃	1	4	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	2	2	no	—
222	C₂	—	4	4	no	—
223	C₂	#20298572	1	2	yes	direct product factor
224	C₂	C₂²×C₂²⋊Dic₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Frattini	C₂³	C₂³×D₁₁₃
Fitting, prime radical	C₂²×#41×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
cosocle	C₂³×C₁₁₃	C₂⁴
hypercenter	C₂²×#41	D₁₁₃
hypocenter	C₁₁₃	C₂×#410
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16813581	C₂	semidirect
2. #20298572	C₂	direct (N × H)
3. C₂×#16813537	C₂	semidirect
6. #16813581	C₂²	semidirect
10. #16813537	C₂²	semidirect
14. C₂×#41×C₁₁₃	C₄	semidirect
16. C₂²×Q₈×C₁₁₃	C₄	semidirect
17. C₂²×D₄×C₁₁₃	C₄	semidirect
19. C₂³×C₄×C₁₁₃	C₄	semidirect
34. C₂×Q₈×C₁₁₃	C₂×C₄	semidirect
40. C₂×D₄×C₁₁₃	C₂×C₄	semidirect
49. C₂²×C₄×C₁₁₃	C₂×C₄	semidirect
110. C₂×C₁₁₃	#410	semidirect
113. C₁₁₃	C₂×#410	semidirect
121. C₂×#41	Dic₁₁₃	semidirect
123. C₂²×Q₈	Dic₁₁₃	semidirect
124. C₂²×D₄	Dic₁₁₃	semidirect
126. C₂³×C₄	Dic₁₁₃	semidirect
142. C₂×Q₈	C₂×Dic₁₁₃	semidirect
148. C₂×D₄	C₂×Dic₁₁₃	semidirect
158. C₂²×C₄	C₂×Dic₁₁₃	semidirect
223. C₂	#20298572	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×#16813581	2
#20298572	2
C₂×#16813537	2
C₂²×#41×C₁₁₃	2
C₂×#410	113

Subgroups — C2×#20298572 #28399031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20298572 #28399031