Subgroups — #16807313

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#6092	C₂	1	8	yes	maximal, complemented
2	#13131193	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#6092	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	2	yes	—
7	C₂²×#6092	C₂²	1	8	yes	complemented
8	C₂×C₂²⋊#1055	C₄	1	2	yes	—
9	D₄×#1055	C₄	1	4	yes	—
10	D₄×#1055	C₂²	1	4	yes	—
11	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂²⋊#6092	C₂²	1	4	yes	—
14	C₂²⋊#6092	C₂²	1	8	yes	complemented
15	C₄⋊#6092	C₂²	1	4	yes	complemented
16	C₄×#6092	C₂²	1	4	yes	complemented
17	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₄×D₄×C₁₃	C₈	1	4	yes	—
20	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
22	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	—	4	4	no	—
25	C₂²×#1055	C₂×C₄	1	4	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂³	1	4	yes	—
29	C₂²×#1055	C₂³	1	1	yes	characteristic
30	C₂×#6092	—	4	8	no	—
31	C₂×#6092	C₂³	1	8	yes	—
32	C₂²⋊#1055	C₂×C₄	1	8	yes	—
33	C₄⋊#1055	C₂×C₄	1	4	yes	—
34	C₄×#1055	C₂×C₄	1	2	yes	—
35	C₄×#1055	C₂³	1	2	yes	—
36	#33486	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
38	C₂×D₄×C₁₃	C₁₆	1	8	yes	complemented
39	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
49	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
50	C₄²×C₁₃	C₂×C₈	1	2	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂×#1055	—	2	8	no	—
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	—	4	4	no	—
55	C₂×#1055	—	4	2	no	—
56	C₂×#1055	—	4	1	no	—
57	C₂×#1055	C₂²×C₄	1	2	yes	—
58	C₂×#1055	C₂²×C₄	1	4	yes	—
59	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
60	C₂×#1055	C₂⁴	1	1	yes	characteristic
61	#6092	—	2	8	no	—
62	C₂×C₈×D₄	—	13	1	no	—
63	D₄×C₁₆	—	13	8	no	—
64	#2162	—	13	2	no	—
65	C₂×C₂²⋊C₁₆	—	13	4	no	—
66	C₂³×C₁₃	—	4	2	no	—
67	C₂³×C₁₃	—	4	2	no	—
68	C₂³×C₁₃	C₂×C₁₆	1	8	yes	complemented
69	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
70	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	D₄×C₁₃	C₂×C₁₆	1	16	yes	complemented
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	#1055	—	2	1	no	—
86	#1055	—	2	4	no	—
87	#1055	—	2	2	no	—
88	#1055	H(5,𝔽₂)	1	2	yes	—
89	C₂³×C₈	—	13	2	no	—
90	C₂×C₄×D₄	#261	1	1	yes	characteristic
91	C₂²×C₁₆	—	13	8	no	—
92	C₈×D₄	—	13	4	no	—
93	C₈×D₄	—	13	4	no	—
94	C₂×C₄⋊C₈	—	13	1	no	—
95	C₂×C₂²⋊C₈	—	13	2	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₄⋊C₁₆	—	13	4	no	—
98	C₂²⋊C₁₆	—	13	4	no	—
99	C₂²⋊C₁₆	—	13	8	no	—
100	C₄×C₁₆	—	13	4	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	—	4	8	no	—
106	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
107	C₂²×C₁₃	C₂²×C₁₆	1	8	yes	—
108	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	C₂²×C₁₆	1	4	yes	—
113	C₄×C₁₃	#518	1	1	yes	characteristic
114	C₄×C₁₃	#568	1	1	yes	characteristic
115	C₂²×D₄	#1057	1	1	yes	characteristic
116	C₂³×C₄	C₂×#261	1	2	yes	—
117	C₂²×C₈	—	13	1	no	—
118	C₂²×C₈	—	13	4	no	—
119	C₂²×C₈	—	13	2	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₂²×C₈	—	13	1	no	—
122	C₂²×C₈	—	52	4	no	—
123	C₄×D₄	#1057	1	4	yes	—
124	C₄×D₄	C₂×#261	1	4	yes	—
