Subgroups — C₂²×#11718565 #16769406

Order 3328.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88158	C₂	1	2	yes	maximal, complemented
2	C₂×#11718565	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88040	C₂	1	2	yes	maximal, complemented
5	C₂²×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6127	—	2	2	no	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₂	C₂²	1	2	yes	—
10	C₂×#88158	—	2	12	no	—
11	C₂×#88158	C₂²	1	24	yes	complemented
12	C₂²×D₁₀₄	C₂²	1	2	yes	—
13	C₂²×#6095	—	2	2	no	—
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	12	yes	complemented
17	C₂×Dic₁₀₄	C₂²	1	12	yes	complemented
18	#11718565	C₂²	1	64	yes	direct product factor
19	C₂×#88040	C₂²	1	24	yes	complemented
20	C₂×D₂₀₈	C₂²	1	12	yes	complemented
21	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₂²⋊D₁₃	—	4	2	no	—
23	C₂×#6127	—	2	24	no	—
24	C₂×#6127	—	4	12	no	—
25	C₂²×D₅₂	—	2	2	no	—
26	C₂²×C₄×D₁₃	—	4	2	no	—
27	C₂²×Dic₂₆	—	2	2	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	6	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
32	C₂×Dic₅₂	—	2	6	no	—
33	C₂×Dic₅₂	C₂³	1	12	yes	—
34	#88158	—	2	48	no	—
35	#88158	C₂³	1	32	yes	complemented
36	C₂×D₁₀₄	—	2	6	no	—
37	C₂×D₁₀₄	C₂³	1	12	yes	—
38	C₂×#6095	—	2	24	no	—
39	C₂×C₁₃×C₁₆	C₂³	1	6	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	6	yes	—
41	C₂×C₁₃×C₁₆	C₂³	1	16	yes	complemented
42	Dic₁₀₄	C₂³	1	16	yes	complemented
43	#88040	C₂³	1	32	yes	complemented
44	D₂₀₈	C₂³	1	16	yes	complemented
45	C₂²×#508	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
47	C₂³×D₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	D₈	1	6	yes	—
49	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
52	C₂×C₂²⋊D₁₃	—	4	24	no	—
53	C₂²×Dic₁₃	—	4	2	no	—
54	#6127	—	2	32	no	—
55	#6127	—	4	48	no	—
56	C₂×D₅₂	—	2	12	no	—
57	C₂×D₅₂	—	4	6	no	—
58	C₂×C₄×D₁₃	—	4	24	no	—
59	C₂×Dic₂₆	—	2	12	no	—
60	C₂×Dic₂₆	—	4	6	no	—
61	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
63	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
65	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
66	Dic₅₂	—	2	12	no	—
67	Dic₅₂	C₂⁴	1	8	yes	—
68	D₁₀₄	—	2	12	no	—
69	D₁₀₄	C₂⁴	1	8	yes	—
70	#6095	—	2	32	no	—
71	C₁₃×C₁₆	C₂⁴	1	4	yes	—
72	C₁₃×C₁₆	C₂⁴	1	4	yes	—
73	C₂²×#135	—	13	2	no	—
74	C₂×#508	—	13	24	no	—
75	C₂²×Q₃₂	—	13	1	no	—
76	C₂²×SD₃₂	—	13	2	no	—
77	C₂²×D₁₆	—	13	1	no	—
78	C₂³×C₁₆	D₁₃	1	1	yes	characteristic, complemented
79	C₂³×C₁₃	—	2	4	no	—
80	C₂³×C₁₃	C₂×D₈	1	6	yes	—
