Subgroups — C₂²×#11718340 #16727276

Order 3328.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718340	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88054	C₂	1	2	yes	maximal
5	C₂²×#88055	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
12	C₂²×#6101	C₄	1	2	yes	—
13	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
14	C₂³×#1055	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₆	C₄	1	12	yes	—
16	C₂×C₄×Dic₂₆	C₂²	1	12	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	—
18	C₂×#88055	C₂²	1	24	yes	—
19	C₂×#88054	C₂²	1	24	yes	—
20	C₂×C₄⋊#1055	C₂²	1	12	yes	—
21	#11718340	C₂²	1	64	yes	direct product factor
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
26	C₂³×Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂²×Dic₂₆	—	2	8	no	—
28	C₂²×Dic₂₆	C₂×C₄	1	12	yes	—
29	C₂²×C₈×C₁₃	—	2	4	no	—
30	C₂²×C₈×C₁₃	Q₈	1	8	yes	complemented
31	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	12	yes	—
36	C₂×#6101	C₂×C₄	1	24	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
38	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
39	C₂²×#1055	—	2	8	no	—
40	C₂²×#1055	C₂³	1	12	yes	—
41	C₄×Dic₂₆	C₂×C₄	1	48	yes	—
42	C₄×Dic₂₆	C₂³	1	16	yes	—
43	C₄×C₈×C₁₃	C₂³	1	16	yes	—
44	#88055	C₂³	1	32	yes	—
45	#88054	C₂³	1	32	yes	—
46	C₄⋊#1055	C₂³	1	16	yes	—
47	C₂²×#446	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₁₃	#41	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
57	C₂²×Dic₁₃	—	2	8	no	—
58	C₂²×Dic₁₃	—	2	8	no	—
59	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
60	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
61	C₂×Dic₂₆	—	2	48	no	—
62	C₂×Dic₂₆	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₁₃	—	2	12	no	—
64	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	complemented
65	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
66	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
67	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
68	C₄²×C₁₃	C₂⁴	1	4	yes	—
69	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
70	#6101	C₂²×C₄	1	32	yes	—
71	C₄×Dic₁₃	C₂²×C₄	1	24	yes	—
72	C₄×Dic₁₃	C₂⁴	1	8	yes	—
73	C₂×#1055	—	2	24	no	—
74	C₂×#1055	C₂⁴	1	8	yes	—
75	C₂²×C₄×Q₈	—	13	1	no	—
76	C₂×#446	—	13	24	no	—
77	C₂²×C₄⋊C₈	—	13	1	no	—
78	C₂²×C₄⋊C₈	—	13	2	no	—
79	C₂²×#97	—	13	2	no	—
80	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
81	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
82	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
83	C₂³×C₁₃	#131	1	4	yes	—
84	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
85	C₂×C₄×C₁₃	—	2	12	no	—
86	C₂×C₄×C₁₃	C₄×Q₈	1	12	yes	—
87	C₂×C₄×C₁₃	C₂×M₁₆	1	24	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
92	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
93	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
94	C₂×Dic₁₃	—	2	24	no	—
95	C₂×Dic₁₃	—	2	24	no	—
96	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
97	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
98	Dic₂₆	—	2	32	no	—
99	C₈×C₁₃	—	2	4	no	—
100	C₈×C₁₃	C₂²×Q₈	1	8	yes	complemented
101	#1055	—	2	8	no	—
102	C₂³×Q₈	—	13	1	no	—
103	C₂³×C₈	—	13	2	no	—
104	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄×Q₈	—	13	12	no	—
107	C₂×C₄×Q₈	—	13	12	no	—
108	C₂²×C₄⋊C₄	—	13	2	no	—
109	C₂²×C₄⋊C₄	—	13	1	no	—
110	C₂²×C₄²	—	13	2	no	—
111	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
112	#446	—	13	64	no	—
113	C₂×C₄⋊C₈	—	13	24	no	—
114	C₂×C₄⋊C₈	—	13	12	no	—
115	C₂×#97	—	13	24	no	—
116	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	—
117	C₂²×C₁₃	#446	1	16	yes	complemented
118	C₂²×C₁₃	C₂×C₄×Q₈	1	6	yes	—
119	C₂²×C₁₃	C₂²×M₁₆	1	6	yes	—
120	C₂²×C₁₃	C₂×#131	1	6	yes	—
121	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	C₂×C₄×Q₈	1	6	yes	—
124	C₄×C₁₃	C₂²×M₁₆	1	8	yes	—
125	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
126	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
127	Dic₁₃	—	2	8	no	—
128	Dic₁₃	—	2	8	no	—
