Subgroups — #28404460

Order 14464.

992 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299326	C₂	1	4	yes	maximal, complemented
2	#20299318	C₂	1	4	yes	maximal, complemented
3	#20299288	C₂	1	8	yes	maximal, complemented
4	#20299210	C₂	1	4	yes	maximal, complemented
5	#20299169	C₂	1	2	yes	maximal
6	#20299155	C₂	1	4	yes	maximal, complemented
7	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16813622	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299075	C₂	1	2	yes	maximal, complemented
10	C₄×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
12	#16813622	C₂²	1	4	yes	—
13	#16813622	C₂²	1	2	yes	—
14	#16813622	C₂²	1	2	yes	—
15	#16813622	C₂²	1	2	yes	—
16	#16813620	C₂²	1	4	yes	—
17	#16813615	C₂²	1	16	yes	—
18	#16813614	C₂²	1	8	yes	—
19	#16813614	C₂²	1	2	yes	—
20	#16813612	C₂²	1	16	yes	—
21	#16813609	C₂²	1	4	yes	—
22	#16813603	C₂²	1	8	yes	—
23	#16813601	C₂²	1	4	yes	—
24	#16813601	C₂²	1	8	yes	—
25	#16813601	C₂²	1	8	yes	—
26	C₄×Dic₂₂₆	C₂²	1	8	yes	—
27	C₄×Dic₂₂₆	C₂²	1	4	yes	—
28	C₄×Dic₂₂₆	C₂²	1	8	yes	—
29	#16813587	C₂²	1	8	yes	—
30	#16813581	C₂²	1	8	yes	—
31	#16813581	C₂²	1	2	yes	—
32	#16813578	C₂²	1	8	yes	—
33	C₂×C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
34	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
36	C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
37	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
38	#117×C₁₁₃	C₂²	1	1	yes	characteristic
39	#117×C₁₁₃	C₂²	1	2	yes	—
40	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
42	C₂×Dic₂₂₆	—	2	8	no	—
43	C₂×Dic₂₂₆	C₂³	1	4	yes	—
44	C₂×Dic₂₂₆	C₂³	1	4	yes	—
45	C₂×Dic₂₂₆	C₂³	1	4	yes	—
46	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
47	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
48	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
49	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
50	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
51	#14268626	C₂³	1	16	yes	—
52	#14268626	C₂³	1	16	yes	—
53	#14268626	C₂³	1	8	yes	—
54	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
55	C₂²⋊Dic₁₁₃	C₂³	1	16	yes	—
56	C₄×Dic₁₁₃	C₂³	1	16	yes	—
57	C₄×Dic₁₁₃	C₂³	1	4	yes	—
58	#41×C₁₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
61	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
69	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
70	C₄²×C₁₁₃	C₂³	1	2	yes	—
71	C₄²×C₁₁₃	C₂³	1	4	yes	—
72	C₄²×C₁₁₃	C₂³	1	2	yes	—
73	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
74	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
75	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
76	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
77	Q₈×C₁₁₃	—	2	2	no	—
78	D₄×C₁₁₃	—	2	4	no	—
79	D₄×C₁₁₃	—	2	2	no	—
80	C₂×C₄×C₁₁₃	—	2	4	no	—
81	C₂×C₄×C₁₁₃	—	2	4	no	—
82	C₂×C₄×C₁₁₃	—	2	8	no	—
83	C₂×C₄×C₁₁₃	#41	1	4	yes	—
84	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
85	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
90	C₂×Dic₁₁₃	—	2	8	no	—
91	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
92	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
93	Dic₂₂₆	—	2	8	no	—
94	Dic₂₂₆	—	2	8	no	—
95	Dic₂₂₆	—	2	8	no	—
96	C₂²×C₁₁₃	—	2	2	no	—
97	C₂²×C₁₁₃	—	2	2	no	—
98	C₂²×C₁₁₃	—	2	4	no	—
99	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
100	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
101	C₄×C₁₁₃	—	2	2	no	—
102	C₄×C₁₁₃	—	2	4	no	—
103	C₄×C₁₁₃	—	2	2	no	—
104	C₄×C₁₁₃	C₂×#41	1	2	yes	—
105	C₄×C₁₁₃	C₂×#41	1	2	yes	—
106	C₄×C₁₁₃	2₋¹⁺⁴	1	4	yes	—
107	Dic₁₁₃	—	2	8	no	—
108	Dic₁₁₃	—	2	8	no	—
109	C₂×C₁₁₃	—	2	2	no	—
110	C₂×C₁₁₃	—	4	2	no	—
111	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
112	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
113	#3509	—	113	1	no	maximal, Sylow
114	C₁₁₃	#3509	1	1	yes	characteristic, complemented, Sylow
