Subgroups — C₂²×#11718535 #16732516

Order 3328.

2398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88142	C₂	1	2	yes	maximal
2	C₂×#11718535	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88047	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	—	2	2	no	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#6101	—	2	2	no	—
12	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂³×#1055	C₂²	1	1	yes	characteristic
14	C₂×#88142	—	2	12	no	—
15	C₂×#88142	C₂²	1	24	yes	—
16	C₂×C₁₃×#107	C₂²	1	12	yes	—
17	C₂×#88058	C₂²	1	12	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	12	yes	—
19	C₂×#88047	C₂²	1	24	yes	—
20	#11718535	C₂²	1	64	yes	direct product factor
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	—	2	2	no	—
24	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
25	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
26	C₂²×Dic₂₆	—	2	24	no	—
27	C₂²×Dic₂₆	—	4	16	no	—
28	C₂²×C₈×C₁₃	Q₈	1	8	yes	complemented
29	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
30	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
32	C₂×C₄⋊Dic₁₃	—	2	6	no	—
33	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
34	C₂×#6101	—	2	24	no	—
35	C₂×C₄×Dic₁₃	D₄	1	6	yes	—
36	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
37	C₂²×#1055	Q₈	1	8	yes	complemented
38	C₂²×#1055	C₂³	1	6	yes	—
39	#88142	—	2	48	no	—
40	#88142	C₂³	1	32	yes	—
41	C₁₃×#107	C₂³	1	16	yes	—
42	#88058	C₂³	1	16	yes	—
43	C₈×Dic₁₃	C₂³	1	16	yes	—
44	#88047	C₂³	1	32	yes	—
45	C₂²×#500	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	12	no	—
48	C₂²×C₄×C₁₃	—	4	8	no	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
53	C₂²×Dic₁₃	—	2	12	no	—
54	C₂²×Dic₁₃	—	4	8	no	—
55	C₂²×Dic₁₃	D₈	1	8	yes	—
56	C₂²×Dic₁₃	Q₁₆	1	8	yes	—
57	C₂²×Dic₁₃	C₂×D₄	1	6	yes	—
58	C₂²×Dic₁₃	C₂×D₄	1	6	yes	—
59	C₂×Dic₂₆	—	2	32	no	—
60	C₂×Dic₂₆	—	4	96	no	—
61	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	complemented
62	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
63	C₁₃×C₄⋊C₄	—	2	12	no	—
64	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
65	C₄⋊Dic₁₃	—	2	12	no	—
66	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
67	#6101	—	2	32	no	—
68	C₄×Dic₁₃	C₂×D₄	1	12	yes	—
69	C₄×Dic₁₃	C₂⁴	1	4	yes	—
70	C₂×#1055	C₂×Q₈	1	24	yes	complemented
71	C₂×#1055	C₂⁴	1	4	yes	—
72	C₂²×C₄⋊Q₈	—	13	2	no	—
73	C₂×#500	—	13	24	no	—
74	C₂²×#107	—	13	1	no	—
75	C₂²×#107	—	13	2	no	—
76	C₂²×#107	D₁₃	1	1	yes	characteristic
77	C₂²×C₄×C₈	—	13	1	no	—
78	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
79	C₂³×C₁₃	C₂×D₈	1	4	yes	—
80	C₂³×C₁₃	C₂×Q₁₆	1	4	yes	—
81	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
82	C₂×C₄×C₁₃	—	2	8	no	—
83	C₂×C₄×C₁₃	—	4	24	no	—
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
85	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
86	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
87	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
88	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
89	C₂×Dic₁₃	—	2	8	no	—
90	C₂×Dic₁₃	—	4	24	no	—
91	C₂×Dic₁₃	C₂×D₈	1	24	yes	—
92	C₂×Dic₁₃	C₂×Q₁₆	1	24	yes	—
93	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
94	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
95	Dic₂₆	—	4	64	no	—
96	C₈×C₁₃	C₂²×Q₈	1	8	yes	complemented
97	#1055	C₂²×Q₈	1	8	yes	complemented
98	C₂³×Q₈	—	26	2	no	—
99	C₂³×C₈	—	13	1	no	—
100	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
101	C₂×C₄⋊Q₈	—	13	24	no	—
102	C₂×C₄⋊Q₈	—	26	12	no	—
103	C₂²×C₄⋊C₄	—	13	2	no	—
104	C₂²×C₄⋊C₄	—	26	2	no	—
105	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
106	C₂²×C₄²	—	13	1	no	—
107	#500	—	13	64	no	—
108	C₂×#107	—	13	12	no	—
109	C₂×#107	—	13	24	no	—
110	C₂×#107	C₂×D₁₃	1	12	yes	—
111	C₂×C₄×C₈	—	13	12	no	—
112	C₂²×C₁₃	#500	1	16	yes	complemented
113	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
114	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
115	C₂²×C₁₃	C₂²×Q₁₆	1	6	yes	—
116	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₁₃	—	4	8	no	—
118	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
