Subgroups — C₂×#20299412 #28405338

Order 14464.

1408 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813669	C₂	1	2	yes	maximal, complemented
2	#20299412	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16813652	C₂	1	2	yes	maximal, complemented
4	C₂×#16813651	C₂	1	4	yes	maximal, complemented
5	C₂×#16813641	C₂	1	2	yes	maximal, complemented
6	C₂²×Dic₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₁₆×D₁₁₃	C₂	1	2	yes	maximal, complemented
8	C₂×#16813628	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#16813669	—	2	8	no	—
11	#16813669	C₂²	1	16	yes	complemented
12	C₂×#14268647	—	2	2	no	—
13	C₂×#14268647	C₂²	1	1	yes	characteristic
14	C₂×#14268646	—	2	4	no	—
15	C₂×#14268646	C₂²	1	4	yes	—
16	C₂²×Dic₂₂₆	—	2	2	no	—
17	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
18	C₂×Q₈×D₁₁₃	C₂²	1	4	yes	—
19	#16813652	C₂²	1	16	yes	complemented
20	#16813651	C₂²	1	32	yes	complemented
21	#16813641	C₂²	1	16	yes	complemented
22	C₂×Dic₄₅₂	C₂²	1	2	yes	—
23	C₂×Dic₄₅₂	C₂²	1	2	yes	—
24	C₂×Dic₄₅₂	C₂²	1	8	yes	complemented
25	C₂×#14268637	C₂²	1	4	yes	—
26	Q₁₆×D₁₁₃	C₂²	1	16	yes	complemented
27	C₂×#14268635	C₂²	1	4	yes	—
28	#16813628	C₂²	1	8	yes	complemented
29	C₂×#14268631	C₂²	1	1	yes	characteristic
30	C₂×#14268630	C₂²	1	2	yes	—
31	C₂×C₈×D₁₁₃	C₂²	1	2	yes	—
32	C₂×#41×C₁₁₃	C₂²	1	2	yes	—
33	#135×C₁₁₃	C₂²	1	8	yes	complemented
34	C₂×Q₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₂×SD₁₆×C₁₁₃	C₂²	1	2	yes	—
36	C₂×D₈×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
38	#14268647	—	2	16	no	—
39	#14268647	D₄	1	4	yes	—
40	#14268647	C₂³	1	4	yes	—
41	#14268646	—	2	32	no	—
42	#14268646	—	2	8	no	—
43	#14268646	C₂³	1	16	yes	—
44	C₂×Dic₂₂₆	—	2	2	no	—
45	C₂×Dic₂₂₆	—	2	4	no	—
46	C₂×Dic₂₂₆	—	2	4	no	—
47	C₂×Dic₂₂₆	—	2	16	no	—
48	C₂×Dic₂₂₆	D₄	1	1	yes	characteristic
49	C₂×Dic₂₂₆	C₂³	1	4	yes	—
50	C₂×Dic₂₂₆	C₂³	1	8	yes	—
51	Q₈×D₁₁₃	—	2	8	no	—
52	Q₈×D₁₁₃	C₂³	1	16	yes	—
53	C₂×D₄₅₂	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
55	C₂×C₂²⋊D₁₁₃	D₄	1	2	yes	—
56	C₂²×Dic₁₁₃	—	2	2	no	—
57	C₂²×Dic₁₁₃	—	2	4	no	—
58	C₂×C₄×D₁₁₃	—	2	4	no	—
59	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
60	Dic₄₅₂	C₂³	1	8	yes	—
61	Dic₄₅₂	C₂³	1	8	yes	—
62	#14268637	C₂³	1	16	yes	—
63	#14268635	C₂³	1	16	yes	—
64	#14268631	C₂³	1	4	yes	—
65	#14268630	C₂³	1	8	yes	—
66	C₂×#11744751	C₂³	1	2	yes	—
67	C₈×D₁₁₃	C₂³	1	8	yes	—
68	#41×C₁₁₃	—	2	4	no	—
69	#41×C₁₁₃	C₂³	1	8	yes	—
70	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
71	C₂×D₄×C₁₁₃	—	2	2	no	—
72	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
75	Q₁₆×C₁₁₃	C₂³	1	4	yes	—
76	SD₁₆×C₁₁₃	C₂³	1	8	yes	—
77	D₈×C₁₁₃	C₂³	1	4	yes	—
78	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₁₃	C₂³	1	4	yes	—
81	C₂³×C₁₁₃	—	2	2	no	—
82	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₁₁₃	C₂×D₄	1	2	yes	—
84	Q₈×C₁₁₃	—	2	2	no	—
85	Q₈×C₁₁₃	C₂⁴	1	4	yes	—
86	D₄×C₁₁₃	—	2	8	no	—
87	D₄×C₁₁₃	—	2	2	no	—
88	D₄×C₁₁₃	C₂⁴	1	4	yes	—
89	C₂×C₄×C₁₁₃	—	2	2	no	—
