Subgroups — C₂×#13123232 #16069015

Order 3328.

3774 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88198	C₂	1	4	yes	maximal
3	#13123232	C₂	1	16	yes	maximal, direct product factor
4	C₂×#11718637	C₂	1	8	yes	maximal
5	C₂×C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×Dic₂₆	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊Q₈	—	2	8	no	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	8	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88198	—	2	32	no	—
14	C₂×#88198	C₂²	1	32	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
16	C₂²×#6101	C₂²	1	16	yes	—
17	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
18	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
19	C₂×#88123	—	2	8	no	—
20	C₂×#88123	C₂²	1	8	yes	—
21	#11718637	C₂²	1	64	yes	—
22	C₄²×Dic₁₃	C₂²	1	8	yes	—
23	C₂²×Q₈×C₁₃	—	2	16	no	—
24	C₂²×Q₈×C₁₃	—	2	8	no	—
25	C₂²×Q₈×C₁₃	C₂³	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂³×Dic₁₃	C₂³	1	4	yes	—
30	C₂³×Dic₁₃	C₂³	1	4	yes	—
31	C₂²×Dic₂₆	—	2	8	no	—
32	C₂²×Dic₂₆	—	2	16	no	—
33	C₂²×Dic₂₆	C₂³	1	8	yes	—
34	C₁₃×C₄⋊Q₈	—	2	32	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
37	C₂×C₄²×C₁₃	D₄	1	8	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
39	#88198	—	2	128	no	—
40	C₂×C₄⋊Dic₁₃	—	2	16	no	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
42	C₂×#6101	—	2	64	no	—
43	C₂×#6101	C₂³	1	64	yes	—
44	C₂×C₄×Dic₁₃	D₄	1	16	yes	—
45	C₂×C₄×Dic₁₃	Q₈	1	32	yes	complemented
46	C₂×C₄×Dic₁₃	C₂³	1	8	yes	—
47	C₂×C₄×Dic₁₃	C₂³	1	16	yes	—
48	#88123	—	2	32	no	—
49	C₂×#2868	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×Q₈×C₁₃	—	2	32	no	—
52	C₂×Q₈×C₁₃	—	2	16	no	—
53	C₂×Q₈×C₁₃	—	4	32	no	—
54	C₂²×C₄×C₁₃	—	2	16	no	—
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
63	C₂²×Dic₁₃	—	2	16	no	—
64	C₂²×Dic₁₃	—	2	8	no	—
65	C₂²×Dic₁₃	C₂×D₄	1	16	yes	—
66	C₂²×Dic₁₃	C₂×Q₈	1	32	yes	—
67	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
68	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
69	C₂×Dic₂₆	—	2	16	no	—
70	C₂×Dic₂₆	—	2	32	no	—
71	C₂×Dic₂₆	—	4	32	no	—
72	C₁₃×C₄⋊C₄	—	2	32	no	—
73	C₄²×C₁₃	C₂×D₄	1	16	yes	—
74	C₄⋊Dic₁₃	—	2	32	no	—
75	#6101	—	2	128	no	—
76	C₄×Dic₁₃	C₂×D₄	1	32	yes	—
77	C₄×Dic₁₃	C₂×Q₈	1	64	yes	complemented
78	C₂²×C₄⋊Q₈	—	13	1	no	—
79	C₂²×C₄⋊Q₈	—	13	4	no	—
80	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
81	#2868	—	13	16	no	—
82	C₂×#384	—	13	8	no	—
83	C₂×C₄³	—	13	1	no	—
84	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
85	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
86	C₂³×C₁₃	C₂²×Q₈	1	8	yes	—
