Subgroups — C₂×#20299197 #28403927

Order 14464.

1144 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299197	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813617	C₂	1	2	yes	maximal, complemented
3	C₂×#16813613	C₂	1	2	yes	maximal, complemented
4	C₂×#16813607	C₂	1	2	yes	maximal, complemented
5	C₂×#16813606	C₂	1	2	yes	maximal, complemented
6	C₂×#16813598	C₂	1	2	yes	maximal, complemented
7	C₂×#16813596	C₂	1	1	yes	maximal, characteristic
8	C₂×#16813582	C₂	1	2	yes	maximal, complemented
9	C₂²×#14268626	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
13	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
16	#16813617	C₂²	1	16	yes	complemented
17	#16813613	C₂²	1	16	yes	complemented
18	#16813607	C₂²	1	16	yes	complemented
19	#16813606	C₂²	1	16	yes	complemented
20	#16813598	C₂²	1	16	yes	complemented
21	#16813596	C₂²	1	8	yes	—
22	#16813582	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
26	C₂×#14268626	C₂²	1	2	yes	—
27	C₂×#14268626	C₂²	1	2	yes	—
28	C₂×#14268626	C₂²	1	2	yes	—
29	C₂×#14268626	C₂²	1	8	yes	complemented
30	C₂×#14268624	C₂²	1	2	yes	—
31	C₂×#14268624	C₂²	1	2	yes	—
32	C₂×#14268624	C₂²	1	2	yes	—
33	C₂×#14268624	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₁₁₃	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
41	C₂³×D₁₁₃	C₂³	1	2	yes	—
42	C₂×D₄₅₂	—	4	2	no	—
43	C₂×D₄₅₂	C₂³	1	4	yes	—
44	C₂×C₂²⋊D₁₁₃	—	2	8	no	—
45	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
46	C₂²×Dic₁₁₃	—	2	4	no	—
47	C₂²×Dic₁₁₃	—	4	2	no	—
48	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
49	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
51	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
52	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
53	C₂×C₄×D₁₁₃	—	4	4	no	—
54	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
55	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
56	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
57	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
58	#14268626	C₂³	1	8	yes	—
59	#14268626	C₂³	1	8	yes	—
60	#14268626	C₂³	1	8	yes	—
61	#14268624	C₂³	1	8	yes	—
62	#14268624	C₂³	1	8	yes	—
63	#14268624	C₂³	1	8	yes	—
64	#14268624	C₂³	1	8	yes	—
65	C₄×Dic₁₁₃	C₂³	1	8	yes	—
66	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
67	C₂×Q₈×C₁₁₃	—	4	2	no	—
68	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
69	C₂²×C₄×C₁₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
75	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
76	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
77	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
78	C₂³×C₁₁₃	—	2	2	no	—
79	C₂³×C₁₁₃	—	2	2	no	—
80	C₂³×C₁₁₃	#41	1	4	yes	—
81	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
82	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
83	C₂²×D₁₁₃	—	2	4	no	—
84	C₂²×D₁₁₃	—	2	4	no	—
85	C₂²×D₁₁₃	—	2	4	no	—
86	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
87	Q₈×C₁₁₃	—	4	4	no	—
88	C₂×C₄×C₁₁₃	—	2	2	no	—
89	C₂×C₄×C₁₁₃	—	2	2	no	—
90	C₂×C₄×C₁₁₃	—	2	4	no	—
91	C₂×C₄×C₁₁₃	—	2	4	no	—
92	C₂×C₄×C₁₁₃	—	4	4	no	—