90	C₂×C₄×C₁₁₃	—	2	8	no	—
91	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
95	C₂²⋊D₁₁₃	—	2	16	no	—
96	C₂²⋊D₁₁₃	C₂×D₄	1	8	yes	—
97	C₂×Dic₁₁₃	—	2	4	no	—
98	C₂×Dic₁₁₃	—	2	8	no	—
99	C₂×Dic₁₁₃	—	2	16	no	—
100	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
101	D₄₅₂	C₂×D₄	1	4	yes	—
102	C₄×D₁₁₃	—	2	16	no	—
103	C₄×D₁₁₃	C₂×D₄	1	4	yes	—
104	C₄×D₁₁₃	C₂⁴	1	4	yes	—
105	Dic₂₂₆	—	2	4	no	—
106	Dic₂₂₆	—	2	8	no	—
107	Dic₂₂₆	—	2	16	no	—
108	Dic₂₂₆	C₂×D₄	1	4	yes	—
109	Dic₂₂₆	C₂⁴	1	8	yes	—
110	C₈×C₁₁₃	C₂⁴	1	2	yes	—
111	C₈×C₁₁₃	C₂⁴	1	2	yes	—
112	#11744751	C₂⁴	1	4	yes	—
113	C₂²×C₁₁₃	—	2	2	no	—
114	C₂²×C₁₁₃	—	2	4	no	—
115	C₂²×C₁₁₃	—	4	4	no	—
116	C₂²×C₁₁₃	C₂²×D₄	1	2	yes	—
117	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₁₃	—	2	4	no	—
119	C₂×D₁₁₃	C₂²×D₄	1	4	yes	—
120	C₄×C₁₁₃	—	2	4	no	—
121	C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
122	C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₁₁₃	C₂⁵	1	1	yes	characteristic
124	Dic₁₁₃	—	2	8	no	—
125	Dic₁₁₃	C₂²×D₄	1	4	yes	—
126	C₂×C₁₁₃	—	2	2	no	—
127	C₂×C₁₁₃	—	4	4	no	—
128	C₂×C₁₁₃	#578	1	2	yes	complemented
129	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
130	D₁₁₃	—	2	4	no	—
131	C₂×#578	—	113	1	no	maximal, Sylow
132	C₁₁₃	C₂×#578	1	1	yes	characteristic, complemented, Sylow
133	C₂×2₋¹⁺⁴	—	113	2	no	—
134	#578	—	113	16	no	—
135	C₂×#137	—	113	2	no	—
136	C₂×#137	—	113	4	no	—
137	C₂×#135	—	113	2	no	—
138	C₂×#135	D₁₁₃	1	1	yes	characteristic, complemented
139	C₂²×Q₁₆	—	113	2	no	—
140	C₂²×Q₁₆	—	113	1	no	—
141	C₂×#131	—	113	1	no	—
142	2₋¹⁺⁴	—	113	16	no	—
143	2₋¹⁺⁴	—	226	8	no	—
144	C₂×#41	—	113	1	no	—
145	C₂×#41	—	113	4	no	—
146	C₂×#41	—	226	2	no	—
147	C₂×#41	—	226	4	no	—
148	C₂×#41	C₂×D₁₁₃	1	2	yes	—
149	C₂²×Q₈	—	113	4	no	—
150	C₂²×Q₈	—	113	2	no	—
151	C₂²×Q₈	—	226	2	no	—
152	#137	—	113	16	no	—
153	#137	—	113	32	no	—
154	#135	—	113	16	no	—
155	#135	C₂×D₁₁₃	1	8	yes	complemented
156	C₂×Q₁₆	—	113	4	no	—
157	C₂×Q₁₆	—	113	16	no	—
158	C₂×Q₁₆	—	113	2	no	—
159	C₂×Q₁₆	—	113	2	no	—
160	C₂×Q₁₆	—	113	8	no	—
161	C₂×Q₁₆	C₂×D₁₁₃	1	1	yes	characteristic
162	C₂×SD₁₆	—	113	4	no	—
163	C₂×SD₁₆	C₂×D₁₁₃	1	2	yes	—
164	C₂×D₈	C₂×D₁₁₃	1	1	yes	characteristic
165	#131	—	113	8	no	—
166	C₂×M₁₆	—	113	1	no	—
167	C₂×M₁₆	—	113	2	no	—
168	C₂²×C₈	—	113	2	no	—
169	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
170	#41	—	2	4	no	—
171	#41	—	113	4	no	—
172	#41	—	113	4	no	—
173	#41	—	113	16	no	—
174	#41	—	226	16	no	—
175	#41	—	226	8	no	—
176	#41	—	226	32	no	—
177	#41	C₂²×D₁₁₃	1	8	yes	—
178	C₂×Q₈	—	113	1	no	—
179	C₂×Q₈	—	113	16	no	—
180	C₂×Q₈	—	113	4	no	—
181	C₂×Q₈	—	113	8	no	—
182	C₂×Q₈	—	226	4	no	—
183	C₂×Q₈	—	226	16	no	—
184	C₂×Q₈	—	226	8	no	—
185	C₂×Q₈	—	226	2	no	—
186	C₂×Q₈	—	226	4	no	—
187	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	113	2	no	—
190	C₂×D₄	—	113	1	no	—
191	C₂×D₄	—	226	4	no	—
192	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	113	2	no	—
195	C₂²×C₄	—	226	4	no	—
196	C₂²×C₄	—	226	4	no	—
