Subgroups — C₂²×#11718585 #16726531

Order 3328.

3438 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	C₂³×#6099	C₂	1	2	yes	maximal
2	C₂×#11718585	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#1055	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₄	1	2	yes	—
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₄	1	12	yes	—
9	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	12	yes	—
12	C₂²×#6099	—	2	16	no	—
13	C₂²×#6099	C₂²	1	24	yes	—
14	C₂³×#1055	C₂²	1	4	yes	—
15	#11718585	C₂²	1	64	yes	direct product factor
16	C₂×C₄⋊#1055	C₂²	1	48	yes	complemented
17	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	—	2	16	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	24	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	D₄	1	8	yes	—
25	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	48	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
34	C₂×#6099	—	2	96	no	—
35	C₂×#6099	C₂³	1	32	yes	—
36	C₂²×#1055	—	2	16	no	—
37	C₂²×#1055	C₂³	1	24	yes	—
38	C₄⋊#1055	C₂³	1	64	yes	complemented
39	C₂²×#423	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	—	2	8	no	—
41	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
42	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
43	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	8	no	—
45	C₂²×C₄×C₁₃	—	2	96	no	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	48	yes	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₁₃	M₁₆	1	16	yes	complemented
49	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	32	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
61	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
62	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
64	C₄²×C₁₃	C₂⁴	1	8	yes	—
65	#6099	—	2	64	no	—
66	C₂×#1055	—	2	48	no	—
67	C₂×#1055	C₂⁴	1	16	yes	—
68	C₂³×M₁₆	—	13	2	no	—
69	C₂³×C₄²	D₁₃	1	1	yes	characteristic
70	C₂×#423	—	13	24	no	—
71	C₂²×C₄⋊C₈	—	13	4	no	—
72	C₂³×C₁₃	—	2	24	no	—
73	C₂³×C₁₃	—	2	32	no	—
74	C₂³×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
75	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂³×C₁₃	C₂×M₁₆	1	8	yes	—
77	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
78	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
79	C₂×C₄×C₁₃	—	2	24	no	—
80	C₂×C₄×C₁₃	—	2	64	no	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
84	C₂×C₄×C₁₃	C₂×M₁₆	1	48	yes	complemented
85	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
90	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
91	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
92	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
93	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
94	#1055	—	2	16	no	—
95	C₂⁴×C₄	Dic₁₃	1	2	yes	—
96	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
97	C₂²×M₁₆	—	13	24	no	—
98	C₂²×M₁₆	—	26	16	no	—
99	C₂³×C₈	—	13	4	no	—
100	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
101	C₂²×C₄²	Dic₁₃	1	12	yes	—
102	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
103	C₂²×C₄²	C₂×D₁₃	1	12	yes	—
104	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
105	#423	—	13	64	no	—
106	C₂×C₄⋊C₈	—	13	48	no	—
107	C₂²×C₁₃	—	2	48	no	—
108	C₂²×C₁₃	—	2	8	no	—
109	C₂²×C₁₃	#423	1	16	yes	complemented
110	C₂²×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
111	C₂²×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
112	C₂²×C₁₃	C₂²×M₁₆	1	12	yes	—
113	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
114	C₄×C₁₃	—	2	8	no	—
115	C₄×C₁₃	C₂²×C₄⋊C₄	1	6	yes	—
116	C₄×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
117	C₄×C₁₃	C₂²×M₁₆	1	16	yes	complemented
118	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
120	C₂⁵	C₂×Dic₁₃	1	1	yes	characteristic
121	C₂³×C₄	—	2	16	no	—
122	C₂³×C₄	Dic₂₆	1	8	yes	—
123	C₂³×C₄	D₅₂	1	8	yes	—
124	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
125	C₂³×C₄	C₂×Dic₁₃	1	24	yes	—
126	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
127	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
128	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