87	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
88	Q₈×C₁₃	—	4	32	no	—
89	C₂×C₄×C₁₃	—	2	16	no	—
90	C₂×C₄×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	4	16	no	—
92	C₂×C₄×C₁₃	#127	1	16	yes	—
93	C₂×C₄×C₁₃	C₄⋊Q₈	1	32	yes	complemented
94	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
95	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
97	C₂×C₄×C₁₃	C₂²×Q₈	1	16	yes	—
98	C₂×Dic₁₃	—	2	16	no	—
99	C₂×Dic₁₃	—	2	8	no	—
100	C₂×Dic₁₃	—	4	16	no	—
101	C₂×Dic₁₃	C₄⋊Q₈	1	64	yes	complemented
102	C₂×Dic₁₃	C₂²×D₄	1	16	yes	—
103	C₂×Dic₁₃	C₂²×Q₈	1	32	yes	—
104	Dic₂₆	—	4	32	no	—
105	C₂³×Q₈	—	13	2	no	—
106	C₂³×Q₈	C₂×D₁₃	1	2	yes	—
107	C₂×C₄⋊Q₈	—	2	8	no	—
108	C₂×C₄⋊Q₈	—	13	32	no	—
109	C₂×C₄⋊Q₈	—	13	8	no	—
110	C₂×C₄⋊Q₈	—	26	32	no	—
111	C₂×C₄⋊Q₈	—	26	8	no	—
112	C₂×C₄⋊Q₈	C₂×D₁₃	1	8	yes	—
113	C₂²×C₄⋊C₄	—	13	16	no	—
114	C₂²×C₄⋊C₄	—	13	4	no	—
115	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
116	C₂²×C₄²	—	13	2	no	—
117	C₂²×C₄²	—	13	4	no	—
118	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
119	#384	—	13	64	no	—
120	C₄³	—	13	8	no	—
121	C₂²×C₁₃	C₂×#127	1	4	yes	—
122	C₂²×C₁₃	C₂×C₄⋊Q₈	1	16	yes	—
123	C₂²×C₁₃	C₂×C₄⋊Q₈	1	8	yes	—
124	C₂²×C₁₃	C₂³×D₄	1	2	yes	—
125	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
126	C₂²×C₁₃	C₂³×Q₈	1	4	yes	—
127	C₄×C₁₃	—	4	8	no	—
128	C₄×C₁₃	C₂×#127	1	8	yes	—
129	C₄×C₁₃	C₂×C₄⋊Q₈	1	16	yes	complemented
130	Dic₁₃	—	4	8	no	—
131	Dic₁₃	C₂×C₄⋊Q₈	1	32	yes	complemented
132	C₂²×Q₈	—	2	16	no	—
133	C₂²×Q₈	—	2	8	no	—
134	C₂²×Q₈	—	13	8	no	—
135	C₂²×Q₈	—	26	16	no	—
136	C₂²×Q₈	—	26	8	no	—
137	C₂²×Q₈	C₂²×D₁₃	1	8	yes	—
138	C₂³×C₄	—	13	4	no	—
139	C₂³×C₄	—	13	4	no	—
140	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
141	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
143	C₄⋊Q₈	—	2	32	no	—
144	C₄⋊Q₈	—	26	128	no	—
145	C₄⋊Q₈	—	26	32	no	—
146	C₂×C₄⋊C₄	—	2	16	no	—
147	C₂×C₄⋊C₄	—	13	64	no	—
148	C₂×C₄⋊C₄	—	13	16	no	—
149	C₂×C₄⋊C₄	—	26	64	no	—
150	C₂×C₄⋊C₄	—	26	16	no	—
151	C₂×C₄⋊C₄	C₂²×D₁₃	1	16	yes	—
152	C₂×C₄²	—	13	16	no	—
153	C₂×C₄²	—	13	32	no	—
154	C₂×C₄²	—	13	8	no	—
155	C₂×C₄²	—	13	16	no	—
156	C₂×C₄²	C₂²⋊D₁₃	1	8	yes	—
157	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
158	C₂×C₁₃	#2868	1	8	yes	complemented
159	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
160	C₂×C₁₃	C₂²×C₄⋊Q₈	1	4	yes	—
161	C₂×C₁₃	C₂²×C₄⋊Q₈	1	2	yes	—
162	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
163	C₂×Q₈	—	2	32	no	—
164	C₂×Q₈	—	2	16	no	—
165	C₂×Q₈	—	4	32	no	—
166	C₂×Q₈	—	26	32	no	—
167	C₂×Q₈	—	26	16	no	—
