Subgroups — #28393222

Order 14464.

444 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×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	#20298502	C₂	1	2	yes	maximal
3	#20298493	C₂	1	2	yes	maximal
4	#20298481	C₂	1	2	yes	maximal
5	#20298479	C₂	1	2	yes	maximal
6	#20298466	C₂	1	1	yes	maximal, characteristic
7	#20298465	C₂	1	1	yes	maximal, characteristic
8	#20298462	C₂	1	1	yes	maximal, characteristic
9	#20298449	C₂	1	1	yes	maximal, characteristic
10	#20298443	C₂	1	1	yes	maximal, characteristic
11	#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₂₆	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×#14268626	C₂²	1	2	yes	—
16	C₂×#14268626	C₂²	1	2	yes	—
17	C₂×#14268626	C₂²	1	1	yes	characteristic
18	C₂×#14268626	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
23	#16813529	C₂²	1	2	yes	—
24	#16813529	C₂²	1	2	yes	—
25	#16813529	C₂²	1	2	yes	—
26	#16813529	C₂²	1	1	yes	characteristic
27	#16813529	C₂²	1	2	yes	—
28	#16813529	C₂²	1	1	yes	characteristic
29	#16813529	C₂²	1	2	yes	—
30	#16813529	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
33	#95×C₁₁₃	C₂²	1	2	yes	—
34	#95×C₁₁₃	C₂²	1	1	yes	characteristic
35	#95×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂×Dic₂₂₆	—	2	4	no	—
37	C₂×Dic₂₂₆	C₂×C₄	1	4	yes	—
38	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
39	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
45	#14268626	—	2	4	no	—
46	#14268626	—	4	2	no	—
47	#14268626	C₂×C₄	1	4	yes	—
48	#14268626	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₁₃	—	2	4	no	—
50	C₄×Dic₁₁₃	—	4	2	no	—
51	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₁₃	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₁₃	—	4	2	no	—
59	C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
60	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₁₁₃	—	2	2	no	—
62	C₂×C₄×C₁₁₃	—	2	1	no	—
63	C₂×C₄×C₁₁₃	—	2	2	no	—
64	C₂×C₄×C₁₁₃	—	2	1	no	—
65	C₂×C₄×C₁₁₃	—	2	2	no	—
66	C₂×C₄×C₁₁₃	—	2	2	no	—
67	C₂×C₄×C₁₁₃	—	2	2	no	—
68	C₂×C₄×C₁₁₃	—	2	2	no	—
69	C₂×C₄×C₁₁₃	—	2	2	no	—
70	C₂×C₄×C₁₁₃	—	2	2	no	—
71	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₁₃	—	2	2	no	—
75	C₂×Dic₁₁₃	—	2	1	no	—
76	C₂×Dic₁₁₃	—	2	1	no	—
77	C₂×Dic₁₁₃	—	2	1	no	—
78	C₂×Dic₁₁₃	—	2	1	no	—
79	C₂×Dic₁₁₃	—	2	2	no	—
80	C₂×Dic₁₁₃	—	2	2	no	—
81	C₂×Dic₁₁₃	—	2	2	no	—
82	C₂×Dic₁₁₃	—	2	2	no	—
83	C₂×Dic₁₁₃	—	2	2	no	—
84	C₂×Dic₁₁₃	—	2	2	no	—
85	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₁₃	#41	1	4	yes	—
90	Dic₂₂₆	—	2	8	no	—
91	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
93	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
95	C₂²×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
96	C₄×C₁₁₃	—	2	2	no	—
97	C₄×C₁₁₃	—	2	2	no	—
98	C₄×C₁₁₃	—	2	2	no	—
99	C₄×C₁₁₃	—	4	2	no	—
100	C₄×C₁₁₃	—	4	2	no	—
101	Dic₁₁₃	—	2	2	no	—
102	Dic₁₁₃	—	2	2	no	—
103	Dic₁₁₃	—	2	2	no	—
104	Dic₁₁₃	—	4	2	no	—
105	Dic₁₁₃	—	4	2	no	—
106	Dic₁₁₃	#117	1	4	yes	—
107	C₂×C₁₁₃	C₂×#117	1	1	yes	characteristic
108	C₂×C₁₁₃	#519	1	1	yes	characteristic
109	C₂×C₁₁₃	#520	1	1	yes	characteristic
110	C₂×C₁₁₃	#541	1	2	yes	—
111	C₂×C₁₁₃	#542	1	2	yes	—
112	#2320	—	113	1	no	maximal, Sylow