81	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
82	C₂²×D₁₃	—	4	12	no	—
83	C₂²×D₁₃	—	8	8	no	—
84	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
86	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
87	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
89	C₂²⋊D₁₃	—	4	32	no	—
90	C₂×Dic₁₃	—	4	12	no	—
91	D₅₂	—	2	8	no	—
92	D₅₂	—	4	12	no	—
93	C₄×D₁₃	—	4	32	no	—
94	Dic₂₆	—	2	8	no	—
95	Dic₂₆	—	4	12	no	—
96	C₈×C₁₃	C₂²×D₄	1	4	yes	—
97	C₈×C₁₃	C₂²×D₄	1	3	yes	—
98	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
99	C₂²×#41	—	26	2	no	—
100	C₂×#135	—	13	24	no	—
101	C₂×#135	—	26	12	no	—
102	C₂²×Q₁₆	—	13	2	no	—
103	C₂²×SD₁₆	—	26	2	no	—
104	C₂²×D₈	—	13	2	no	—
105	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
106	#508	—	13	64	no	—
107	C₂×Q₃₂	—	13	12	no	—
108	C₂×SD₃₂	—	13	24	no	—
109	C₂×D₁₆	—	13	12	no	—
110	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
111	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₁₃	1	12	yes	complemented
113	C₂²×C₁₃	—	2	12	no	—
114	C₂²×C₁₃	#508	1	4	yes	complemented
115	C₂²×C₁₃	C₂²×D₈	1	4	yes	—
116	C₂²×C₁₃	C₂²×D₈	1	3	yes	—
117	C₂×D₁₃	—	4	8	no	—
118	C₂×D₁₃	—	8	24	no	—
119	C₄×C₁₃	C₂²×D₈	1	3	yes	—
120	C₄×C₁₃	C₂²×D₈	1	4	yes	—
121	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
122	Dic₁₃	—	4	8	no	—
123	C₂×#41	—	26	24	no	—
124	C₂×#41	—	52	12	no	—
125	C₂²×Q₈	—	26	2	no	—
126	C₂²×D₄	—	26	2	no	—
127	C₂²×D₄	—	52	2	no	—
128	C₂³×C₄	—	52	2	no	—
129	C₂³×C₄	D₅₂	1	1	yes	characteristic
130	#135	—	13	32	no	—
131	#135	—	26	48	no	—
132	C₂×Q₁₆	—	13	12	no	—
133	C₂×Q₁₆	—	26	6	no	—
134	C₂×SD₁₆	—	26	24	no	—
135	C₂×D₈	—	13	12	no	—
136	C₂×D₈	—	26	6	no	—
137	C₂²×C₈	D₅₂	1	1	yes	characteristic
138	C₂²×C₈	D₅₂	1	6	yes	—
139	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
141	Q₃₂	—	13	16	no	—
142	SD₃₂	—	13	32	no	—
143	D₁₆	—	13	16	no	—
144	C₂×C₁₆	C₂²×D₁₃	1	6	yes	—
145	C₂×C₁₆	C₂²×D₁₃	1	6	yes	—
146	C₂×C₁₆	C₂²×D₁₃	1	16	yes	complemented
147	C₂×C₁₃	—	2	4	no	—
148	C₂×C₁₃	C₂×#508	1	6	yes	complemented
149	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
150	D₁₃	—	8	8	no	—
151	C₂⁴	—	52	2	no	—
152	C₂⁴	D₁₀₄	1	1	yes	characteristic
153	#41	—	26	32	no	—
154	#41	—	52	48	no	—
155	C₂×Q₈	—	26	12	no	—
156	C₂×Q₈	—	52	6	no	—
157	C₂×D₄	—	26	12	no	—
158	C₂×D₄	—	52	24	no	—
159	C₂×D₄	—	52	6	no	—
160	C₂²×C₄	—	52	2	no	—
161	C₂²×C₄	—	52	24	no	—
162	C₂²×C₄	D₁₀₄	1	6	yes	—
163	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₅₂	1	6	yes	—
166	Q₁₆	—	13	8	no	—
167	Q₁₆	—	26	12	no	—
168	SD₁₆	—	26	32	no	—
