Subgroups — #28399603

Order 14464.

456 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20298670	C₂	1	8	yes	maximal, complemented
2	#20298655	C₂	1	2	yes	maximal, complemented
3	#20298562	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#11744751	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14268631	C₂²	1	4	yes	complemented
7	C₂²×#11744751	C₂²	1	4	yes	complemented
8	C₄⋊#11744751	C₂²	1	4	yes	complemented
9	C₂²⋊#11744751	C₂²	1	4	yes	—
10	C₂²⋊#11744751	C₂²	1	4	yes	complemented
11	C₂²⋊#11744751	C₂²	1	4	yes	complemented
12	#16813533	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₁₁₃	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₁₃	C₄	1	4	yes	—
17	C₄×D₄×C₁₁₃	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
22	#14268631	—	4	4	no	—
23	C₂×#11744751	—	4	4	no	—
24	C₂×#11744751	C₂³	1	8	yes	—
25	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₁₁₃	—	2	4	no	—
28	C₂×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₁₃	—	4	2	no	—
30	C₂²×C₄×C₁₁₃	—	4	2	no	—
31	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
42	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
43	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
44	C₄²×C₁₁₃	C₂³	1	2	yes	—
45	C₂³×C₁₁₃	—	2	2	no	—
46	C₂³×C₁₁₃	—	4	1	no	—
47	C₂³×C₁₁₃	—	4	1	no	—
48	C₂³×C₁₁₃	—	4	1	no	—
49	C₂³×C₁₁₃	—	4	1	no	—
50	C₂³×C₁₁₃	M₁₆	1	4	yes	complemented
51	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
52	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
54	D₄×C₁₁₃	—	2	8	no	—
55	C₂×C₄×C₁₁₃	—	2	2	no	—
56	C₂×C₄×C₁₁₃	—	2	2	no	—
57	C₂×C₄×C₁₁₃	—	2	4	no	—
58	C₂×C₄×C₁₁₃	—	2	4	no	—
59	C₂×C₄×C₁₁₃	—	4	2	no	—
60	C₂×C₄×C₁₁₃	—	4	2	no	—
61	C₂×C₄×C₁₁₃	—	4	1	no	—
62	C₂×C₄×C₁₁₃	—	4	1	no	—
63	C₂×C₄×C₁₁₃	—	4	1	no	—
64	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
70	#11744751	—	2	8	no	—
71	C₂²×C₁₁₃	—	2	1	no	—
72	C₂²×C₁₁₃	—	2	2	no	—
73	C₂²×C₁₁₃	—	2	1	no	—
74	C₂²×C₁₁₃	—	2	2	no	—
75	C₂²×C₁₁₃	—	2	2	no	—
76	C₂²×C₁₁₃	—	2	2	no	—
77	C₂²×C₁₁₃	—	2	2	no	—
78	C₂²×C₁₁₃	—	4	4	no	—
79	C₂²×C₁₁₃	—	4	4	no	—
80	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
81	C₂²×C₁₁₃	C₂×M₁₆	1	4	yes	—
82	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₁₁₃	—	2	1	no	—
84	C₄×C₁₁₃	—	2	2	no	—
85	C₄×C₁₁₃	—	2	2	no	—
86	C₄×C₁₁₃	—	2	2	no	—
87	C₄×C₁₁₃	—	2	2	no	—
88	C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
89	C₂×C₁₁₃	—	2	2	no	—
90	C₂×C₁₁₃	—	2	4	no	—
91	C₂×C₁₁₃	—	4	2	no	—
92	C₂×C₁₁₃	#518	1	1	yes	characteristic
93	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
94	C₂×C₁₁₃	#568	1	1	yes	characteristic
95	#2973	—	113	1	no	maximal, Sylow
96	C₁₁₃	#2973	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄×D₄	D₁₁₃	1	1	yes	characteristic
98	#435	—	113	8	no	—
99	#432	—	113	2	no	—
100	#407	—	113	2	no	—
101	C₂×C₂²⋊C₈	—	113	2	no	—
102	C₂²×D₄	Dic₁₁₃	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
105	C₂×M₁₆	—	113	4	no	—
106	C₂²×C₈	—	113	4	no	—
107	C₄×D₄	Dic₁₁₃	1	4	yes	—
108	C₄×D₄	C₂×D₁₁₃	1	4	yes	—
109	C₂×C₄⋊C₄	Dic₁₁₃	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	Dic₁₁₃	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	Dic₁₁₃	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
113	C₄⋊C₈	—	113	4	no	—
114	C₂²⋊C₈	—	113	4	no	—
115	C₂²⋊C₈	—	113	4	no	—
116	C₂²⋊C₈	—	113	4	no	—
117	#97	—	113	4	no	—
118	C₂⁴	C₂×Dic₁₁₃	1	1	yes	characteristic
119	C₂⁴	C₂×Dic₁₁₃	1	1	yes	characteristic
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	C₂×Dic₁₁₃	1	4	yes	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
125	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
129	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
130	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
132	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
133	M₁₆	—	452	4	no	—
134	C₂×C₈	—	113	8	no	—
135	C₂×C₈	—	452	4	no	—
136	C₄⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
137	C₂²⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
138	C₂²⋊C₄	C₂×Dic₁₁₃	1	4	yes	—
139	C₄²	C₂×Dic₁₁₃	1	2	yes	—
140	C₄²	C₂²×D₁₁₃	1	2	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	1	no	—
146	C₂³	#14268631	1	4	yes	complemented
147	C₂³	C₂²×Dic₁₁₃	1	2	yes	—
148	C₂³	C₂²×Dic₁₁₃	1	2	yes	—
149	C₂³	C₂²×Dic₁₁₃	1	1	yes	characteristic
150	D₄	—	2	8	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
162	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
163	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
164	C₂×C₄	C₂²×Dic₁₁₃	1	2	yes	—
165	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
166	C₈	—	226	8	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂×#14268631	1	2	yes	—
177	C₂²	C₂×#14268631	1	4	yes	—
178	C₂²	C₂³×Dic₁₁₃	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	#16813667	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	2	no	—
188	C₂	#20299073	1	1	yes	characteristic
189	C₂	C₂²×#14268631	1	1	yes	characteristic
190	C₂	#20299365	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₁₃	C₂
prime radical	C₂²×D₄×C₁₁₃	C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	#2973
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20298670	C₂	semidirect
2. #20298655	C₂	semidirect
3. #20298562	C₂	semidirect
4. C₂×C₂²⋊#11744751	C₂	semidirect
6. C₂×#14268631	C₂²	semidirect
7. C₂²×#11744751	C₂²	semidirect
8. C₄⋊#11744751	C₂²	semidirect
10. C₂²⋊#11744751	C₂²	semidirect
11. C₂²⋊#11744751	C₂²	semidirect
12. #16813533	C₂²	semidirect
50. C₂³×C₁₁₃	M₁₆	semidirect
96. C₁₁₃	#2973	semidirect
146. C₂³	#14268631	semidirect

Maximal subgroups

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

subgroup	index
#20298670	2
#20298655	2
#20298562	2
C₂×C₂²⋊#11744751	2
C₂×C₄×D₄×C₁₁₃	2
#2973	113