Subgroups — C₂×#13127811 #16743813

Order 3328.

2582 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719282	C₂	1	2	yes	maximal, complemented
2	#13127811	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718356	C₂	1	3	yes	maximal, complemented
4	C₂×#11718504	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₁₀₄	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6131	—	2	2	no	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×D₁₀₄	C₂²	1	3	yes	—
10	C₂²×D₁₀₄	C₂²	1	6	yes	—
11	#11719282	—	2	8	no	—
12	#11719282	C₂²	1	16	yes	complemented
13	C₂×#88149	—	2	6	no	—
14	C₂×#88128	C₂²	1	6	yes	—
15	C₂×C₄×D₅₂	C₂²	1	6	yes	—
16	C₈×Q₈×C₁₃	C₂²	1	8	yes	complemented
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
19	C₂×#88061	C₂²	1	6	yes	—
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	#11718504	C₂²	1	48	yes	complemented
22	#11718356	C₂²	1	24	yes	complemented
23	C₄×D₁₀₄	C₂²	1	24	yes	complemented
24	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂×#6131	—	2	16	no	—
26	C₂×#6131	—	4	16	no	—
27	C₂²×D₅₂	—	2	6	no	—
28	C₂²×D₅₂	—	2	6	no	—
29	C₂²×D₅₂	C₂³	1	6	yes	—
30	C₂²×C₄×D₁₃	—	2	6	no	—
31	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₁₃	D₄	1	4	yes	—
34	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
37	C₂×#6103	—	2	6	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
39	C₂×D₁₀₄	—	2	12	no	—
40	C₂×D₁₀₄	—	4	12	no	—
41	C₂×D₁₀₄	C₂³	1	12	yes	—
42	C₂×D₁₀₄	C₂³	1	24	yes	—
43	#88149	—	2	48	no	—
44	#88128	—	2	12	no	—
45	#88128	C₂³	1	24	yes	—
46	C₄×D₅₂	—	2	12	no	—
47	C₄×D₅₂	C₂³	1	24	yes	—
48	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
49	C₄×C₈×C₁₃	C₂³	1	12	yes	—
50	#88061	C₂³	1	24	yes	—
51	#88058	C₂³	1	4	yes	—
52	C₂×#3391	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	—	2	6	no	—
54	C₂×Q₈×C₁₃	D₈	1	8	yes	complemented
55	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	2	no	—
60	#6131	—	4	64	no	—
61	C₂×D₅₂	—	2	12	no	—
62	C₂×D₅₂	—	2	12	no	—
63	C₂×D₅₂	—	2	6	no	—
64	C₂×D₅₂	—	2	24	no	—
65	C₂×D₅₂	—	2	24	no	—
66	C₂×D₅₂	—	4	24	no	—
67	C₂×D₅₂	—	8	12	no	—
68	C₂×D₅₂	C₂⁴	1	12	yes	—
69	C₂×C₄×D₁₃	—	2	24	no	—
70	C₂×C₄×D₁₃	—	4	24	no	—
71	C₂×C₈×C₁₃	—	2	6	no	—
72	C₂×C₈×C₁₃	#41	1	4	yes	complemented
73	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
75	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
76	C₄²×C₁₃	C₂×D₄	1	6	yes	—
77	C₄²×C₁₃	C₂⁴	1	6	yes	—
78	#6103	—	2	24	no	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
81	D₁₀₄	—	2	24	no	—
82	D₁₀₄	—	4	24	no	—
83	C₂×#550	—	13	2	no	—
84	#3391	—	13	16	no	—
85	C₂×#493	—	13	3	no	—
86	C₂×#459	—	13	6	no	—
87	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic, complemented
88	C₂×C₄×D₈	—	13	3	no	—
89	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×D₁₃	—	2	12	no	—
91	C₂²×D₁₃	—	4	12	no	—
92	C₂²×D₁₃	—	4	12	no	—
93	C₂²×D₁₃	—	4	12	no	—
94	Q₈×C₁₃	C₂×D₈	1	16	yes	complemented
95	C₂×C₄×C₁₃	—	2	6	no	—