129	C₂²×Q₈	—	13	12	no	—
130	C₂²×Q₈	—	26	8	no	—
131	C₂³×C₄	—	13	2	no	—
132	C₂³×C₄	—	13	2	no	—
133	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
134	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	—	13	12	no	—
138	C₂²×C₈	—	26	8	no	—
139	C₂²×C₈	Dic₂₆	1	8	yes	complemented
140	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
141	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
142	C₄×Q₈	—	13	48	no	—
143	C₄×Q₈	—	13	16	no	—
144	C₂×C₄⋊C₄	—	13	24	no	—
145	C₂×C₄⋊C₄	—	13	12	no	—
146	C₂×C₄²	—	13	12	no	—
147	C₂×C₄²	—	13	12	no	—
148	C₂×C₄²	C₄×D₁₃	1	6	yes	—
149	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
150	C₄⋊C₈	—	13	32	no	—
151	C₄⋊C₈	—	13	16	no	—
152	#97	—	13	32	no	—
153	C₄×C₈	C₂²×D₁₃	1	16	yes	—
154	C₂×C₁₃	C₂×#446	1	12	yes	complemented
155	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
156	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
157	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
158	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
159	C₂×Q₈	—	13	16	no	—
160	C₂×Q₈	—	26	48	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	13	12	no	—
163	C₂²×C₄	—	13	12	no	—
164	C₂²×C₄	—	26	8	no	—
165	C₂²×C₄	—	26	8	no	—
166	C₂²×C₄	#6094	1	8	yes	—
167	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
168	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
169	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
170	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
171	C₂²×C₄	#6127	1	4	yes	—
172	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
173	C₂×C₈	—	2	12	no	—
174	C₂×C₈	—	13	8	no	—
175	C₂×C₈	—	26	24	no	—
176	C₂×C₈	C₂×Dic₂₆	1	24	yes	complemented
177	C₂×C₈	C₂³×D₁₃	1	4	yes	—
178	C₂×C₈	C₂³×D₁₃	1	4	yes	—
179	C₄⋊C₄	—	13	32	no	—
180	C₄⋊C₄	—	13	16	no	—
181	C₄²	—	13	24	no	—
182	C₄²	—	13	8	no	—
183	C₄²	C₂×C₄×D₁₃	1	12	yes	—
184	C₄²	C₂³×D₁₃	1	4	yes	—
185	C₁₃	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₄×Dic₂₆	1	4	yes	—
187	C₂³	C₂×#6094	1	4	yes	—
188	C₂³	#88155	1	4	yes	—
189	C₂³	C₂²×C₄×D₁₃	1	3	yes	—
190	Q₈	—	26	32	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	13	8	no	—
193	C₂×C₄	—	13	8	no	—
194	C₂×C₄	—	26	24	no	—
195	C₂×C₄	—	26	24	no	—
196	C₂×C₄	C₄×Dic₂₆	1	12	yes	—
197	C₂×C₄	C₂×#6094	1	24	yes	—
198	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
199	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
200	C₂×C₄	C₂²×C₄×D₁₃	1	3	yes	—
201	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
202	C₂×C₄	C₂×#6127	1	6	yes	—
203	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	—	26	8	no	—
206	C₈	C₂²×Dic₂₆	1	8	yes	complemented
207	C₂²	#11718340	1	16	yes	direct product factor
208	C₂²	C₂×C₄×Dic₂₆	1	6	yes	—
209	C₂²	C₂²×#6094	1	6	yes	—
210	C₂²	C₂×#88155	1	6	yes	—
211	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	26	8	no	—
214	C₄	—	26	8	no	—
215	C₄	C₂×C₄×Dic₂₆	1	6	yes	—
216	C₄	C₂²×#6094	1	8	yes	—
217	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
218	C₄	C₂²×#6127	1	1	yes	characteristic
219	C₂	C₂×#11718340	1	12	yes	direct product factor
220	C₂	C₂²×C₄×Dic₂₆	1	1	yes	characteristic
221	C₂	C₂³×#6094	1	1	yes	characteristic
222	C₂	C₂²×#88155	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#446
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718340	C₂	direct (N × H)
21. #11718340	C₂²	direct (N × H)
30. C₂²×C₈×C₁₃	Q₈	semidirect
64. C₂×C₈×C₁₃	C₂×Q₈	semidirect
100. C₈×C₁₃	C₂²×Q₈	semidirect
117. C₂²×C₁₃	#446	semidirect
139. C₂²×C₈	Dic₂₆	semidirect
154. C₂×C₁₃	C₂×#446	semidirect
176. C₂×C₈	C₂×Dic₂₆	semidirect
185. C₁₃	C₂²×#446	semidirect
206. C₈	C₂²×Dic₂₆	semidirect
207. C₂²	#11718340	direct (N × H)
219. C₂	C₂×#11718340	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×C₄×Dic₂₆	2
C₂×#11718340	2
C₂²×C₄⋊#1055	2
C₂²×#88054	2
C₂²×#88055	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#446	13

Subgroups — C22×#11718340 #16727276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718340 #16727276