169	D₈	—	13	8	no	—
170	D₈	—	26	12	no	—
171	C₂×C₈	C₂×D₅₂	1	3	yes	—
172	C₂×C₈	C₂×D₅₂	1	6	yes	—
173	C₂×C₈	C₂×D₅₂	1	12	yes	—
174	C₂×C₈	C₂³×D₁₃	1	3	yes	—
175	C₂×C₈	C₂³×D₁₃	1	4	yes	—
176	C₁₆	C₂³×D₁₃	1	4	yes	—
177	C₁₆	C₂³×D₁₃	1	4	yes	—
178	C₁₃	C₂²×#508	1	1	yes	characteristic, complemented, Sylow
179	C₂³	—	2	4	no	—
180	C₂³	—	52	12	no	—
181	C₂³	—	104	8	no	—
182	C₂³	C₂×D₁₀₄	1	6	yes	—
183	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
184	Q₈	—	26	8	no	—
185	Q₈	—	52	12	no	—
186	D₄	—	26	8	no	—
187	D₄	—	52	32	no	—
188	D₄	—	52	12	no	—
189	C₂×C₄	—	52	12	no	—
190	C₂×C₄	—	52	32	no	—
191	C₂×C₄	C₂×D₁₀₄	1	6	yes	—
192	C₂×C₄	C₂×D₁₀₄	1	3	yes	—
193	C₂×C₄	C₂×D₁₀₄	1	12	yes	—
194	C₂×C₄	C₂²×D₅₂	1	3	yes	—
195	C₂×C₄	C₂²×D₅₂	1	4	yes	—
196	C₈	C₂²×D₅₂	1	3	yes	—
197	C₈	C₂²×D₅₂	1	4	yes	—
198	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
199	C₂²	—	2	12	no	—
200	C₂²	—	52	8	no	—
201	C₂²	—	104	24	no	—
202	C₂²	#11718565	1	4	yes	direct product factor
203	C₂²	C₂²×D₁₀₄	1	4	yes	—
204	C₂²	C₂²×D₁₀₄	1	3	yes	—
205	C₄	—	52	8	no	—
206	C₄	C₂²×D₁₀₄	1	4	yes	—
207	C₄	C₂²×D₁₀₄	1	3	yes	—
208	C₄	C₂³×D₅₂	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	104	8	no	—
211	C₂	C₂×#11718565	1	6	yes	direct product factor
212	C₂	C₂³×D₁₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₀₄
commutator, cosocle	C₈×C₁₃	C₂⁵
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂³×C₁₆	D₁₃
hypocenter	C₁₃	C₂²×#508
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#88158	C₂	semidirect
2. C₂×#11718565	C₂	direct (N × H)
3. C₂²×Dic₁₀₄	C₂	semidirect
4. C₂²×#88040	C₂	semidirect
5. C₂²×D₂₀₈	C₂	semidirect
6. C₂³×C₁₃×C₁₆	C₂	semidirect
11. C₂×#88158	C₂²	semidirect
16. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. C₂×Dic₁₀₄	C₂²	semidirect
18. #11718565	C₂²	direct (N × H)
19. C₂×#88040	C₂²	semidirect
20. C₂×D₂₀₈	C₂²	semidirect
35. #88158	C₂³	semidirect
41. C₂×C₁₃×C₁₆	C₂³	semidirect
42. Dic₁₀₄	C₂³	semidirect
43. #88040	C₂³	semidirect
44. D₂₀₈	C₂³	semidirect
78. C₂³×C₁₆	D₁₃	semidirect
112. C₂²×C₁₆	C₂×D₁₃	semidirect
114. C₂²×C₁₃	#508	semidirect
146. C₂×C₁₆	C₂²×D₁₃	semidirect
148. C₂×C₁₃	C₂×#508	semidirect
178. C₁₃	C₂²×#508	semidirect
202. C₂²	#11718565	direct (N × H)
211. C₂	C₂×#11718565	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×#88158	2
C₂×#11718565	2
C₂²×Dic₁₀₄	2
C₂²×#88040	2
C₂²×D₂₀₈	2
C₂³×C₁₃×C₁₆	2
C₂²×#508	13

Subgroups — C22×#11718565 #16769406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718565 #16769406