Subgroups — C₂²×#11718514 #16726957

Order 3328.

2790 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	C₂³×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88141	C₂	1	2	yes	maximal
3	C₂×#11718514	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#88050	C₂	1	2	yes	maximal
5	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×#6101	—	2	2	no	—
11	C₂²×C₄×Dic₁₃	—	2	2	no	—
12	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂³×#1055	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₂	C₄	1	16	yes	complemented
15	C₂²×Dic₅₂	C₂²	1	12	yes	—
16	C₂×#88141	—	2	12	no	—
17	C₂×#88141	C₂²	1	24	yes	—
18	C₂×C₁₃×#106	C₂²	1	12	yes	—
19	C₂×#88055	C₂²	1	12	yes	—
20	C₂×#88050	C₂²	1	24	yes	—
21	#11718514	C₂²	1	64	yes	direct product factor
22	C₂³×C₄×C₁₃	—	2	2	no	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂³×Dic₁₃	—	2	2	no	—
25	C₂³×Dic₁₃	D₄	1	2	yes	—
26	C₂²×Dic₂₆	—	2	6	no	—
27	C₂²×Dic₂₆	—	2	8	no	—
28	C₂²×Dic₂₆	C₂×C₄	1	16	yes	—
29	C₂²×Dic₂₆	C₂³	1	12	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
32	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
34	C₂×#6101	—	2	24	no	—
35	C₂×C₄×Dic₁₃	—	2	24	no	—
36	C₂×C₄×Dic₁₃	D₄	1	6	yes	—
37	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
38	C₂²×#1055	—	2	4	no	—
39	C₂²×#1055	C₂³	1	6	yes	—
40	C₂×Dic₅₂	C₂×C₄	1	96	yes	complemented
41	C₂×Dic₅₂	C₂³	1	16	yes	—
42	#88141	—	2	48	no	—
43	#88141	C₂³	1	32	yes	—
44	C₁₃×#106	C₂³	1	16	yes	—
45	#88055	C₂³	1	16	yes	—
46	#88050	C₂³	1	32	yes	—
47	C₂²×#441	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	12	no	—
50	C₂²×C₄×C₁₃	—	4	8	no	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₁₃	#41	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
55	C₂²×Dic₁₃	—	2	16	no	—
56	C₂²×Dic₁₃	—	2	8	no	—
57	C₂²×Dic₁₃	—	2	12	no	—
58	C₂²×Dic₁₃	C₂×D₄	1	12	yes	—
59	C₂×Dic₂₆	—	2	48	no	—
60	C₂×Dic₂₆	—	2	12	no	—
61	C₂×Dic₂₆	—	2	24	no	—
62	C₂×Dic₂₆	C₂²×C₄	1	48	yes	—
63	C₂×Dic₂₆	C₂⁴	1	8	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
65	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
66	C₁₃×C₄⋊C₄	—	2	12	no	—
67	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
68	#6101	—	2	32	no	—
69	C₄×Dic₁₃	—	2	32	no	—
70	C₄×Dic₁₃	C₂×D₄	1	12	yes	—
71	C₄×Dic₁₃	C₂⁴	1	4	yes	—
72	C₂×#1055	—	2	12	no	—
73	C₂×#1055	C₂⁴	1	4	yes	—
74	Dic₅₂	C₂²×C₄	1	64	yes	complemented
75	C₂³×Q₁₆	—	13	1	no	—
76	C₂²×C₄×Q₈	—	13	2	no	—
77	C₂×#441	—	13	24	no	—
78	C₂²×#106	D₁₃	1	1	yes	characteristic
79	C₂²×#103	—	13	2	no	—
80	C₂²×#97	—	13	1	no	—
81	C₂³×C₁₃	C₄×D₄	1	4	yes	—
82	C₂³×C₁₃	#137	1	8	yes	—
83	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
84	C₂×C₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	—	4	24	no	—
86	C₂×C₄×C₁₃	C₄×D₄	1	12	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
89	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
90	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
91	C₂×Dic₁₃	—	2	48	no	—
92	C₂×Dic₁₃	—	2	24	no	—
93	C₂×Dic₁₃	—	2	8	no	—
94	C₂×Dic₁₃	C₂²×D₄	1	8	yes	—
95	Dic₂₆	—	2	48	no	—
96	Dic₂₆	—	2	8	no	—
97	Dic₂₆	C₂³×C₄	1	16	yes	—
98	C₈×C₁₃	C₂³×C₄	1	8	yes	—
99	#1055	—	2	4	no	—
100	C₂³×Q₈	—	13	2	no	—
101	C₂²×Q₁₆	—	13	16	no	—
102	C₂²×Q₁₆	—	13	12	no	—
103	C₂³×C₈	—	13	1	no	—
104	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₄×Q₈	—	13	24	no	—
106	C₂×C₄×Q₈	—	26	12	no	—
107	C₂²×C₄⋊C₄	—	26	2	no	—
108	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
109	C₂²×C₄²	—	13	1	no	—
110	C₂²×C₄²	—	26	2	no	—
111	#441	—	13	64	no	—
112	C₂×#106	C₂×D₁₃	1	12	yes	—
113	C₂×#103	—	13	24	no	—
114	C₂×#97	—	13	12	no	—
115	C₂²×C₁₃	#441	1	16	yes	complemented
116	C₂²×C₁₃	C₂×C₄×D₄	1	6	yes	—
117	C₂²×C₁₃	C₂×#137	1	12	yes	—
