Subgroups — C₂²×#11718354 #16732317

Order 3328.

3950 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×#6095	C₂	1	4	yes	maximal, complemented
2	C₂²×#88123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88128	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718354	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₅₂	—	2	2	no	—
7	C₂³×D₅₂	C₂²	1	2	yes	—
8	C₂³×Dic₂₆	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₃	—	2	2	no	—
12	C₂²×#6095	—	2	32	no	—
13	C₂²×#6095	C₂²	1	48	yes	complemented
14	C₂×#88128	—	2	6	no	—
15	C₂×#88128	C₂²	1	12	yes	—
16	C₂×#88123	—	2	6	no	—
17	C₂×#88123	C₂²	1	12	yes	complemented
18	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
19	#11718354	C₂²	1	64	yes	direct product factor
20	C₂⁴×D₁₃	—	2	2	no	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	—	2	2	no	—
24	C₂²×D₅₂	—	2	16	no	—
25	C₂²×D₅₂	—	2	6	no	—
26	C₂²×D₅₂	—	2	24	no	—
27	C₂²×D₅₂	—	4	16	no	—
28	C₂²×D₅₂	C₂³	1	12	yes	—
29	C₂²×Dic₂₆	—	2	16	no	—
30	C₂²×Dic₂₆	—	2	6	no	—
31	C₂²×Dic₂₆	C₂³	1	12	yes	—
32	C₂²×C₈×C₁₃	D₄	1	16	yes	complemented
33	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
34	C₂×C₄²×C₁₃	D₄	1	6	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
36	C₂×C₄⋊Dic₁₃	—	2	24	no	—
37	C₂×#6095	—	2	192	no	—
38	C₂×#6095	C₂³	1	64	yes	complemented
39	#88128	—	2	24	no	—
40	#88128	C₂³	1	16	yes	—
41	#88123	—	2	24	no	—
42	#88123	C₂³	1	16	yes	complemented
43	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
44	C₂²×#492	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂³×D₁₃	—	2	12	no	—
47	C₂³×D₁₃	—	4	8	no	—
48	C₂³×D₁₃	—	4	16	no	—
49	C₂²×C₄×C₁₃	SD₁₆	1	16	yes	complemented
50	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
54	C₂²×Dic₁₃	—	2	12	no	—
55	C₂²×Dic₁₃	—	4	8	no	—
56	C₂×D₅₂	—	2	48	no	—
57	C₂×D₅₂	—	2	12	no	—
58	C₂×D₅₂	—	2	32	no	—
59	C₂×D₅₂	—	4	24	no	—
60	C₂×D₅₂	—	4	96	no	—
61	C₂×D₅₂	C₂⁴	1	8	yes	—
62	C₂×Dic₂₆	—	2	48	no	—
63	C₂×Dic₂₆	—	2	12	no	—
64	C₂×Dic₂₆	—	4	24	no	—
65	C₂×Dic₂₆	C₂⁴	1	8	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	48	yes	complemented
67	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
68	C₄²×C₁₃	C₂×D₄	1	12	yes	—
69	C₄²×C₁₃	C₂⁴	1	4	yes	—
70	C₄⋊Dic₁₃	—	2	32	no	—
71	#6095	—	2	128	no	—
72	C₂³×SD₁₆	—	13	4	no	—
73	C₂²×C₄⋊Q₈	—	13	1	no	—
74	C₂²×#127	—	13	1	no	—
75	C₂×#492	—	13	24	no	—
76	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
77	C₂³×C₁₃	#127	1	4	yes	—
78	C₂³×C₁₃	C₂×SD₁₆	1	8	yes	—
79	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
80	C₂²×D₁₃	—	2	8	no	—
81	C₂²×D₁₃	—	4	24	no	—
82	C₂²×D₁₃	—	4	48	no	—
83	C₂²×D₁₃	—	8	32	no	—
84	C₂×C₄×C₁₃	#127	1	12	yes	—
