Subgroups — C₂×#20299269 #28404054

Order 14464.

1736 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268647	C₂	1	2	yes	maximal, complemented
2	#20299269	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16813619	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16813618	C₂	1	2	yes	maximal, complemented
5	C₂×#16813596	C₂	1	2	yes	maximal, complemented
6	C₂×#16813595	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16813584	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14268647	—	2	16	no	—
11	C₂×#14268647	C₂²	1	16	yes	complemented
12	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
13	C₂²×D₄₅₂	C₂²	1	2	yes	—
14	C₂²×D₄₅₂	C₂²	1	2	yes	—
15	C₂²×D₄₅₂	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
17	C₂²×C₂²⋊D₁₁₃	C₂²	1	4	yes	—
18	C₂²×C₄×D₁₁₃	C₂²	1	4	yes	—
19	#16813619	C₂²	1	8	yes	complemented
20	#16813618	C₂²	1	16	yes	complemented
21	#16813596	C₂²	1	16	yes	complemented
22	#16813595	C₂²	1	32	yes	complemented
23	C₄×Dic₂₂₆	C₂²	1	8	yes	complemented
24	#16813584	C₂²	1	16	yes	complemented
25	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
26	C₂×#14268626	C₂²	1	2	yes	—
27	C₂×#14268624	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
31	C₄×D₄×C₁₁₃	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₂³×D₁₁₃	C₂³	1	4	yes	—
36	#14268647	—	2	64	no	—
37	C₂×Dic₂₂₆	D₄	1	8	yes	complemented
38	C₂×Dic₂₂₆	C₂³	1	4	yes	—
39	C₂×D₄₅₂	—	2	8	no	—
40	C₂×D₄₅₂	—	2	8	no	—
41	C₂×D₄₅₂	—	4	4	no	—
42	C₂×D₄₅₂	C₂³	1	8	yes	—
43	C₂×D₄₅₂	C₂³	1	8	yes	—
44	C₂×D₄₅₂	C₂³	1	8	yes	—
45	C₂×C₂²⋊D₁₁₃	—	2	16	no	—
46	C₂×C₂²⋊D₁₁₃	—	4	8	no	—
47	C₂×C₂²⋊D₁₁₃	C₂³	1	16	yes	—
48	C₂×C₂²⋊D₁₁₃	C₂³	1	16	yes	—
49	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
51	C₂×C₄×D₁₁₃	—	2	16	no	—
52	C₂×C₄×D₁₁₃	C₂³	1	16	yes	—
53	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
54	#14268626	C₂³	1	8	yes	—
55	#14268624	C₂³	1	16	yes	—
56	C₄×Dic₁₁₃	C₂³	1	8	yes	—
57	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
58	C₂×D₄×C₁₁₃	—	2	4	no	—
59	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
60	C₂²×C₄×C₁₁₃	—	2	8	no	—
61	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₁₃	C₂³	1	8	yes	—
64	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
67	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
68	C₄²×C₁₁₃	C₂³	1	4	yes	—
69	C₂³×C₁₁₃	—	2	4	no	—
70	C₂³×C₁₁₃	—	2	4	no	—
71	C₂³×C₁₁₃	—	2	4	no	—
72	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
73	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
74	C₂²×D₁₁₃	—	2	8	no	—
75	C₂²×D₁₁₃	—	2	8	no	—
76	C₂²×D₁₁₃	—	2	8	no	—
77	C₂²×D₁₁₃	C₂⁴	1	8	yes	—
78	D₄×C₁₁₃	—	2	8	no	—
79	C₂×C₄×C₁₁₃	—	2	4	no	—
80	C₂×C₄×C₁₁₃	—	2	2	no	—
81	C₂×C₄×C₁₁₃	—	2	16	no	—
82	C₂×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₁₁₃	#41	1	4	yes	complemented
84	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
85	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
86	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