197	C₂²×C₄	—	226	2	no	—
198	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
199	Q₁₆	—	113	16	no	—
200	Q₁₆	—	113	8	no	—
201	Q₁₆	—	113	8	no	—
202	Q₁₆	C₂²×D₁₁₃	1	4	yes	—
203	SD₁₆	—	113	16	no	—
204	SD₁₆	C₂²×D₁₁₃	1	8	yes	—
205	D₈	C₂²×D₁₁₃	1	4	yes	—
206	M₁₆	—	113	4	no	—
207	M₁₆	—	113	8	no	—
208	C₂×C₈	—	113	2	no	—
209	C₂×C₈	—	113	8	no	—
210	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₁₃	1	4	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	113	2	no	—
215	C₂³	D₄×D₁₁₃	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	113	4	no	—
218	Q₈	—	113	8	no	—
219	Q₈	—	226	16	no	—
220	Q₈	—	226	8	no	—
221	Q₈	—	226	4	no	—
222	Q₈	C₂³×D₁₁₃	1	4	yes	—
223	D₄	—	2	8	no	—
224	D₄	—	2	2	no	—
225	D₄	—	113	8	no	—
226	D₄	—	113	4	no	—
227	D₄	—	226	16	no	—
228	D₄	C₂³×D₁₁₃	1	4	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	113	2	no	—
232	C₂×C₄	—	113	4	no	—
233	C₂×C₄	—	113	4	no	—
234	C₂×C₄	—	226	16	no	—
235	C₂×C₄	—	226	16	no	—
236	C₂×C₄	—	226	8	no	—
237	C₂×C₄	—	226	4	no	—
238	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
239	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
241	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
242	C₈	—	113	4	no	—
243	C₈	C₂³×D₁₁₃	1	2	yes	—
244	C₈	C₂³×D₁₁₃	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	113	4	no	—
249	C₂²	—	226	4	no	—
250	C₂²	C₂×D₄×D₁₁₃	1	2	yes	—
251	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	113	4	no	—
254	C₄	—	226	8	no	—
255	C₄	C₂×D₄×D₁₁₃	1	2	yes	—
256	C₄	C₂×D₄×D₁₁₃	1	1	yes	characteristic
257	C₄	C₂⁴×D₁₁₃	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	4	4	no	—
260	C₂	—	226	4	no	—
261	C₂	#20299412	1	2	yes	direct product factor
262	C₂	C₂²×D₄×D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₁₃
commutator, cosocle	C₄×C₁₁₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₁₃
Fitting	C₂×#135×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
hypercenter	C₂×#135	D₁₁₃
hypocenter	C₁₁₃	C₂×#578
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#16813669	C₂	semidirect
2. #20299412	C₂	direct (N × H)
3. C₂×#16813652	C₂	semidirect
4. C₂×#16813651	C₂	semidirect
5. C₂×#16813641	C₂	semidirect
6. C₂²×Dic₄₅₂	C₂	semidirect
7. C₂×Q₁₆×D₁₁₃	C₂	semidirect
8. C₂×#16813628	C₂	semidirect
9. C₂×#135×C₁₁₃	C₂	semidirect
11. #16813669	C₂²	semidirect
19. #16813652	C₂²	semidirect
20. #16813651	C₂²	semidirect
21. #16813641	C₂²	semidirect
24. C₂×Dic₄₅₂	C₂²	semidirect
26. Q₁₆×D₁₁₃	C₂²	semidirect
28. #16813628	C₂²	semidirect
33. #135×C₁₁₃	C₂²	semidirect
128. C₂×C₁₁₃	#578	semidirect
132. C₁₁₃	C₂×#578	semidirect
138. C₂×#135	D₁₁₃	semidirect
155. #135	C₂×D₁₁₃	semidirect
261. C₂	#20299412	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×#16813669	2
#20299412	2
C₂×#16813652	2
C₂×#16813651	2
C₂×#16813641	2
C₂²×Dic₄₅₂	2
C₂×Q₁₆×D₁₁₃	2
C₂×#16813628	2
C₂×#135×C₁₁₃	2
C₂×#578	113

Subgroups — C2×#20299412 #28405338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299412 #28405338