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	48	yes	complemented
86	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
87	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
89	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₁₃	—	2	8	no	—
91	C₂×Dic₁₃	—	4	24	no	—
92	D₅₂	—	2	16	no	—
93	D₅₂	—	4	24	no	—
94	D₅₂	—	4	64	no	—
95	Dic₂₆	—	2	16	no	—
96	Dic₂₆	—	4	24	no	—
97	C₈×C₁₃	C₂²×D₄	1	16	yes	complemented
98	C₂³×Q₈	—	13	2	no	—
99	C₂³×D₄	—	13	2	no	—
100	C₂³×D₄	—	26	2	no	—
101	C₂²×SD₁₆	—	13	48	no	—
102	C₂²×SD₁₆	—	26	32	no	—
103	C₂³×C₈	C₂×D₁₃	1	2	yes	—
104	C₂×C₄⋊Q₈	—	13	12	no	—
105	C₂×C₄⋊Q₈	—	26	6	no	—
106	C₂×#127	—	13	12	no	—
107	C₂×#127	—	26	6	no	—
108	C₂²×C₄⋊C₄	—	26	2	no	—
109	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
110	#492	—	13	64	no	—
111	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	complemented
112	C₂²×C₁₃	#492	1	16	yes	complemented
113	C₂²×C₁₃	C₂×#127	1	6	yes	—
114	C₂²×C₁₃	C₂²×SD₁₆	1	12	yes	—
115	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
116	C₂×D₁₃	—	4	8	no	—
117	C₂×D₁₃	—	4	16	no	—
118	C₂×D₁₃	—	8	48	no	—
119	C₄×C₁₃	C₂×#127	1	6	yes	—
120	C₄×C₁₃	C₂²×SD₁₆	1	16	yes	complemented
121	C₄×C₁₃	C₂³×D₄	1	2	yes	—
122	Dic₁₃	—	4	8	no	—
123	C₂⁵	—	26	2	no	—
124	C₂²×Q₈	—	13	12	no	—
125	C₂²×Q₈	—	26	6	no	—
126	C₂²×Q₈	—	26	16	no	—
127	C₂²×D₄	—	13	12	no	—
128	C₂²×D₄	—	26	16	no	—
129	C₂²×D₄	—	26	24	no	—
130	C₂²×D₄	—	26	6	no	—
131	C₂²×D₄	—	52	16	no	—
132	C₂³×C₄	—	26	2	no	—
133	C₂³×C₄	D₅₂	1	2	yes	—
134	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
135	C₂×SD₁₆	—	13	64	no	—
136	C₂×SD₁₆	—	26	192	no	—
137	C₂²×C₈	D₅₂	1	16	yes	complemented
138	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
139	C₄⋊Q₈	—	13	16	no	—
140	C₄⋊Q₈	—	26	24	no	—
141	#127	—	13	16	no	—
142	#127	—	26	24	no	—
143	C₂×C₄⋊C₄	—	26	24	no	—
144	C₂×C₄²	D₅₂	1	6	yes	—
145	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
146	C₄×C₈	C₂²×D₁₃	1	16	yes	complemented
147	C₂×C₁₃	C₂×#492	1	12	yes	complemented
148	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
149	C₂×C₁₃	C₂³×SD₁₆	1	2	yes	—
150	D₁₃	—	8	8	no	—
151	C₂⁴	—	26	12	no	—
152	C₂⁴	—	52	8	no	—
153	C₂⁴	—	52	16	no	—
154	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
155	C₂×Q₈	—	13	8	no	—
156	C₂×Q₈	—	26	12	no	—
157	C₂×Q₈	—	26	48	no	—
158	C₂×Q₈	—	52	24	no	—
159	C₂×D₄	—	13	8	no	—
160	C₂×D₄	—	26	48	no	—
161	C₂×D₄	—	26	32	no	—
162	C₂×D₄	—	26	12	no	—
163	C₂×D₄	—	52	24	no	—
164	C₂×D₄	—	52	96	no	—
165	C₂²×C₄	—	26	12	no	—
166	C₂²×C₄	—	52	8	no	—
167	C₂²×C₄	#6095	1	16	yes	complemented
168	C₂²×C₄	C₂×D₅₂	1	3	yes	—
169	C₂²×C₄	C₂×D₅₂	1	8	yes	—
170	C₂²×C₄	C₂×D₅₂	1	12	yes	—
171	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