125	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	#1057	1	2	yes	—
127	C₂×C₄²	C₂×#261	1	1	yes	characteristic
128	C₂×C₁₆	—	13	8	no	—
129	C₂×C₁₆	—	52	8	no	—
130	C₄⋊C₈	—	13	4	no	—
131	C₂²⋊C₈	—	13	8	no	—
132	C₄×C₈	—	13	2	no	—
133	C₄×C₈	—	13	2	no	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	—	2	8	no	—
136	C₂×C₁₃	#2972	1	1	yes	characteristic
137	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
138	C₂×C₁₃	#3407	1	1	yes	characteristic
139	C₂⁴	C₂×#1057	1	2	yes	—
140	C₂×D₄	#6092	1	8	yes	complemented
141	C₂×D₄	C₂×#1057	1	4	yes	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
145	C₂²×C₄	C₂×#1057	1	2	yes	—
146	C₂²×C₄	C₂×#1057	1	4	yes	—
147	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
148	C₂²×C₄	C₂²×#261	1	4	yes	—
149	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
150	C₂×C₈	—	13	1	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	4	no	—
154	C₂×C₈	—	26	8	no	—
155	C₂×C₈	—	26	2	no	—
156	C₂×C₈	—	52	2	no	—
157	C₂×C₈	—	52	4	no	—
158	C₂×C₈	—	52	1	no	—
159	C₄⋊C₄	C₂×#1057	1	4	yes	—
160	C₂²⋊C₄	C₂×#1057	1	8	yes	—
161	C₄²	C₂×#1057	1	2	yes	—
162	C₄²	C₂²×#261	1	2	yes	—
163	C₁₆	—	26	8	no	—
164	C₁₃	#33486	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	4	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂×#6092	1	8	yes	complemented
168	C₂³	C₂²×#1057	1	4	yes	—
169	C₂³	C₂²×#1057	1	1	yes	characteristic
170	D₄	C₂×#6092	1	16	yes	complemented
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×#6092	1	4	yes	complemented
179	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
180	C₂×C₄	C₂²×#1057	1	2	yes	—
181	C₂×C₄	C₂²×#1057	1	4	yes	—
182	C₂×C₄	C₂²×#1057	1	2	yes	—
183	C₂×C₄	C₂³×#261	1	1	yes	characteristic
184	C₈	—	13	2	no	—
185	C₈	—	26	2	no	—
186	C₈	—	26	4	no	—
187	C₈	—	26	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	C₂²×#6092	1	2	yes	—
194	C₂²	C₂²×#6092	1	8	yes	—
195	C₂²	C₂³×#1057	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂²×#6092	1	4	yes	—
200	C₄	#11719621	1	1	yes	characteristic
201	C₄	#11719623	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	2	8	no	—
204	C₂	#13131683	1	1	yes	characteristic
205	C₂	C₂³×#6092	1	1	yes	characteristic
206	C₂	#13131803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Fitting	C₂×C₄×D₄×C₁₃	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂²×D₄×C₁₃	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×D₄	#261
hypocenter	C₁₃	#33486
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. D₄×#6092	C₂	semidirect
2. #13131193	C₂	semidirect
3. C₂×C₂²⋊#6092	C₂	semidirect
7. C₂²×#6092	C₂²	semidirect
14. C₂²⋊#6092	C₂²	semidirect
15. C₄⋊#6092	C₂²	semidirect
16. C₄×#6092	C₂²	semidirect
38. C₂×D₄×C₁₃	C₁₆	semidirect
68. C₂³×C₁₃	C₂×C₁₆	semidirect
71. D₄×C₁₃	C₂×C₁₆	semidirect
79. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
140. C₂×D₄	#6092	semidirect
164. C₁₃	#33486	semidirect
167. C₂³	C₂×#6092	semidirect
170. D₄	C₂×#6092	semidirect
178. C₂×C₄	C₂×#6092	semidirect

Maximal subgroups

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

subgroup	index
D₄×#6092	2
#13131193	2
C₂×C₂²⋊#6092	2
C₂×D₄×#1055	2
#33486	13