88	C₂²⋊D₁₁₃	—	2	32	no	—
89	C₂²⋊D₁₁₃	—	4	16	no	—
90	C₂×Dic₁₁₃	—	2	4	no	—
91	C₂×Dic₁₁₃	C₂×D₄	1	8	yes	—
92	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
93	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
94	D₄₅₂	—	2	16	no	—
95	D₄₅₂	—	2	16	no	—
96	D₄₅₂	—	4	8	no	—
97	C₄×D₁₁₃	—	2	32	no	—
98	Dic₂₂₆	C₂×D₄	1	16	yes	complemented
99	C₂²×C₁₁₃	—	2	4	no	—
100	C₂²×C₁₁₃	—	2	4	no	—
101	C₂²×C₁₁₃	—	2	4	no	—
102	C₂²×C₁₁₃	—	4	8	no	—
103	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
104	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
105	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
106	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
107	C₂×D₁₁₃	—	2	8	no	—
108	C₂×D₁₁₃	—	2	8	no	—
109	C₂×D₁₁₃	—	2	8	no	—
110	C₂×D₁₁₃	—	4	16	no	—
111	C₄×C₁₁₃	—	2	4	no	—
112	C₄×C₁₁₃	—	2	2	no	—
113	C₄×C₁₁₃	C₂²×D₄	1	4	yes	—
114	C₄×C₁₁₃	C₂×#41	1	4	yes	complemented
115	Dic₁₁₃	—	2	4	no	—
116	Dic₁₁₃	C₂²×D₄	1	8	yes	—
117	C₂×C₁₁₃	—	4	4	no	—
118	C₂×C₁₁₃	#550	1	4	yes	complemented
119	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
120	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
121	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
122	D₁₁₃	—	4	8	no	—
123	C₂×#550	—	113	1	no	maximal, Sylow
124	C₁₁₃	C₂×#550	1	1	yes	characteristic, complemented, Sylow
125	C₂²×#41	—	113	2	no	—
126	#550	—	113	16	no	—
127	C₂×#127	—	113	2	no	—
128	C₂×#127	—	113	1	no	—
129	C₂×#121	—	113	2	no	—
130	C₂×#121	—	113	4	no	—
131	C₂×C₄×Q₈	—	113	1	no	—
132	C₂×C₄×D₄	—	113	2	no	—
133	C₂×C₄×D₄	D₁₁₃	1	1	yes	characteristic, complemented
134	C₂×#41	—	113	16	no	—
135	C₂×#41	—	226	16	no	—
136	C₂²×Q₈	—	113	1	no	—
137	C₂²×D₄	—	113	4	no	—
138	C₂²×D₄	—	113	4	no	—
139	C₂²×D₄	—	113	2	no	—
140	C₂²×D₄	—	113	2	no	—
141	C₂²×D₄	—	113	2	no	—
142	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
143	C₂³×C₄	—	113	4	no	—
144	C₂³×C₄	C₂×D₁₁₃	1	2	yes	—
145	#127	—	113	16	no	—
146	#127	—	113	8	no	—
147	#121	—	113	16	no	—
148	#121	—	113	32	no	—
149	C₄×Q₈	—	113	8	no	—
150	C₄×D₄	—	113	16	no	—
151	C₄×D₄	C₂×D₁₁₃	1	8	yes	complemented
152	C₂×C₄⋊C₄	—	113	2	no	—
153	C₂×C₄⋊C₄	—	113	1	no	—
154	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	—	113	4	no	—
156	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
157	C₂×C₄²	—	113	2	no	—
158	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
159	C₂⁴	—	113	4	no	—
160	C₂⁴	C₂²×D₁₁₃	1	2	yes	—
161	#41	—	226	64	no	—
162	C₂×Q₈	—	113	8	no	—
163	C₂×Q₈	—	113	4	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	113	16	no	—
166	C₂×D₄	—	113	16	no	—
167	C₂×D₄	—	113	8	no	—
168	C₂×D₄	—	113	8	no	—
169	C₂×D₄	—	113	8	no	—
170	C₂×D₄	—	452	4	no	—
171	C₂×D₄	—	452	8	no	—
172	C₂×D₄	—	226	16	no	—
173	C₂×D₄	—	226	8	no	—
174	C₂×D₄	—	226	8	no	—
175	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	113	2	no	—
178	C₂²×C₄	—	113	2	no	—
179	C₂²×C₄	—	113	16	no	—
180	C₂²×C₄	—	226	16	no	—
181	C₂²×C₄	C₂²×D₁₁₃	1	8	yes	—
182	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