172	SD₁₆	—	26	128	no	—
173	C₂×C₈	C₂×D₅₂	1	48	yes	complemented
174	C₂×C₈	C₂³×D₁₃	1	8	yes	—
175	C₄⋊C₄	—	26	32	no	—
176	C₄²	C₂×D₅₂	1	12	yes	—
177	C₄²	C₂³×D₁₃	1	4	yes	—
178	C₁₃	C₂²×#492	1	1	yes	characteristic, complemented, Sylow
179	C₂³	—	26	8	no	—
180	C₂³	—	52	24	no	—
181	C₂³	—	52	48	no	—
182	C₂³	—	104	32	no	—
183	C₂³	#88128	1	4	yes	—
184	C₂³	C₂×#6095	1	8	yes	—
185	C₂³	C₂²×D₅₂	1	3	yes	—
186	Q₈	—	26	16	no	—
187	Q₈	—	52	24	no	—
188	D₄	—	26	16	no	—
189	D₄	—	52	24	no	—
190	D₄	—	52	64	no	—
191	C₂×C₄	—	26	8	no	—
192	C₂×C₄	—	52	24	no	—
193	C₂×C₄	#88128	1	12	yes	—
194	C₂×C₄	C₂×#6095	1	48	yes	complemented
195	C₂×C₄	C₂²×D₅₂	1	3	yes	—
196	C₂×C₄	C₂²×D₅₂	1	12	yes	—
197	C₂×C₄	C₂²×D₅₂	1	8	yes	—
198	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
199	C₈	C₂²×D₅₂	1	16	yes	complemented
200	C₂²	—	52	16	no	—
201	C₂²	—	52	8	no	—
202	C₂²	—	104	48	no	—
203	C₂²	#11718354	1	16	yes	direct product factor
204	C₂²	C₂×#88128	1	6	yes	—
205	C₂²	C₂²×#6095	1	12	yes	—
206	C₂²	C₂³×D₅₂	1	1	yes	characteristic
207	C₄	—	52	8	no	—
208	C₄	C₂×#88128	1	6	yes	—
209	C₄	C₂²×#6095	1	16	yes	complemented
210	C₄	C₂³×D₅₂	1	2	yes	—
211	C₂	—	104	8	no	—
212	C₂	C₂×#11718354	1	12	yes	direct product factor
213	C₂	C₂²×#88128	1	1	yes	characteristic
214	C₂	C₂³×#6095	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂²×#88128	C₂
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#492
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#6095	C₂	semidirect
2. C₂²×#88123	C₂	semidirect
4. C₂×#11718354	C₂	direct (N × H)
5. C₂²×C₄×C₈×C₁₃	C₂	semidirect
13. C₂²×#6095	C₂²	semidirect
17. C₂×#88123	C₂²	semidirect
18. C₂×C₄×C₈×C₁₃	C₂²	semidirect
19. #11718354	C₂²	direct (N × H)
32. C₂²×C₈×C₁₃	D₄	semidirect
38. C₂×#6095	C₂³	semidirect
42. #88123	C₂³	semidirect
43. C₄×C₈×C₁₃	C₂³	semidirect
49. C₂²×C₄×C₁₃	SD₁₆	semidirect
66. C₂×C₈×C₁₃	C₂×D₄	semidirect
76. C₂²×C₄×C₈	D₁₃	semidirect
85. C₂×C₄×C₁₃	C₂×SD₁₆	semidirect
97. C₈×C₁₃	C₂²×D₄	semidirect
111. C₂×C₄×C₈	C₂×D₁₃	semidirect
112. C₂²×C₁₃	#492	semidirect
120. C₄×C₁₃	C₂²×SD₁₆	semidirect
137. C₂²×C₈	D₅₂	semidirect
146. C₄×C₈	C₂²×D₁₃	semidirect
147. C₂×C₁₃	C₂×#492	semidirect
167. C₂²×C₄	#6095	semidirect
173. C₂×C₈	C₂×D₅₂	semidirect
178. C₁₃	C₂²×#492	semidirect
194. C₂×C₄	C₂×#6095	semidirect
199. C₈	C₂²×D₅₂	semidirect
203. C₂²	#11718354	direct (N × H)
209. C₄	C₂²×#6095	semidirect
212. C₂	C₂×#11718354	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×#6095	2
C₂²×#88123	2
C₂²×#88128	2
C₂×#11718354	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#492	13

Subgroups — C22×#11718354 #16732317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718354 #16732317