118	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₁₃	—	4	8	no	—
120	C₄×C₁₃	C₂×C₄×D₄	1	6	yes	—
121	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
122	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
123	Dic₁₃	—	2	16	no	—
124	Dic₁₃	—	2	8	no	—
125	C₂²×Q₈	—	13	16	no	—
126	C₂²×Q₈	—	13	12	no	—
127	C₂²×Q₈	—	26	6	no	—
128	C₂²×Q₈	—	26	8	no	—
129	C₂³×C₄	—	2	2	no	—
130	C₂³×C₄	—	13	2	no	—
131	C₂³×C₄	—	26	2	no	—
132	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
133	C₂×Q₁₆	—	13	96	no	—
134	C₂×Q₁₆	—	13	16	no	—
135	C₂²×C₈	—	13	6	no	—
136	C₂²×C₈	—	26	4	no	—
137	C₂²×C₈	C₄×D₁₃	1	8	yes	—
138	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
139	C₄×Q₈	—	13	32	no	—
140	C₄×Q₈	—	26	48	no	—
141	C₂×C₄⋊C₄	—	2	6	no	—
142	C₂×C₄⋊C₄	—	26	24	no	—
143	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
144	C₂×C₄²	—	13	6	no	—
145	C₂×C₄²	—	13	6	no	—
146	C₂×C₄²	—	26	24	no	—
147	#106	C₂²×D₁₃	1	16	yes	—
148	#103	—	13	32	no	—
149	#97	—	13	16	no	—
150	C₂×C₁₃	C₂×#441	1	12	yes	complemented
151	C₂×C₁₃	C₂²×C₄×D₄	1	1	yes	characteristic
152	C₂×C₁₃	C₂²×#137	1	2	yes	—
153	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
154	C₂×Q₈	—	13	48	no	—
155	C₂×Q₈	—	13	8	no	—
156	C₂×Q₈	—	26	48	no	—
157	C₂×Q₈	—	26	12	no	—
158	C₂×Q₈	—	26	24	no	—
159	C₂²×C₄	—	2	12	no	—
160	C₂²×C₄	—	4	8	no	—
161	C₂²×C₄	—	13	12	no	—
162	C₂²×C₄	—	26	16	no	—
163	C₂²×C₄	—	26	8	no	—
164	C₂²×C₄	—	26	12	no	—
165	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
166	C₂²×C₄	D₄×D₁₃	1	3	yes	—
167	C₂²×C₄	#6131	1	4	yes	—
168	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
169	Q₁₆	—	13	64	no	—
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	26	12	no	—
172	C₂×C₈	C₂×C₄×D₁₃	1	24	yes	—
173	C₂×C₈	C₂³×D₁₃	1	4	yes	—
174	C₄⋊C₄	—	2	12	no	—
175	C₄⋊C₄	—	26	32	no	—
176	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
177	C₄²	—	13	12	no	—
178	C₄²	—	13	4	no	—
179	C₄²	—	26	32	no	—
180	C₁₃	C₂²×#441	1	1	yes	characteristic, complemented, Sylow
181	C₂³	#88147	1	4	yes	—
182	C₂³	#88169	1	8	yes	—
183	C₂³	C₂×D₄×D₁₃	1	3	yes	—
184	Q₈	—	13	16	no	—
185	Q₈	—	26	48	no	—
186	Q₈	—	26	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	4	24	no	—
189	C₂×C₄	—	13	8	no	—
190	C₂×C₄	—	26	48	no	—
191	C₂×C₄	—	26	24	no	—
192	C₂×C₄	—	26	8	no	—
193	C₂×C₄	#88147	1	12	yes	—
194	C₂×C₄	C₂²×C₄×D₁₃	1	6	yes	—
195	C₂×C₄	C₂×D₄×D₁₃	1	3	yes	—
196	C₂×C₄	C₂×#6131	1	6	yes	—
197	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
198	C₈	—	26	4	no	—
199	C₈	C₂²×C₄×D₁₃	1	8	yes	—
200	C₂²	#11718514	1	16	yes	direct product factor
201	C₂²	C₂×#88147	1	6	yes	—
202	C₂²	C₂×#88169	1	12	yes	—
203	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	26	16	no	—
206	C₄	—	26	8	no	—
207	C₄	C₂×#88147	1	6	yes	—
208	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
209	C₄	C₂²×#6131	1	1	yes	characteristic
210	C₂	C₂×#11718514	1	12	yes	direct product factor
211	C₂	C₂²×#88147	1	1	yes	characteristic
212	C₂	C₂²×#88169	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#106	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#106	D₁₃
hypocenter	C₁₃	C₂²×#441
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. C₂×#11718514	C₂	direct (N × H)
14. C₂²×Dic₅₂	C₄	semidirect
21. #11718514	C₂²	direct (N × H)
40. C₂×Dic₅₂	C₂×C₄	semidirect
74. Dic₅₂	C₂²×C₄	semidirect
115. C₂²×C₁₃	#441	semidirect
150. C₂×C₁₃	C₂×#441	semidirect
180. C₁₃	C₂²×#441	semidirect
200. C₂²	#11718514	direct (N × H)
210. C₂	C₂×#11718514	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₅₂	2
C₂²×#88141	2
C₂×#11718514	2
C₂²×#88050	2
C₂²×#88055	2
C₂²×C₁₃×#106	2
C₂²×#441	13

Subgroups — C22×#11718514 #16726957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718514 #16726957