119	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
120	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
121	Dic₁₃	—	4	8	no	—
122	Dic₁₃	C₂²×D₈	1	8	yes	—
123	Dic₁₃	C₂²×Q₁₆	1	8	yes	—
124	C₂²×Q₈	—	26	24	no	—
125	C₂²×Q₈	—	52	16	no	—
126	C₂³×C₄	—	2	2	no	—
127	C₂³×C₄	—	13	1	no	—
128	C₂³×C₄	—	13	1	no	—
129	C₂³×C₄	—	26	2	no	—
130	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
131	C₂²×C₈	—	13	8	no	—
132	C₂²×C₈	—	13	6	no	—
133	C₂²×C₈	Dic₂₆	1	8	yes	complemented
134	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
135	C₄⋊Q₈	—	13	32	no	—
136	C₄⋊Q₈	—	26	48	no	—
137	C₂×C₄⋊C₄	—	2	6	no	—
138	C₂×C₄⋊C₄	—	13	12	no	—
139	C₂×C₄⋊C₄	—	26	24	no	—
140	C₂×C₄⋊C₄	—	26	6	no	—
141	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
142	C₂×C₄²	—	13	6	no	—
143	C₂×C₄²	—	13	6	no	—
144	#107	—	13	16	no	—
145	#107	—	13	32	no	—
146	#107	C₂²×D₁₃	1	16	yes	—
147	C₄×C₈	—	13	16	no	—
148	C₂×C₁₃	C₂×#500	1	12	yes	complemented
149	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
151	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
152	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
153	C₂×Q₈	—	26	32	no	—
154	C₂×Q₈	—	52	96	no	—
155	C₂²×C₄	—	2	12	no	—
156	C₂²×C₄	—	4	8	no	—
157	C₂²×C₄	—	13	8	no	—
158	C₂²×C₄	—	13	8	no	—
159	C₂²×C₄	—	13	6	no	—
160	C₂²×C₄	—	13	6	no	—
161	C₂²×C₄	—	26	12	no	—
162	C₂²×C₄	—	52	8	no	—
163	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
164	C₂²×C₄	D₄×D₁₃	1	3	yes	—
165	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
166	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
167	C₂×C₈	—	13	24	no	—
168	C₂×C₈	—	13	4	no	—
169	C₂×C₈	C₂×Dic₂₆	1	24	yes	complemented
170	C₂×C₈	C₂³×D₁₃	1	4	yes	—
171	C₄⋊C₄	—	2	12	no	—
172	C₄⋊C₄	—	13	8	no	—
173	C₄⋊C₄	—	26	32	no	—
174	C₄⋊C₄	—	26	12	no	—
175	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
176	C₄²	—	13	12	no	—
177	C₄²	—	13	4	no	—
178	C₁₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
179	C₂³	#88142	1	4	yes	—
180	C₂³	D₈×D₁₃	1	4	yes	—
181	C₂³	Q₁₆×D₁₃	1	4	yes	—
182	C₂³	C₂×D₄×D₁₃	1	3	yes	—
183	Q₈	—	52	64	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	4	24	no	—
186	C₂×C₄	—	13	24	no	—
187	C₂×C₄	—	13	24	no	—
188	C₂×C₄	—	13	4	no	—
189	C₂×C₄	—	13	4	no	—
190	C₂×C₄	—	26	8	no	—
191	C₂×C₄	—	52	24	no	—
192	C₂×C₄	#88142	1	12	yes	—
193	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
194	C₂×C₄	C₂×D₄×D₁₃	1	3	yes	—
195	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
196	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
197	C₈	—	13	8	no	—
198	C₈	C₂²×Dic₂₆	1	8	yes	complemented
199	C₂²	#11718535	1	16	yes	direct product factor
200	C₂²	C₂×#88142	1	6	yes	—
201	C₂²	C₂×D₈×D₁₃	1	6	yes	—
202	C₂²	C₂×Q₁₆×D₁₃	1	6	yes	—
203	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	13	8	no	—
206	C₄	—	13	8	no	—
207	C₄	—	52	8	no	—
208	C₄	C₂×#88142	1	6	yes	—
209	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
210	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
211	C₂	C₂×#11718535	1	12	yes	direct product factor
212	C₂	C₂²×#88142	1	1	yes	characteristic
213	C₂	C₂²×D₈×D₁₃	1	1	yes	characteristic
214	C₂	C₂²×Q₁₆×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#107	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×#107	D₁₃
hypocenter	C₁₃	C₂²×#500
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#11718535	C₂	direct (N × H)
20. #11718535	C₂²	direct (N × H)
28. C₂²×C₈×C₁₃	Q₈	semidirect
37. C₂²×#1055	Q₈	semidirect
61. C₂×C₈×C₁₃	C₂×Q₈	semidirect
70. C₂×#1055	C₂×Q₈	semidirect
96. C₈×C₁₃	C₂²×Q₈	semidirect
97. #1055	C₂²×Q₈	semidirect
112. C₂²×C₁₃	#500	semidirect
133. C₂²×C₈	Dic₂₆	semidirect
148. C₂×C₁₃	C₂×#500	semidirect
169. C₂×C₈	C₂×Dic₂₆	semidirect
178. C₁₃	C₂²×#500	semidirect
198. C₈	C₂²×Dic₂₆	semidirect
199. C₂²	#11718535	direct (N × H)
211. C₂	C₂×#11718535	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88142	2
C₂×#11718535	2
C₂²×#88058	2
C₂²×#88047	2
C₂²×C₈×Dic₁₃	2
C₂²×C₁₃×#107	2
C₂²×#500	13

Subgroups — C22×#11718535 #16732516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718535 #16732516