93	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
94	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
95	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
97	C₂²⋊D₁₁₃	—	2	16	no	—
98	C₂×Dic₁₁₃	—	2	8	no	—
99	C₂×Dic₁₁₃	—	2	4	no	—
100	C₂×Dic₁₁₃	—	2	4	no	—
101	C₂×Dic₁₁₃	—	2	4	no	—
102	C₂×Dic₁₁₃	—	4	4	no	—
103	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
104	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
105	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
106	D₄₅₂	—	4	4	no	—
107	C₄×D₁₁₃	—	4	8	no	—
108	C₂²×C₁₁₃	—	2	8	no	—
109	C₂²×C₁₁₃	—	2	2	no	—
110	C₂²×C₁₁₃	—	2	2	no	—
111	C₂²×C₁₁₃	C₂×#41	1	4	yes	—
112	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
113	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
116	C₂×D₁₁₃	—	2	4	no	—
117	C₂×D₁₁₃	—	2	4	no	—
118	C₂×D₁₁₃	—	2	4	no	—
119	C₂×D₁₁₃	—	4	8	no	—
120	C₄×C₁₁₃	—	2	4	no	—
121	C₄×C₁₁₃	—	2	2	no	—
122	C₄×C₁₁₃	—	2	2	no	—
123	C₄×C₁₁₃	—	2	4	no	—
124	Dic₁₁₃	—	2	4	no	—
125	Dic₁₁₃	—	2	4	no	—
126	Dic₁₁₃	—	2	4	no	—
127	C₂×C₁₁₃	—	2	4	no	—
128	C₂×C₁₁₃	#539	1	4	yes	complemented
129	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
130	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
132	D₁₁₃	—	4	4	no	—
133	C₂×#539	—	113	1	no	maximal, Sylow
134	C₁₁₃	C₂×#539	1	1	yes	characteristic, complemented, Sylow
135	#539	—	113	16	no	—
136	C₂×#126	—	113	2	no	—
137	C₂×#125	—	113	2	no	—
138	C₂×#123	—	113	2	no	—
139	C₂×#123	—	113	2	no	—
140	C₂×C₂²⋊Q₈	—	113	2	no	—
141	C₂×C₂²⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
142	C₂×#121	—	113	1	no	—
143	C₂×C₄×D₄	—	113	2	no	—
144	C₂²×C₄⋊C₄	—	113	1	no	—
145	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
146	C₂²×D₄	—	113	2	no	—
147	C₂²×D₄	—	113	1	no	—
148	C₂³×C₄	—	113	1	no	—
149	C₂³×C₄	—	113	1	no	—
150	C₂³×C₄	—	113	2	no	—
151	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
152	#126	—	113	16	no	—
153	#125	—	113	16	no	—
154	#123	—	113	16	no	—
155	#123	—	113	16	no	—
156	C₂²⋊Q₈	—	113	16	no	—
157	C₂²⋊Q₈	C₂×D₁₁₃	1	8	yes	complemented
158	#121	—	113	8	no	—
159	C₄×D₄	—	113	16	no	—
160	C₂×C₄⋊C₄	—	113	2	no	—
161	C₂×C₄⋊C₄	—	113	2	no	—
162	C₂×C₄⋊C₄	—	113	8	no	—
163	C₂×C₄⋊C₄	—	113	2	no	—
164	C₂×C₄⋊C₄	—	113	1	no	—
165	C₂×C₄⋊C₄	—	113	2	no	—
166	C₂×C₄⋊C₄	—	113	2	no	—
167	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	113	2	no	—
170	C₂×C₂²⋊C₄	—	113	2	no	—
171	C₂×C₂²⋊C₄	—	113	2	no	—
172	C₂×C₂²⋊C₄	—	113	2	no	—
173	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
174	C₂×C₄²	—	113	2	no	—
175	C₂⁴	—	113	2	no	—
176	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
179	C₂×D₄	—	113	8	no	—
180	C₂×D₄	—	113	4	no	—
181	C₂×D₄	—	452	2	no	—
182	C₂×D₄	—	226	8	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	113	4	no	—
185	C₂²×C₄	—	113	4	no	—
186	C₂²×C₄	—	113	2	no	—
187	C₂²×C₄	—	113	2	no	—
188	C₂²×C₄	—	113	2	no	—
189	C₂²×C₄	—	113	8	no	—
190	C₂²×C₄	—	452	4	no	—
191	C₂²×C₄	—	452	2	no	—
192	C₂²×C₄	—	226	4	no	—
193	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
194	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