113	C₁₁₃	#2320	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄×Q₈	—	113	1	no	—
115	#391	—	113	2	no	—
116	#389	—	113	2	no	—
117	#387	—	113	2	no	—
118	#387	—	113	2	no	—
119	#383	—	113	1	no	—
120	#383	—	113	1	no	—
121	#382	—	113	1	no	—
122	#382	D₁₁₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	—	113	1	no	—
124	#376	—	113	1	no	—
125	C₂²×Q₈	—	113	1	no	—
126	C₄×Q₈	—	113	8	no	—
127	C₂×C₄⋊C₄	—	113	1	no	—
128	C₂×C₄⋊C₄	—	113	1	no	—
129	C₂×C₄⋊C₄	—	113	1	no	—
130	C₂×C₄⋊C₄	—	113	2	no	—
131	C₂×C₄⋊C₄	—	113	2	no	—
132	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
133	C₂×C₄²	—	113	1	no	—
134	C₂×C₄²	—	113	1	no	—
135	C₂×C₄²	—	113	2	no	—
136	C₂×C₄²	—	113	2	no	—
137	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
138	#95	—	113	1	no	—
139	#95	—	113	2	no	—
140	#95	—	113	1	no	—
141	#95	—	113	2	no	—
142	#95	—	113	2	no	—
143	#95	—	113	2	no	—
144	#95	—	113	2	no	—
145	#95	—	113	2	no	—
146	#95	C₂×D₁₁₃	1	1	yes	characteristic
147	#95	C₂×D₁₁₃	1	1	yes	characteristic
148	#95	C₂×D₁₁₃	1	2	yes	—
149	C₂×Q₈	—	113	4	no	—
150	C₂×Q₈	—	226	4	no	—
151	C₂²×C₄	—	113	1	no	—
152	C₂²×C₄	—	113	1	no	—
153	C₂²×C₄	—	113	1	no	—
154	C₂²×C₄	—	113	1	no	—
155	C₂²×C₄	—	113	2	no	—
156	C₂²×C₄	—	113	2	no	—
157	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
158	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	113	4	no	—
164	C₄⋊C₄	—	113	4	no	—
165	C₄⋊C₄	—	113	4	no	—
166	C₄⋊C₄	—	452	2	no	—
167	C₄⋊C₄	—	226	4	no	—
168	C₄²	—	113	4	no	—
169	C₄²	—	113	4	no	—
170	C₄²	—	452	2	no	—
171	C₄²	—	226	4	no	—
172	C₄²	C₄×D₁₁₃	1	4	yes	—
173	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
174	Q₈	—	226	8	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	113	2	no	—
186	C₂×C₄	—	113	4	no	—
187	C₂×C₄	—	113	2	no	—
188	C₂×C₄	—	113	2	no	—
189	C₂×C₄	—	113	2	no	—
190	C₂×C₄	—	226	2	no	—
191	C₂×C₄	—	226	2	no	—
192	C₂×C₄	—	226	2	no	—
193	C₂×C₄	—	226	2	no	—
194	C₂×C₄	—	226	2	no	—
195	C₂×C₄	—	226	2	no	—
196	C₂×C₄	—	226	2	no	—
197	C₂×C₄	—	226	1	no	—
198	C₂×C₄	—	226	1	no	—
199	C₂×C₄	—	226	1	no	—
200	C₂×C₄	—	226	1	no	—
201	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
204	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
205	C₂²	#41×D₁₁₃	1	2	yes	—
206	C₂²	#16813667	1	1	yes	characteristic
207	C₂²	#16813669	1	2	yes	—
208	C₂²	#16813670	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	113	4	no	—
215	C₄	—	452	2	no	—
216	C₄	—	452	2	no	—
217	C₄	—	226	2	no	—
218	C₄	—	226	2	no	—
219	C₄	—	226	2	no	—
220	C₂	#117×D₁₁₃	1	1	yes	characteristic
221	C₂	#20299075	1	1	yes	characteristic
222	C₂	#20299079	1	1	yes	characteristic
223	C₂	#20299207	1	2	yes	—
224	C₂	#20299212	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Fitting	#382×C₁₁₃	C₂
socle, prime radical, cosocle	C₂³×C₁₁₃	C₂⁴
hypercenter	#382	D₁₁₃
hypocenter	C₁₁₃	#2320
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ #382

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₂₂₆	C₄	semidirect
113. C₁₁₃	#2320	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₂₆	2
#20298502	2
#20298493	2
#20298481	2
#20298479	2
#20298466	2
#20298465	2
#20298462	2
#20298449	2
#20298443	2
#382×C₁₁₃	2
#2320	113