Subgroups — C₂×#13116887 #16017170

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₄	C₂	1	2	yes	maximal
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
4	#13116887	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718095	C₂	1	2	yes	maximal
6	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₂	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	2	yes	—
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂×#88123	—	2	1	no	—
11	C₂×C₄×Dic₂₆	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88058	C₂²	1	1	yes	characteristic
15	C₂×#88056	—	2	1	no	—
16	C₂×Dic₁₀₄	—	2	8	no	—
17	C₂×Dic₁₀₄	C₂²	1	8	yes	—
18	#11718360	—	2	2	no	—
19	#11718360	C₂²	1	4	yes	—
20	C₄×Dic₅₂	—	2	2	no	—
21	C₄×Dic₅₂	C₂²	1	4	yes	—
22	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
23	#11718095	C₂²	1	8	yes	—
24	C₂²×Dic₂₆	—	2	1	no	—
25	C₂²×Dic₂₆	—	2	2	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	4	1	no	—
31	C₂×C₄⋊Dic₁₃	—	4	1	no	—
32	C₂×#6101	—	4	1	no	—
33	C₂×C₄×Dic₁₃	—	4	1	no	—
34	C₂×Dic₅₂	—	2	1	no	—
35	C₂×Dic₅₂	—	2	2	no	—
36	C₂×Dic₅₂	—	2	4	no	—
37	C₂×Dic₅₂	—	2	4	no	—
38	C₂×Dic₅₂	—	2	4	no	—
39	C₂×Dic₅₂	—	2	2	no	—
40	C₂×Dic₅₂	—	4	4	no	—
41	C₂×Dic₅₂	D₄	1	4	yes	—
42	C₂×Dic₅₂	C₂³	1	4	yes	—
43	C₂×Dic₅₂	C₂³	1	2	yes	—
44	#88123	—	2	4	no	—
45	#88123	—	4	2	no	—
46	C₄×Dic₂₆	—	2	4	no	—
47	C₄×Dic₂₆	—	4	2	no	—
48	C₂×C₁₃×C₁₆	—	2	4	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
50	C₄×C₈×C₁₃	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	#88058	—	2	1	no	—
53	#88058	C₂³	1	2	yes	—
54	#88056	—	2	4	no	—
55	Dic₁₀₄	—	2	16	no	—
56	C₂×#2209	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	4	1	no	—
61	C₂²×Dic₁₃	—	4	1	no	—
62	C₂²×Dic₁₃	—	4	2	no	—
63	C₂×Dic₂₆	—	2	4	no	—
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	—	2	4	no	—
66	C₂×Dic₂₆	—	4	1	no	—
67	C₂×Dic₂₆	—	4	2	no	—
68	C₂×Dic₂₆	—	4	4	no	—
69	C₂×Dic₂₆	—	4	4	no	—
70	C₂×Dic₂₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₄²×C₁₃	D₈	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	—	4	1	no	—
81	C₄⋊Dic₁₃	—	4	4	no	—
82	C₄⋊Dic₁₃	—	4	4	no	—
83	#6101	—	4	4	no	—
84	C₄×Dic₁₃	—	4	4	no	—
85	Dic₅₂	—	2	4	no	—
86	Dic₅₂	—	2	4	no	—
87	Dic₅₂	—	2	2	no	—
88	Dic₅₂	—	2	4	no	—
89	Dic₅₂	—	4	8	no	—
90	Dic₅₂	—	4	4	no	—
91	Dic₅₂	C₂×D₄	1	4	yes	—
92	C₁₃×C₁₆	—	2	4	no	—
93	C₂²×Q₃₂	—	13	2	no	—
94	C₂×#494	—	13	1	no	—
95	C₂×C₄×Q₁₆	—	13	1	no	—
96	#2209	—	13	8	no	—
97	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic
98	C₂×#358	—	13	2	no	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	Q₃₂	1	4	yes	complemented
102	C₂×C₄×C₁₃	#121	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	4	no	—
112	C₂×Dic₁₃	—	8	4	no	—
113	C₂×Dic₁₃	—	8	2	no	—
114	Dic₂₆	—	2	4	no	—
115	Dic₂₆	—	4	2	no	—
116	Dic₂₆	—	4	4	no	—
117	Dic₂₆	—	4	4	no	—
118	Dic₂₆	—	4	4	no	—
119	Dic₂₆	—	8	4	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	#121	1	2	yes	—
122	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂²×Q₁₆	—	13	1	no	—
125	C₂²×Q₁₆	—	13	2	no	—
126	C₂²×Q₁₆	—	26	1	no	—
127	C₂×C₄⋊Q₈	—	26	1	no	—
128	C₂×C₄×Q₈	—	26	1	no	—
129	C₂×Q₃₂	—	13	8	no	—
130	C₂×Q₃₂	—	26	8	no	—
131	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
132	#494	—	13	4	no	—
133	#494	—	26	2	no	—
134	C₄×Q₁₆	—	13	4	no	—
135	C₄×Q₁₆	—	26	2	no	—
136	C₂×#107	—	13	1	no	—
137	C₂×#103	—	26	1	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	—
140	#358	—	13	8	no	—
141	C₂²×C₁₃	#459	1	2	yes	—
142	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₁₃	#510	1	2	yes	—
144	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	#459	1	2	yes	—
147	C₄×C₁₃	C₂×Q₃₂	1	4	yes	complemented
148	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	Dic₁₃	—	4	4	no	—
151	Dic₁₃	—	8	2	no	—
152	Dic₁₃	—	8	4	no	—
153	C₂²×Q₈	—	26	1	no	—
154	C₂²×Q₈	—	26	2	no	—
155	C₂×Q₁₆	—	13	4	no	—
156	C₂×Q₁₆	—	13	2	no	—