96	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
97	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
98	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
101	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
102	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
103	C₂×Dic₁₃	—	2	4	no	—
104	C₂×Dic₁₃	—	4	4	no	—
105	D₅₂	—	2	12	no	—
106	D₅₂	—	2	12	no	—
107	D₅₂	—	4	12	no	—
108	D₅₂	—	4	48	no	—
109	D₅₂	—	8	24	no	—
110	C₄×D₁₃	—	4	48	no	—
111	C₈×C₁₃	—	2	6	no	—
112	C₈×C₁₃	C₂×#41	1	4	yes	complemented
113	C₂²×#41	—	26	2	no	—
114	C₂²×D₈	—	13	3	no	—
115	C₂²×D₈	—	13	6	no	—
116	#550	—	13	16	no	—
117	#550	—	26	8	no	—
118	C₂×#127	—	13	6	no	—
119	C₂×#121	—	26	6	no	—
120	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
121	C₂×C₄×D₄	—	13	6	no	—
122	#493	—	13	24	no	—
123	#459	—	13	48	no	—
124	C₈×Q₈	C₂×D₁₃	1	8	yes	complemented
125	C₄×D₈	—	13	24	no	—
126	C₂×#107	—	13	1	no	—
127	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
128	C₂×#102	—	13	6	no	—
129	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
130	C₂²×C₁₃	#550	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	2	yes	—
132	C₂²×C₁₃	#576	1	2	yes	—
133	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
134	C₂×D₁₃	—	4	12	no	—
135	C₂×D₁₃	—	4	12	no	—
136	C₂×D₁₃	—	4	12	no	—
137	C₂×D₁₃	—	8	24	no	—
138	C₄×C₁₃	—	2	6	no	—
139	C₄×C₁₃	#550	1	2	yes	—
140	C₄×C₁₃	C₂²×D₈	1	12	yes	—
141	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
142	C₄×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
143	Dic₁₃	—	4	4	no	—
144	C₂×#41	—	26	16	no	—
145	C₂×#41	—	52	16	no	—
146	C₂²×Q₈	D₅₂	1	1	yes	characteristic
147	C₂²×D₄	—	13	6	no	—
148	C₂²×D₄	—	26	6	no	—
149	C₂²×D₄	—	26	6	no	—
150	C₂³×C₄	—	26	6	no	—
151	C₂×D₈	—	13	12	no	—
152	C₂×D₈	—	13	24	no	—
153	C₂×D₈	—	26	12	no	—
154	C₂×D₈	—	52	12	no	—
155	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
156	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
157	#127	—	13	24	no	—
158	#127	—	26	12	no	—
159	#121	—	26	48	no	—
160	C₄×Q₈	D₅₂	1	4	yes	—
161	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
162	C₄×D₄	—	13	24	no	—
163	C₄×D₄	—	26	12	no	—
164	C₂×C₄⋊C₄	—	13	2	no	—
165	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
166	C₂×C₂²⋊C₄	—	26	6	no	—
167	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
168	#107	—	13	4	no	—
169	C₄⋊C₈	C₂²×D₁₃	1	12	yes	—
170	#102	—	13	24	no	—
171	C₄×C₈	C₂²×D₁₃	1	12	yes	—
172	C₂×C₁₃	#3391	1	4	yes	complemented
173	C₂×C₁₃	C₂×#550	1	1	yes	characteristic
174	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
175	C₂×C₁₃	C₂×#576	1	1	yes	characteristic
176	D₁₃	—	8	12	no	—
177	C₂⁴	—	26	6	no	—
178	#41	—	52	64	no	—
179	C₂×Q₈	D₁₀₄	1	8	yes	complemented
180	C₂×Q₈	C₂×D₅₂	1	4	yes	—
181	C₂×D₄	—	13	12	no	—
182	C₂×D₄	—	26	6	no	—
183	C₂×D₄	—	26	24	no	—
184	C₂×D₄	—	26	24	no	—
185	C₂×D₄	—	26	12	no	—
186	C₂×D₄	—	26	12	no	—
187	C₂×D₄	—	52	24	no	—
188	C₂×D₄	—	104	12	no	—
189	C₂²×C₄	—	26	2	no	—