129	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
130	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
131	C₂×M₁₆	—	13	32	no	—
132	C₂×M₁₆	—	26	96	no	—
133	C₂²×C₈	—	13	24	no	—
134	C₂²×C₈	—	26	16	no	—
135	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
136	C₂×C₄²	C₂×Dic₁₃	1	48	yes	—
137	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
138	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
139	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
140	C₂×C₄²	C₂²×D₁₃	1	16	yes	—
141	C₄⋊C₈	—	13	64	no	—
142	C₂×C₁₃	—	2	8	no	—
143	C₂×C₁₃	C₂×#423	1	12	yes	complemented
144	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₁₃	C₂³×M₁₆	1	2	yes	—
146	C₂⁴	—	2	8	no	—
147	C₂⁴	C₄⋊Dic₁₃	1	8	yes	—
148	C₂⁴	C₂²×Dic₁₃	1	6	yes	—
149	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	96	no	—
152	C₂²×C₄	#6099	1	16	yes	complemented
153	C₂²×C₄	C₄⋊Dic₁₃	1	48	yes	—
154	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
155	C₂²×C₄	C₂×Dic₂₆	1	24	yes	—
156	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
157	C₂²×C₄	C₂×D₅₂	1	4	yes	—
158	C₂²×C₄	C₂×D₅₂	1	24	yes	—
159	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
160	C₂²×C₄	C₂²×Dic₁₃	1	32	yes	—
161	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
162	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
163	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
164	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
165	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
166	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
167	M₁₆	—	26	64	no	—
168	C₂×C₈	—	13	16	no	—
169	C₂×C₈	—	26	48	no	—
170	C₄²	C₂²×Dic₁₃	1	16	yes	—
171	C₄²	C₂²×Dic₁₃	1	24	yes	—
172	C₄²	C₂²×Dic₁₃	1	16	yes	—
173	C₄²	C₂³×D₁₃	1	8	yes	—
174	C₁₃	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	2	24	no	—
176	C₂³	—	2	32	no	—
177	C₂³	C₂×#6099	1	8	yes	—
178	C₂³	C₂×C₄⋊Dic₁₃	1	24	yes	—
179	C₂³	C₂×C₄⋊Dic₁₃	1	4	yes	—
180	C₂³	C₂³×Dic₁₃	1	4	yes	—
181	C₂³	C₂³×Dic₁₃	1	3	yes	—
182	C₂×C₄	—	2	24	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	C₂×#6099	1	48	yes	complemented
185	C₂×C₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
188	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
189	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
190	C₂×C₄	C₂²×D₅₂	1	6	yes	—
191	C₂×C₄	C₂²×D₅₂	1	8	yes	—
192	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
193	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
194	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
195	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
196	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
197	C₈	—	26	16	no	—
198	C₂²	—	2	48	no	—
199	C₂²	—	2	8	no	—
200	C₂²	#11718585	1	16	yes	direct product factor
201	C₂²	C₂²×#6099	1	12	yes	—
202	C₂²	C₂²×C₄⋊Dic₁₃	1	8	yes	—
203	C₂²	C₂²×C₄⋊Dic₁₃	1	6	yes	—
204	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
205	C₄	—	2	8	no	—
206	C₄	C₂²×#6099	1	16	yes	complemented
207	C₄	C₂²×C₄⋊Dic₁₃	1	6	yes	—
208	C₄	C₂²×C₄⋊Dic₁₃	1	8	yes	—
209	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
210	C₄	C₂³×D₅₂	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#11718585	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
214	C₂	C₂³×#6099	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂²×#423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#11718585	C₂	direct (N × H)
3. C₂²×C₄⋊#1055	C₂	semidirect
15. #11718585	C₂²	direct (N × H)
16. C₂×C₄⋊#1055	C₂²	semidirect
38. C₄⋊#1055	C₂³	semidirect
48. C₂²×C₄×C₁₃	M₁₆	semidirect
84. C₂×C₄×C₁₃	C₂×M₁₆	semidirect
109. C₂²×C₁₃	#423	semidirect
117. C₄×C₁₃	C₂²×M₁₆	semidirect
143. C₂×C₁₃	C₂×#423	semidirect
152. C₂²×C₄	#6099	semidirect
174. C₁₃	C₂²×#423	semidirect
184. C₂×C₄	C₂×#6099	semidirect
200. C₂²	#11718585	direct (N × H)
206. C₄	C₂²×#6099	semidirect
212. C₂	C₂×#11718585	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×#6099	2
C₂×#11718585	2
C₂²×C₄⋊#1055	2
C₂³×C₄²×C₁₃	2
C₂²×#423	13

Subgroups — C22×#11718585 #16726531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718585 #16726531