186	C₄⋊C₄	—	113	8	no	—
187	C₄⋊C₄	—	113	4	no	—
188	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
189	C₂²⋊C₄	—	113	16	no	—
190	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
191	C₄²	—	113	8	no	—
192	C₄²	C₂²×D₁₁₃	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	113	8	no	—
197	C₂³	—	226	8	no	—
198	C₂³	—	226	8	no	—
199	C₂³	—	226	8	no	—
200	C₂³	C₂³×D₁₁₃	1	4	yes	—
201	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
202	Q₈	—	113	16	no	—
203	D₄	—	2	8	no	—
204	D₄	—	452	8	no	—
205	D₄	—	452	16	no	—
206	D₄	—	226	32	no	—
207	D₄	—	226	16	no	—
208	D₄	—	226	16	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	113	8	no	—
213	C₂×C₄	—	113	4	no	—
214	C₂×C₄	—	113	4	no	—
215	C₂×C₄	—	226	32	no	—
216	C₂×C₄	—	226	4	no	—
217	C₂×C₄	D₄×D₁₁₃	1	4	yes	—
218	C₂×C₄	#14268647	1	4	yes	complemented
219	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
220	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
221	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
222	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	8	no	—
227	C₂²	—	452	16	no	—
228	C₂²	—	226	8	no	—
229	C₂²	—	226	8	no	—
230	C₂²	—	226	8	no	—
231	C₂²	C₂×D₄×D₁₁₃	1	2	yes	—
232	C₂²	C₂×#14268647	1	2	yes	—
233	C₂²	#16813668	1	2	yes	—
234	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	2	2	no	—
237	C₄	—	113	8	no	—
238	C₄	—	226	4	no	—
239	C₄	C₂×D₄×D₁₁₃	1	4	yes	—
240	C₄	C₂×#14268647	1	4	yes	complemented
241	C₂	—	4	4	no	—
242	C₂	—	452	8	no	—
243	C₂	#20299269	1	4	yes	direct product factor
244	C₂	C₂²×D₄×D₁₁₃	1	1	yes	characteristic
245	C₂	C₂²×#14268647	1	1	yes	characteristic
246	C₂	C₂×#16813668	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×C₄×D₄×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	C₂×#550
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14268647	C₂	semidirect
2. #20299269	C₂	direct (N × H)
3. C₂×#16813619	C₂	semidirect
4. C₂×#16813618	C₂	semidirect
5. C₂×#16813596	C₂	semidirect
6. C₂×#16813595	C₂	semidirect
7. C₂×C₄×Dic₂₂₆	C₂	semidirect
8. C₂×#16813584	C₂	semidirect
9. C₂×C₄×D₄×C₁₁₃	C₂	semidirect
11. C₂×#14268647	C₂²	semidirect
19. #16813619	C₂²	semidirect
20. #16813618	C₂²	semidirect
21. #16813596	C₂²	semidirect
22. #16813595	C₂²	semidirect
23. C₄×Dic₂₂₆	C₂²	semidirect
24. #16813584	C₂²	semidirect
31. C₄×D₄×C₁₁₃	C₂²	semidirect
37. C₂×Dic₂₂₆	D₄	semidirect
83. C₂×C₄×C₁₁₃	#41	semidirect
98. Dic₂₂₆	C₂×D₄	semidirect
114. C₄×C₁₁₃	C₂×#41	semidirect
118. C₂×C₁₁₃	#550	semidirect
124. C₁₁₃	C₂×#550	semidirect
133. C₂×C₄×D₄	D₁₁₃	semidirect
151. C₄×D₄	C₂×D₁₁₃	semidirect
218. C₂×C₄	#14268647	semidirect
240. C₄	C₂×#14268647	semidirect
243. C₂	#20299269	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×#14268647	2
#20299269	2
C₂×#16813619	2
C₂×#16813618	2
C₂×#16813596	2
C₂×#16813595	2
C₂×C₄×Dic₂₂₆	2
C₂×#16813584	2
C₂×C₄×D₄×C₁₁₃	2
C₂×#550	113

Subgroups — C2×#20299269 #28404054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299269 #28404054