168	C₂×Q₈	—	52	32	no	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	13	16	no	—
172	C₂²×C₄	—	13	32	no	—
173	C₂²×C₄	—	13	8	no	—
174	C₂²×C₄	—	13	8	no	—
175	C₂²×C₄	—	26	16	no	—
176	C₂²×C₄	—	26	8	no	—
177	C₂²×C₄	D₄×D₁₃	1	8	yes	—
178	C₂²×C₄	Q₈×D₁₃	1	16	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₁₃	1	8	yes	—
181	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
182	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
183	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
184	C₄⋊C₄	—	2	32	no	—
185	C₄⋊C₄	—	26	128	no	—
186	C₄⋊C₄	—	26	32	no	—
187	C₄²	—	13	32	no	—
188	C₄²	—	13	64	no	—
189	C₄²	C₂×C₂²⋊D₁₃	1	16	yes	—
190	C₁₃	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
191	C₂³	C₂×D₄×D₁₃	1	4	yes	—
192	C₂³	C₂×Q₈×D₁₃	1	8	yes	—
193	C₂³	C₂²×C₂²⋊D₁₃	1	2	yes	—
194	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
195	Q₈	—	4	32	no	—
196	Q₈	—	52	32	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	16	no	—
200	C₂×C₄	—	13	64	no	—
201	C₂×C₄	—	13	16	no	—
202	C₂×C₄	—	13	32	no	—
203	C₂×C₄	—	26	16	no	—
204	C₂×C₄	—	26	8	no	—
205	C₂×C₄	—	52	16	no	—
206	C₂×C₄	#88194	1	16	yes	—
207	C₂×C₄	#88198	1	32	yes	complemented
208	C₂×C₄	C₂×D₄×D₁₃	1	8	yes	—
209	C₂×C₄	C₂×Q₈×D₁₃	1	16	yes	—
210	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
211	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
212	C₂²	D₁₃×C₄⋊Q₈	1	16	yes	—
213	C₂²	C₂×#88194	1	4	yes	—
214	C₂²	C₂×#88198	1	8	yes	—
215	C₂²	C₂²×D₄×D₁₃	1	2	yes	—
216	C₂²	C₂²×Q₈×D₁₃	1	4	yes	—
217	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
218	C₄	—	4	8	no	—
219	C₄	—	13	32	no	—
220	C₄	—	52	8	no	—
221	C₄	C₂×#88194	1	8	yes	—
222	C₄	C₂×#88198	1	16	yes	complemented
223	C₂	#13123232	1	8	yes	direct product factor
224	C₂	C₂²×#88194	1	1	yes	characteristic
225	C₂	C₂×D₁₃×C₄⋊Q₈	1	4	yes	—
226	C₂	C₂²×#88198	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂×#2868
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #13123232	C₂	direct (N × H)
45. C₂×C₄×Dic₁₃	Q₈	semidirect
77. C₄×Dic₁₃	C₂×Q₈	semidirect
93. C₂×C₄×C₁₃	C₄⋊Q₈	semidirect
101. C₂×Dic₁₃	C₄⋊Q₈	semidirect
129. C₄×C₁₃	C₂×C₄⋊Q₈	semidirect
131. Dic₁₃	C₂×C₄⋊Q₈	semidirect
158. C₂×C₁₃	#2868	semidirect
190. C₁₃	C₂×#2868	semidirect
207. C₂×C₄	#88198	semidirect
222. C₄	C₂×#88198	semidirect
223. C₂	#13123232	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×#88123	2
C₂²×#88198	2
#13123232	2
C₂×#11718637	2
C₂×C₄²×Dic₁₃	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂×#2868	13

Subgroups — C2×#13123232 #16069015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123232 #16069015