Subgroups — C₂×#13116939 #16017365

Order 3328.

786 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718776	C₂	1	2	yes	maximal, complemented
3	#13116939	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718097	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718101	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88184	—	2	2	no	—
9	C₂²×D₁₀₄	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	—	2	4	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	4	yes	complemented
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718776	—	2	4	no	—
15	#11718776	C₂²	1	8	yes	complemented
16	C₂×#88061	—	2	2	no	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×D₂₀₈	—	2	4	no	—
19	C₂×D₂₀₈	C₂²	1	4	yes	complemented
20	#11718101	C₂²	1	8	yes	complemented
21	#11718097	C₂²	1	4	yes	complemented
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₂²⋊D₁₃	—	4	2	no	—
24	C₂²×D₅₂	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	#88184	—	2	8	no	—
32	#88184	—	4	4	no	—
33	C₂×#6103	—	4	2	no	—
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₂×D₁₀₄	—	2	4	no	—
36	C₂×D₁₀₄	—	2	4	no	—
37	C₂×D₁₀₄	—	2	2	no	—
38	C₂×D₁₀₄	C₂³	1	4	yes	—
39	C₁₃×M₃₂	—	2	8	no	—
40	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
42	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	#88061	—	2	8	no	—
45	#88058	—	2	2	no	—
46	#88058	C₂³	1	4	yes	—
47	D₂₀₈	—	2	8	no	—
48	C₂×#2218	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
50	C₂³×D₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	D₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₃	—	4	8	no	—
58	C₂×C₂²⋊D₁₃	—	8	8	no	—
59	C₂²×Dic₁₃	—	4	2	no	—
60	C₂×D₅₂	—	2	4	no	—
61	C₂×D₅₂	—	4	4	no	—
62	C₂×D₅₂	—	4	4	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	#6103	—	4	8	no	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	D₁₀₄	—	2	4	no	—
77	D₁₀₄	—	2	4	no	—
78	D₁₀₄	—	4	4	no	—
79	C₁₃×C₁₆	—	2	2	no	—
80	C₁₃×C₁₆	C₂×D₄	1	4	yes	complemented
81	C₂²×D₁₆	—	13	1	no	—
82	C₂²×M₃₂	D₁₃	1	1	yes	characteristic, complemented
83	C₂×#466	—	13	2	no	—
84	#2218	—	13	8	no	—
85	C₂×#367	—	13	1	no	—
86	C₂×#357	—	13	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	C₂×D₈	1	2	yes	—
90	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₁₃	—	4	4	no	—
92	C₂²×D₁₃	—	8	4	no	—
93	C₂²×D₁₃	—	8	4	no	—
94	C₂²×D₁₃	—	8	4	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	#121	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#135	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₁₃	—	8	16	no	—
105	C₂×Dic₁₃	—	4	4	no	—
106	C₂×Dic₁₃	—	8	4	no	—
107	D₅₂	—	4	4	no	—
108	D₅₂	—	4	4	no	—
109	D₅₂	—	8	4	no	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	#121	1	2	yes	—
112	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂²×D₈	—	13	2	no	—
115	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
116	C₂×#121	—	26	2	no	—
117	C₂×D₁₆	—	13	4	no	—
118	C₂×D₁₆	—	26	4	no	—
119	C₂×M₃₂	—	2	4	no	—
120	C₂×M₃₂	C₂×D₁₃	1	4	yes	complemented
121	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
123	#466	—	13	8	no	—
124	#466	—	26	4	no	—
125	C₂×#107	—	13	2	no	—
126	C₂×#102	—	26	2	no	—
127	#367	—	13	4	no	—
128	#357	—	13	8	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	#466	1	2	yes	—
133	C₂²×C₁₃	#509	1	4	yes	—
134	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
135	C₂×D₁₃	—	8	4	no	—
136	C₂×D₁₃	—	8	4	no	—
137	C₂×D₁₃	—	8	4	no	—
138	C₂×D₁₃	—	16	8	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	#466	1	2	yes	—
141	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
142	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
143	Dic₁₃	—	8	4	no	—
144	C₂²×D₄	—	26	2	no	—
145	C₂²×D₄	—	52	2	no	—
146	C₂³×C₄	D₅₂	1	1	yes	characteristic
147	C₂×D₈	—	13	4	no	—
148	C₂×D₈	—	26	2	no	—
149	C₂×D₈	—	26	4	no	—