115	#557	—	113	4	no	—
116	#556	—	113	4	no	—
117	#552	—	113	8	no	—
118	#541	—	113	4	no	—
119	#536	—	113	2	no	—
120	#533	—	113	1	no	—
121	#533	—	113	4	no	—
122	C₂×C₄⋊Q₈	—	113	1	no	—
123	#519	—	113	2	no	—
124	C₄×#41	D₁₁₃	1	1	yes	characteristic
125	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
126	C₂²×Q₈	—	113	2	no	—
127	C₄⋊Q₈	—	113	4	no	—
128	C₄⋊Q₈	—	113	2	no	—
129	C₄⋊Q₈	—	113	2	no	—
130	C₄⋊Q₈	—	113	4	no	—
131	C₄⋊Q₈	—	113	2	no	—
132	#126	—	113	16	no	—
133	#125	—	113	16	no	—
134	#125	—	113	2	no	—
135	#125	—	113	8	no	—
136	#124	—	113	4	no	—
137	#123	—	113	8	no	—
138	C₂²⋊Q₈	—	113	8	no	—
139	C₂²⋊Q₈	—	113	4	no	—
140	C₂²⋊Q₈	—	113	8	no	—
141	C₄×Q₈	—	113	8	no	—
142	C₄×Q₈	—	113	8	no	—
143	C₄×Q₈	—	113	8	no	—
144	C₄×Q₈	—	113	4	no	—
145	C₄×Q₈	C₂×D₁₁₃	1	2	yes	—
146	C₄×D₄	C₂×D₁₁₃	1	4	yes	—
147	C₄×D₄	C₂×D₁₁₃	1	2	yes	—
148	#117	—	113	2	no	—
149	#117	—	113	8	no	—
150	#117	—	113	8	no	—
151	#117	C₂×D₁₁₃	1	1	yes	characteristic
152	#117	C₂×D₁₁₃	1	2	yes	—
153	C₂×C₄⋊C₄	—	113	4	no	—
154	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
155	C₂×C₄²	C₂×D₁₁₃	1	2	yes	—
156	#41	—	2	4	no	—
157	C₂×Q₈	—	113	4	no	—
158	C₂×Q₈	—	113	4	no	—
159	C₂×Q₈	—	113	4	no	—
160	C₂×Q₈	—	226	8	no	—
161	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
162	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
163	C₂×D₄	C₂²×D₁₁₃	1	1	yes	characteristic
164	C₂²×C₄	—	113	4	no	—
165	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
166	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
167	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
169	C₄⋊C₄	—	113	8	no	—
170	C₄⋊C₄	—	113	8	no	—
171	C₄⋊C₄	—	113	4	no	—
172	C₄⋊C₄	—	113	4	no	—
173	C₄⋊C₄	—	113	16	no	—
174	C₄⋊C₄	—	113	16	no	—
175	C₄⋊C₄	—	113	8	no	—
176	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
177	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
178	C₂²⋊C₄	—	113	4	no	—
179	C₂²⋊C₄	—	113	16	no	—
180	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
181	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
182	C₄²	—	113	16	no	—
183	C₄²	—	113	4	no	—
184	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
185	C₄²	C₂²×D₁₁₃	1	2	yes	—
186	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
187	C₄²	C₂²×D₁₁₃	1	4	yes	—
188	C₄²	C₂²×D₁₁₃	1	2	yes	—
189	C₂³	C₂³×D₁₁₃	1	2	yes	—
190	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	226	8	no	—
193	Q₈	—	226	8	no	—
194	Q₈	—	226	8	no	—
195	D₄	—	2	4	no	—
196	D₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	113	8	no	—
201	C₂×C₄	—	113	8	no	—
202	C₂×C₄	—	226	8	no	—
203	C₂×C₄	#14268647	1	4	yes	—
204	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
205	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
206	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
207	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₂×#14268647	1	2	yes	—
214	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	226	8	no	—
219	C₄	—	226	8	no	—
220	C₄	C₂×#14268647	1	2	yes	—
221	C₄	C₂×#14268647	1	2	yes	—
222	C₄	#16813669	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂²×#14268647	1	1	yes	characteristic
226	C₂	C₂×#16813669	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₄×#41×C₁₁₃	C₂
prime radical	C₂×#41×C₁₁₃	C₂²
hypercenter	C₄×#41	D₁₁₃
hypocenter	C₁₁₃	#3509
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #20299326	C₂	semidirect
2. #20299318	C₂	semidirect
3. #20299288	C₂	semidirect
4. #20299210	C₂	semidirect
6. #20299155	C₂	semidirect
7. #20299155	C₂	semidirect
8. C₂×#16813622	C₂	semidirect
9. #20299075	C₂	semidirect
114. C₁₁₃	#3509	semidirect

Maximal subgroups

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

subgroup	index
#20299326	2
#20299318	2
#20299288	2
#20299210	2
#20299169	2
#20299155	2
C₂×#16813622	2
#20299075	2
C₄×#41×C₁₁₃	2
#3509	113