190	C₂²×C₄	—	26	24	no	—
191	C₂²×C₄	—	52	24	no	—
192	C₂²×C₄	C₂×D₅₂	1	3	yes	—
193	C₂²×C₄	C₂×D₅₂	1	3	yes	—
194	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
195	D₈	—	26	24	no	—
196	D₈	—	52	24	no	—
197	C₂×C₈	—	2	6	no	—
198	C₂×C₈	#6131	1	4	yes	complemented
199	C₂×C₈	C₂³×D₁₃	1	6	yes	—
200	C₂×C₈	C₂³×D₁₃	1	2	yes	—
201	C₄⋊C₄	—	13	4	no	—
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	C₂×D₅₂	1	12	yes	—
204	C₂²⋊C₄	—	26	24	no	—
205	C₄²	C₂×D₅₂	1	6	yes	—
206	C₄²	C₂³×D₁₃	1	6	yes	—
207	C₁₃	C₂×#3391	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	26	12	no	—
209	C₂³	—	52	12	no	—
210	C₂³	—	52	12	no	—
211	C₂³	—	52	12	no	—
212	C₂³	C₂²×D₅₂	1	1	yes	characteristic
213	Q₈	C₂×D₁₀₄	1	16	yes	complemented
214	D₄	—	26	12	no	—
215	D₄	—	26	12	no	—
216	D₄	—	52	48	no	—
217	D₄	—	52	12	no	—
218	D₄	—	104	24	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	26	4	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	—	52	48	no	—
223	C₂×C₄	C₂×D₁₀₄	1	12	yes	—
224	C₂×C₄	C₂²×D₅₂	1	6	yes	—
225	C₂×C₄	C₂²×D₅₂	1	6	yes	—
226	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
227	C₂×C₄	C₂×#6131	1	2	yes	—
228	C₂×C₄	#88256	1	2	yes	—
229	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
230	C₈	—	2	6	no	—
231	C₈	C₂×#6131	1	4	yes	complemented
232	C₂²	—	52	12	no	—
233	C₂²	—	52	12	no	—
234	C₂²	—	52	12	no	—
235	C₂²	—	104	24	no	—
236	C₂²	#11719282	1	2	yes	—
237	C₂²	C₂²×D₁₀₄	1	2	yes	—
238	C₂²	#11719457	1	2	yes	—
239	C₂²	C₂³×D₅₂	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	52	4	no	—
242	C₄	#11719282	1	2	yes	—
243	C₄	C₂²×D₁₀₄	1	12	yes	—
244	C₄	C₂²×#6131	1	1	yes	characteristic
245	C₄	C₂×#88256	1	1	yes	characteristic
246	C₂	—	104	12	no	—
247	C₂	#13127811	1	4	yes	direct product factor
248	C₂	C₂×#11719282	1	1	yes	characteristic
249	C₂	C₂³×D₁₀₄	1	1	yes	characteristic
250	C₂	C₂×#11719457	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×#3391
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11719282	C₂	semidirect
2. #13127811	C₂	direct (N × H)
3. C₂×#11718356	C₂	semidirect
4. C₂×#11718504	C₂	semidirect
5. C₂×C₄×D₁₀₄	C₂	semidirect
6. C₂×C₈×Q₈×C₁₃	C₂	semidirect
12. #11719282	C₂²	semidirect
16. C₈×Q₈×C₁₃	C₂²	semidirect
21. #11718504	C₂²	semidirect
22. #11718356	C₂²	semidirect
23. C₄×D₁₀₄	C₂²	semidirect
54. C₂×Q₈×C₁₃	D₈	semidirect
72. C₂×C₈×C₁₃	#41	semidirect
87. C₂×C₈×Q₈	D₁₃	semidirect
94. Q₈×C₁₃	C₂×D₈	semidirect
112. C₈×C₁₃	C₂×#41	semidirect
124. C₈×Q₈	C₂×D₁₃	semidirect
172. C₂×C₁₃	#3391	semidirect
179. C₂×Q₈	D₁₀₄	semidirect
198. C₂×C₈	#6131	semidirect
207. C₁₃	C₂×#3391	semidirect
213. Q₈	C₂×D₁₀₄	semidirect
231. C₈	C₂×#6131	semidirect
247. C₂	#13127811	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719282	2
#13127811	2
C₂×#11718356	2
C₂×#11718504	2
C₂×C₄×D₁₀₄	2
C₂×C₈×Q₈×C₁₃	2
C₂×#3391	13

Subgroups — C2×#13127811 #16743813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127811 #16743813