Subgroups — C₂×#13131168 #16806991

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13131168	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#6092	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	—	2	8	no	—
9	C₂×#88106	C₂²	1	8	yes	—
10	C₂²×#6092	C₂²	1	4	yes	—
11	C₂×C₄×#1055	C₄	1	4	yes	—
12	C₂×C₄×#1055	C₄	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	4	yes	—
15	C₄⋊#6092	C₂²	1	16	yes	complemented
16	C₂³×C₄×C₁₃	C₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₈	1	4	yes	—
20	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×#1055	—	2	4	no	—
24	C₂²×#1055	C₂×C₄	1	4	yes	—
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	D₄	1	4	yes	—
29	C₂²×#1055	Q₈	1	4	yes	—
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	2	yes	—
32	#88106	—	2	16	no	—
33	C₂×#6092	—	2	8	no	—
34	C₂×#6092	C₂³	1	8	yes	—
35	C₄×#1055	C₂×C₄	1	4	yes	—
36	C₄×#1055	C₂×C₄	1	8	yes	—
37	C₄×#1055	C₂³	1	4	yes	—
38	C₂×#2150	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	—	2	8	no	—
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×C₈	1	4	yes	—
51	C₄²×C₁₃	C₂×C₈	1	4	yes	—
52	C₄²×C₁₃	C₂×C₈	1	4	yes	—
53	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
54	C₂×#1055	—	2	8	no	—
55	C₂×#1055	—	2	4	no	—
56	C₂×#1055	C₄⋊C₄	1	8	yes	—
57	C₂×#1055	C₄⋊C₄	1	4	yes	—
58	C₂×#1055	C₂²×C₄	1	4	yes	—
59	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
60	C₂×#1055	C₂²×C₄	1	2	yes	—
61	C₂×#1055	C₂×D₄	1	2	yes	—
62	C₂×#1055	C₂×D₄	1	4	yes	—
63	C₂×#1055	C₂×Q₈	1	2	yes	—
64	C₂×#1055	C₂×Q₈	1	4	yes	—
65	C₂×#1055	C₂⁴	1	1	yes	characteristic
66	#6092	—	2	8	no	—
67	C₂²×M₃₂	—	13	2	no	—
68	C₂²×C₄×C₈	—	13	1	no	—
69	#2150	—	13	8	no	—
70	C₂×C₄⋊C₁₆	—	13	4	no	—
71	C₂³×C₁₃	—	2	4	no	—
72	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
73	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
74	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	16	no	—
77	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
78	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
79	C₂×C₄×C₁₃	M₃₂	1	8	yes	complemented
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	#1055	—	2	4	no	—
90	#1055	C₂×C₄⋊C₄	1	2	yes	—
91	#1055	C₂×C₄⋊C₄	1	4	yes	—
92	#1055	C₂²×D₄	1	1	yes	characteristic
93	#1055	C₂²×Q₈	1	1	yes	characteristic
94	C₂³×C₈	—	13	1	no	—
95	C₂³×C₈	—	13	1	no	—
96	C₂²×C₄²	#261	1	1	yes	characteristic
97	C₂×M₃₂	—	13	8	no	—
98	C₂×M₃₂	—	26	8	no	—
99	C₂²×C₁₆	—	13	4	no	—
100	C₂×C₄×C₈	—	13	2	no	—
101	C₂×C₄×C₈	—	13	4	no	—
102	C₂×C₄×C₈	—	13	4	no	—
103	C₂×C₄×C₈	—	13	2	no	—
104	C₄⋊C₁₆	—	13	16	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	8	no	—
107	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
108	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
109	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
113	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
114	C₄×C₁₃	C₂×M₃₂	1	8	yes	complemented
115	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₂³×C₄	#1057	1	2	yes	—
118	C₂³×C₄	C₂×#261	1	1	yes	characteristic
119	C₂²×C₈	—	13	4	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₂²×C₈	—	13	2	no	—
122	C₂²×C₈	—	13	2	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	—	13	1	no	—