150	C₂×D₈	—	26	4	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	D₅₂	1	1	yes	characteristic
154	C₂²×C₈	D₅₂	1	2	yes	—
155	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
156	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
157	#121	—	26	8	no	—
158	#121	—	52	4	no	—
159	C₂×C₄⋊C₄	—	26	2	no	—
160	C₂×C₂²⋊C₄	—	52	2	no	—
161	D₁₆	—	26	8	no	—
162	M₃₂	—	2	8	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	complemented
165	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
166	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
167	#107	—	13	4	no	—
168	#107	—	26	2	no	—
169	#102	—	26	8	no	—
170	C₂×C₁₃	—	4	2	no	—
171	C₂×C₁₃	#2218	1	4	yes	complemented
172	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
173	C₂×C₁₃	C₂×#509	1	2	yes	—
174	D₁₃	—	16	4	no	—
175	C₂⁴	—	52	2	no	—
176	C₂⁴	D₁₀₄	1	1	yes	characteristic
177	C₂×D₄	—	26	4	no	—
178	C₂×D₄	—	52	8	no	—
179	C₂×D₄	—	52	2	no	—
180	C₂×D₄	—	52	4	no	—
181	C₂×D₄	—	52	4	no	—
182	C₂×D₄	—	104	8	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	52	2	no	—
186	C₂²×C₄	D₁₀₄	1	2	yes	—
187	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₅₂	1	2	yes	—
189	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
190	D₈	—	26	4	no	—
191	D₈	—	26	4	no	—
192	D₈	—	52	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	C₂×D₅₂	1	2	yes	—
198	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
199	C₂×C₈	#6127	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
201	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
202	C₄⋊C₄	—	26	4	no	—
203	C₄⋊C₄	—	52	2	no	—
204	C₂²⋊C₄	—	52	8	no	—
205	C₁₆	—	2	2	no	—
206	C₁₆	C₂×C₂²⋊D₁₃	1	4	yes	complemented
207	C₁₃	C₂×#2218	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	52	4	no	—
211	C₂³	—	104	4	no	—
212	C₂³	—	104	4	no	—
213	C₂³	—	104	4	no	—
214	C₂³	C₂×D₁₀₄	1	2	yes	—
215	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
216	D₄	—	52	4	no	—
217	D₄	—	52	4	no	—
218	D₄	—	104	16	no	—
219	D₄	—	104	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	52	4	no	—
225	C₂×C₄	—	104	4	no	—
226	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
227	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
228	C₂×C₄	#88158	1	2	yes	—
229	C₂×C₄	#88184	1	2	yes	—
230	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	#88184	1	2	yes	—
233	C₈	C₂×#6127	1	1	yes	characteristic
234	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	104	4	no	—
239	C₂²	—	104	4	no	—
240	C₂²	—	104	4	no	—
241	C₂²	—	208	8	no	—
242	C₂²	#11718569	1	4	yes	—
243	C₂²	#11718776	1	2	yes	—
244	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	104	4	no	—
247	C₄	#11718776	1	2	yes	—
248	C₄	C₂×#88158	1	1	yes	characteristic
249	C₄	C₂×#88184	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	208	4	no	—
252	C₂	#13116939	1	4	yes	direct product factor
253	C₂	C₂×#11718776	1	1	yes	characteristic
254	C₂	C₂×#11718569	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂×#2218
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₀₈	C₂	semidirect
2. C₂×#11718776	C₂	semidirect
3. #13116939	C₂	direct (N × H)
4. C₂×#11718097	C₂	semidirect
5. C₂×#11718101	C₂	semidirect
6. C₂²×C₁₃×M₃₂	C₂	semidirect
11. C₂×C₁₃×M₃₂	C₂²	semidirect
15. #11718776	C₂²	semidirect
19. C₂×D₂₀₈	C₂²	semidirect
20. #11718101	C₂²	semidirect
21. #11718097	C₂²	semidirect
41. C₂×C₁₃×C₁₆	D₄	semidirect
80. C₁₃×C₁₆	C₂×D₄	semidirect
82. C₂²×M₃₂	D₁₃	semidirect
120. C₂×M₃₂	C₂×D₁₃	semidirect
164. C₂×C₁₆	C₂²⋊D₁₃	semidirect
171. C₂×C₁₃	#2218	semidirect
206. C₁₆	C₂×C₂²⋊D₁₃	semidirect
207. C₁₃	C₂×#2218	semidirect
252. C₂	#13116939	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×D₂₀₈	2
C₂×#11718776	2
#13116939	2
C₂×#11718097	2
C₂×#11718101	2
C₂²×C₁₃×M₃₂	2
C₂×#2218	13

Subgroups — C2×#13116939 #16017365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116939 #16017365