157	C₂×Q₁₆	—	13	4	no	—
158	C₂×Q₁₆	—	26	1	no	—
159	C₂×Q₁₆	—	26	2	no	—
160	C₂×Q₁₆	—	26	4	no	—
161	C₂×Q₁₆	—	26	4	no	—
162	C₂×Q₁₆	—	26	2	no	—
163	C₂×Q₁₆	—	26	4	no	—
164	C₂×Q₁₆	—	52	4	no	—
165	C₂²×C₈	D₅₂	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
167	C₄⋊Q₈	—	26	4	no	—
168	C₄⋊Q₈	—	52	2	no	—
169	C₄×Q₈	—	26	4	no	—
170	C₄×Q₈	—	52	2	no	—
171	C₂×C₄⋊C₄	—	26	1	no	—
172	C₂×C₄⋊C₄	—	52	1	no	—
173	C₂×C₄⋊C₄	—	52	1	no	—
174	C₂×C₄⋊C₄	—	52	1	no	—
175	C₂×C₄²	—	52	1	no	—
176	C₂×C₄²	D₅₂	1	1	yes	characteristic
177	Q₃₂	—	26	16	no	—
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
180	#107	—	13	2	no	—
181	#107	—	26	1	no	—
182	#103	—	26	4	no	—
183	C₄×C₈	D₅₂	1	2	yes	—
184	C₄×C₈	C₂²×D₁₃	1	2	yes	—
185	C₂×C₁₃	#2209	1	4	yes	complemented
186	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
187	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
189	C₂×Q₈	—	26	4	no	—
190	C₂×Q₈	—	26	2	no	—
191	C₂×Q₈	—	26	4	no	—
192	C₂×Q₈	—	52	1	no	—
193	C₂×Q₈	—	52	2	no	—
194	C₂×Q₈	—	52	2	no	—
195	C₂×Q₈	—	52	4	no	—
196	C₂×Q₈	—	52	4	no	—
197	C₂²×C₄	—	52	1	no	—
198	C₂²×C₄	—	52	1	no	—
199	C₂²×C₄	—	52	2	no	—
200	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
201	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
203	Q₁₆	—	13	4	no	—
204	Q₁₆	—	26	4	no	—
205	Q₁₆	—	26	4	no	—
206	Q₁₆	—	26	2	no	—
207	Q₁₆	—	26	4	no	—
208	Q₁₆	—	52	8	no	—
209	Q₁₆	—	52	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₅₂	1	2	yes	—
212	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
213	C₂×C₈	D₄×D₁₃	1	2	yes	—
214	C₂×C₈	#6131	1	2	yes	—
215	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	26	2	no	—
217	C₄⋊C₄	—	52	1	no	—
218	C₄⋊C₄	—	52	4	no	—
219	C₄⋊C₄	—	52	4	no	—
220	C₄⋊C₄	—	52	4	no	—
221	C₄²	—	52	4	no	—
222	C₄²	D₁₀₄	1	2	yes	—
223	C₄²	C₂×D₅₂	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₃	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
226	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
227	Q₈	—	26	4	no	—
228	Q₈	—	52	2	no	—
229	Q₈	—	52	4	no	—
230	Q₈	—	52	4	no	—
231	Q₈	—	52	4	no	—
232	Q₈	—	104	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	52	2	no	—
235	C₂×C₄	—	52	4	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	52	4	no	—
238	C₂×C₄	—	104	4	no	—
239	C₂×C₄	—	104	2	no	—
240	C₂×C₄	Dic₁₀₄	1	4	yes	complemented
241	C₂×C₄	#88149	1	2	yes	—
242	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
243	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
244	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
245	C₂×C₄	#88162	1	2	yes	—
246	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#88149	1	2	yes	—
249	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
250	C₈	C₂×#6131	1	1	yes	characteristic
251	C₂²	#11718504	1	2	yes	—
252	C₂²	C₂×Dic₁₀₄	1	2	yes	—
253	C₂²	#11718570	1	2	yes	—
254	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	52	4	no	—
257	C₄	—	104	2	no	—
258	C₄	—	104	4	no	—
259	C₄	#11718504	1	2	yes	—
260	C₄	C₂×Dic₁₀₄	1	4	yes	complemented
261	C₄	C₂×#88149	1	1	yes	characteristic
262	C₄	C₂×#88162	1	1	yes	characteristic
263	C₂	#13116887	1	4	yes	direct product factor
264	C₂	C₂×#11718504	1	1	yes	characteristic
265	C₂	C₂²×Dic₁₀₄	1	1	yes	characteristic
266	C₂	C₂×#11718570	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2209
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #13116887	C₂	direct (N × H)
101. C₂×C₄×C₁₃	Q₃₂	semidirect
147. C₄×C₁₃	C₂×Q₃₂	semidirect
185. C₂×C₁₃	#2209	semidirect
225. C₁₃	C₂×#2209	semidirect
240. C₂×C₄	Dic₁₀₄	semidirect
260. C₄	C₂×Dic₁₀₄	semidirect
263. C₂	#13116887	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×Dic₁₀₄	2
C₂×#11718360	2
C₂×C₄×Dic₅₂	2
#13116887	2
C₂×#11718095	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×#2209	13

Subgroups — C2×#13116887 #16017170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116887 #16017170