Subgroups — C₂×#13123921 #16727863

Order 3328.

1978 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123921	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11718685	C₂	1	8	yes	maximal, complemented
3	C₂×#11718587	C₂	1	4	yes	maximal
4	C₂×#11718586	C₂	1	2	yes	maximal
5	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₄	1	4	yes	—
8	C₂×C₄×Q₈×C₁₃	C₄	1	8	yes	—
9	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
16	#11718685	C₂²	1	64	yes	complemented
17	#11718587	C₂²	1	32	yes	—
18	#11718586	C₂²	1	16	yes	—
19	C₂×C₄⋊#1055	C₂²	1	4	yes	—
20	C₂×C₄⋊#1055	C₂²	1	8	yes	—
21	C₂×#88043	C₂²	1	8	yes	—
22	C₂×C₄×#1055	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	—	2	8	no	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	C₂×C₄	1	32	yes	—
32	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
33	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
46	C₂²×#1055	C₂³	1	8	yes	—
47	C₂²⋊#1055	C₂³	1	16	yes	—
48	C₄⋊#1055	C₂³	1	16	yes	—
49	C₄⋊#1055	C₂³	1	32	yes	—
50	#88043	C₂³	1	32	yes	—
51	C₄×#1055	C₂³	1	16	yes	—
52	C₂×#2983	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Q₈×C₁₃	—	2	32	no	—
55	C₂×Q₈×C₁₃	—	2	16	no	—
56	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
75	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₁₃	C₂⁴	1	4	yes	—
81	C₄²×C₁₃	C₂⁴	1	8	yes	—
82	C₂×#1055	—	2	16	no	—
83	C₂×#1055	C₂⁴	1	16	yes	—
84	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
85	#2983	—	13	16	no	—
86	C₂×#446	—	13	8	no	—
87	C₂×#432	—	13	4	no	—
88	C₂×#431	—	13	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
93	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₁₃	—	2	32	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	16	no	—
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	8	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	4	8	no	—
104	C₂×C₄×C₁₃	C₂×M₁₆	1	16	yes	—
105	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
112	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
113	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
114	#1055	—	2	16	no	—
115	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
116	C₂×C₄×Q₈	Dic₁₃	1	4	yes	—
117	C₂×C₄×Q₈	Dic₁₃	1	8	yes	—
118	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
119	C₂×C₄×Q₈	C₂×D₁₃	1	8	yes	—
120	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
121	C₂²×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
122	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
124	#446	—	13	64	no	—
125	#432	—	13	32	no	—
126	#431	—	13	16	no	—
127	C₂×C₄⋊C₈	—	13	4	no	—
128	C₂×C₄⋊C₈	—	13	8	no	—
129	C₂×C₂²⋊C₈	—	13	4	no	—
130	C₂×#97	—	13	8	no	—
131	C₂×C₄×C₈	—	13	4	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	8	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	#519	1	2	yes	—
136	C₂²×C₁₃	C₂²×M₁₆	1	4	yes	—
137	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
138	C₂²×C₁₃	#568	1	2	yes	—
139	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	8	no	—
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	—	2	8	no	—
144	C₄×C₁₃	#519	1	2	yes	—
145	C₄×C₁₃	C₂²×M₁₆	1	8	yes	—
146	C₄×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
147	C₂²×Q₈	—	2	8	no	—
148	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
149	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
150	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
151	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
152	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
153	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
155	C₂²×C₈	—	13	8	no	—
156	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
157	C₄×Q₈	C₂×Dic₁₃	1	32	yes	—
158	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
161	C₂×C₄⋊C₄	C₂×Dic₁₃	1	16	yes	—
162	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
163	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
164	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
165	C₂×C₄²	C₂×Dic₁₃	1	8	yes	—
166	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
167	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
168	C₂×C₄²	C₂²×D₁₃	1	8	yes	—
169	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
170	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
171	C₄⋊C₈	—	13	16	no	—
172	C₄⋊C₈	—	13	32	no	—
173	C₂²⋊C₈	—	13	16	no	—
174	#97	—	13	32	no	—
175	C₄×C₈	—	13	16	no	—
176	C₂×C₁₃	—	2	4	no	—
177	C₂×C₁₃	#2983	1	4	yes	complemented
178	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
179	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#568	1	1	yes	characteristic
181	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
182	C₂×Q₈	—	2	32	no	—
183	C₂×Q₈	—	2	16	no	—
184	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	#6099	1	8	yes	—
190	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
195	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
196	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
200	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
201	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
202	C₂×C₈	—	13	16	no	—
203	C₂×C₈	—	26	16	no	—
204	C₄⋊C₄	C₂²×Dic₁₃	1	32	yes	—
205	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
206	C₄²	C₂²×Dic₁₃	1	16	yes	—
207	C₄²	C₂²×Dic₁₃	1	8	yes	—
208	C₄²	C₂²×Dic₁₃	1	8	yes	—
209	C₄²	C₂²×Dic₁₃	1	4	yes	—
210	C₄²	C₂³×D₁₃	1	4	yes	—
211	C₄²	C₂³×D₁₃	1	8	yes	—
212	C₁₃	C₂×#2983	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×#6099	1	4	yes	—
216	C₂³	C₂³×Dic₁₃	1	2	yes	—
217	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
218	Q₈	—	2	32	no	—
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	C₂×#6099	1	16	yes	—
229	C₂×C₄	C₂×#6099	1	8	yes	—
230	C₂×C₄	#88254	1	4	yes	—
231	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
232	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
233	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
234	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
235	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
236	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
238	C₈	—	26	16	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₂²×#6099	1	4	yes	—
243	C₂²	C₂²×#6099	1	2	yes	—
244	C₂²	#11719517	1	2	yes	—
245	C₂²	#11719524	1	2	yes	—
246	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
247	C₄	—	2	8	no	—
248	C₄	—	2	4	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	2	no	—
251	C₄	C₂²×#6099	1	8	yes	—
252	C₄	#11719517	1	2	yes	—
253	C₄	C₂×#88254	1	2	yes	—
254	C₂	—	2	4	no	—
255	C₂	#13123921	1	4	yes	direct product factor
256	C₂	C₂×#11719517	1	1	yes	characteristic
257	C₂	C₂³×#6099	1	1	yes	characteristic
258	C₂	C₂×#11719524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#2983
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13123921	C₂	direct (N × H)
2. C₂×#11718685	C₂	semidirect
16. #11718685	C₂²	semidirect
177. C₂×C₁₃	#2983	semidirect
212. C₁₃	C₂×#2983	semidirect
255. C₂	#13123921	direct (N × H)

Maximal subgroups

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

subgroup	index
#13123921	2
C₂×#11718685	2
C₂×#11718587	2
C₂×#11718586	2
C₂²×C₄×Q₈×C₁₃	2
C₂×#2983	13

Subgroups — C2×#13123921 #16727863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123921 #16727863