196	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
198	C₄⋊C₄	—	113	8	no	—
199	C₄⋊C₄	—	113	8	no	—
200	C₄⋊C₄	—	113	8	no	—
201	C₄⋊C₄	—	113	4	no	—
202	C₄⋊C₄	—	113	8	no	—
203	C₄⋊C₄	—	113	8	no	—
204	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
205	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
206	C₂²⋊C₄	—	113	8	no	—
207	C₂²⋊C₄	—	113	8	no	—
208	C₂²⋊C₄	—	113	8	no	—
209	C₂²⋊C₄	—	113	8	no	—
210	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
211	C₄²	—	113	8	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	113	4	no	—
215	C₂³	—	226	4	no	—
216	C₂³	—	226	4	no	—
217	C₂³	—	226	4	no	—
218	C₂³	#14268648	1	4	yes	—
219	C₂³	C₂³×D₁₁₃	1	2	yes	—
220	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	D₄	—	452	4	no	—
223	D₄	—	226	16	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	113	4	no	—
230	C₂×C₄	—	113	4	no	—
231	C₂×C₄	—	113	4	no	—
232	C₂×C₄	—	452	8	no	—
233	C₂×C₄	—	452	4	no	—
234	C₂×C₄	—	226	8	no	—
235	C₂×C₄	—	226	4	no	—
236	C₂×C₄	—	226	4	no	—
237	C₂×C₄	—	226	4	no	—
238	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
239	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
240	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
241	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	452	8	no	—
246	C₂²	—	226	4	no	—
247	C₂²	—	226	4	no	—
248	C₂²	—	226	4	no	—
249	C₂²	C₂×#14268648	1	4	yes	—
250	C₂²	C₂×#14268648	1	2	yes	—
251	C₂²	#16813667	1	2	yes	—
252	C₂²	#16813669	1	2	yes	—
253	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	4	no	—
258	C₄	—	226	4	no	—
259	C₄	—	226	4	no	—
260	C₄	—	226	4	no	—
261	C₂	—	2	4	no	—
262	C₂	—	452	4	no	—
263	C₂	#20299197	1	4	yes	direct product factor
264	C₂	C₂²×#14268648	1	1	yes	characteristic
265	C₂	C₂×#16813667	1	1	yes	characteristic
266	C₂	C₂×#16813669	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×C₂²⋊Q₈×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂×#16813596	C₂
hypercenter	C₂×C₂²⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	C₂×#539
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20299197	C₂	direct (N × H)
2. C₂×#16813617	C₂	semidirect
3. C₂×#16813613	C₂	semidirect
4. C₂×#16813607	C₂	semidirect
5. C₂×#16813606	C₂	semidirect
6. C₂×#16813598	C₂	semidirect
8. C₂×#16813582	C₂	semidirect
9. C₂²×#14268626	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₁₁₃	C₂	semidirect
16. #16813617	C₂²	semidirect
17. #16813613	C₂²	semidirect
18. #16813607	C₂²	semidirect
19. #16813606	C₂²	semidirect
20. #16813598	C₂²	semidirect
22. #16813582	C₂²	semidirect
29. C₂×#14268626	C₂²	semidirect
37. C₂²⋊Q₈×C₁₁₃	C₂²	semidirect
128. C₂×C₁₁₃	#539	semidirect
134. C₁₁₃	C₂×#539	semidirect
141. C₂×C₂²⋊Q₈	D₁₁₃	semidirect
157. C₂²⋊Q₈	C₂×D₁₁₃	semidirect
263. C₂	#20299197	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299197	2
C₂×#16813617	2
C₂×#16813613	2
C₂×#16813607	2
C₂×#16813606	2
C₂×#16813598	2
C₂×#16813596	2
C₂×#16813582	2
C₂²×#14268626	2
C₂×C₂²⋊Q₈×C₁₁₃	2
C₂×#539	113

Subgroups — C2×#20299197 #28403927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299197 #28403927