125	C₂²×C₈	—	13	4	no	—
126	C₂²×C₈	—	13	2	no	—
127	C₂²×C₈	—	26	4	no	—
128	C₂×C₄²	#1057	1	1	yes	characteristic
129	C₂×C₄²	#1057	1	4	yes	—
130	C₂×C₄²	#1057	1	1	yes	characteristic
131	C₂×C₄²	C₂×#261	1	4	yes	—
132	C₂×C₄²	C₂×#261	1	2	yes	—
133	M₃₂	—	26	16	no	—
134	C₂×C₁₆	—	13	8	no	—
135	C₂×C₁₆	—	26	8	no	—
136	C₄×C₈	—	13	4	no	—
137	C₄×C₈	—	13	4	no	—
138	C₄×C₈	—	13	8	no	—
139	C₂×C₁₃	—	2	4	no	—
140	C₂×C₁₃	#2150	1	4	yes	complemented
141	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
143	C₂⁴	C₂×#1057	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	#6118	1	4	yes	—
146	C₂²×C₄	C₄⋊#261	1	4	yes	—
147	C₂²×C₄	C₂×#1057	1	2	yes	—
148	C₂²×C₄	C₂×#1057	1	8	yes	—
149	C₂²×C₄	C₂×#1057	1	2	yes	—
150	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
151	C₂²×C₄	C₂×#1057	1	2	yes	—
152	C₂²×C₄	C₂²×#261	1	2	yes	—
153	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	13	4	no	—
156	C₂×C₈	—	13	8	no	—
157	C₂×C₈	—	13	4	no	—
158	C₂×C₈	—	13	4	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	13	1	no	—
161	C₂×C₈	—	13	1	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	26	8	no	—
165	C₂×C₈	—	26	4	no	—
166	C₄²	C₂×#1057	1	4	yes	—
167	C₄²	C₂×#1057	1	4	yes	—
168	C₄²	C₂×#1057	1	4	yes	—
169	C₄²	C₂²×#261	1	4	yes	—
170	C₁₆	—	26	8	no	—
171	C₁₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
172	C₂³	—	2	4	no	—
173	C₂³	C₄⋊#1057	1	4	yes	—
174	C₂³	C₂²×#1057	1	2	yes	—
175	C₂³	C₂²×#1057	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	#88106	1	8	yes	complemented
179	C₂×C₄	C₄⋊#1057	1	8	yes	—
180	C₂×C₄	C₄⋊#1057	1	4	yes	—
181	C₂×C₄	C₂×#6118	1	4	yes	—
182	C₂×C₄	C₂×#6118	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
185	C₂×C₄	C₂²×#1057	1	4	yes	—
186	C₂×C₄	C₂²×#1057	1	4	yes	—
187	C₂×C₄	C₂²×#1057	1	2	yes	—
188	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
189	C₂×C₄	C₂³×#261	1	1	yes	characteristic
190	C₈	—	13	1	no	—
191	C₈	—	13	2	no	—
192	C₈	—	13	4	no	—
193	C₈	—	13	1	no	—
194	C₈	—	26	4	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	C₂×#88106	1	4	yes	—
198	C₂²	C₂×C₄⋊#1057	1	4	yes	—
199	C₂²	C₂×C₄⋊#1057	1	2	yes	—
200	C₂²	C₂³×#1057	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	C₂×#88106	1	8	yes	complemented
203	C₄	C₂×C₄⋊#1057	1	4	yes	—
204	C₄	C₂×C₄⋊#1057	1	2	yes	—
205	C₄	C₂²×#6118	1	1	yes	characteristic
206	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#13131168	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊#1057	1	1	yes	characteristic
210	C₂	C₂²×#88106	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	C₂×#2150
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #13131168	C₂	direct (N × H)
3. C₂×C₄⋊#6092	C₂	semidirect
15. C₄⋊#6092	C₂²	semidirect
79. C₂×C₄×C₁₃	M₃₂	semidirect
114. C₄×C₁₃	C₂×M₃₂	semidirect
140. C₂×C₁₃	#2150	semidirect
171. C₁₃	C₂×#2150	semidirect
178. C₂×C₄	#88106	semidirect
202. C₄	C₂×#88106	semidirect
208. C₂	#13131168	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×#88106	2
#13131168	2
C₂×C₄⋊#6092	2
C₂²×C₄×#1055	2
C₂×#2150	13

Subgroups — C2×#13131168 #16806991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131168 #16806991