Subgroups — C₂×#13116944 #16017371

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 240 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	1	yes	maximal, characteristic, complemented
2	C₂×#11718777	C₂	1	2	yes	maximal
3	#13116944	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718097	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718095	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88178	—	2	2	no	—
9	C₂²×Dic₅₂	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	—	2	4	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718777	—	2	4	no	—
15	#11718777	C₂²	1	8	yes	—
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×#88056	—	2	2	no	—
18	C₂×Dic₁₀₄	—	2	4	no	—
19	C₂×Dic₁₀₄	C₂²	1	4	yes	complemented
20	#11718097	C₂²	1	4	yes	complemented
21	#11718095	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×Dic₂₆	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₁₃	—	4	2	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	#88178	—	2	8	no	—
33	#88178	—	4	4	no	—
34	C₂×#6101	—	4	2	no	—
35	C₂×Dic₅₂	—	2	4	no	—
36	C₂×Dic₅₂	—	2	4	no	—
37	C₂×Dic₅₂	—	2	2	no	—
38	C₂×Dic₅₂	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	—
42	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	#88058	—	2	2	no	—
45	#88058	C₂³	1	4	yes	—
46	#88056	—	2	8	no	—
47	Dic₁₀₄	—	2	8	no	—
48	C₂×#2219	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	D₈	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	4	2	no	—
57	C₂²×Dic₁₃	—	4	2	no	—
58	C₂×Dic₂₆	—	2	4	no	—
59	C₂×Dic₂₆	—	4	4	no	—
60	C₂×Dic₂₆	—	4	4	no	—
61	C₂×Dic₂₆	—	4	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	#41	1	2	yes	—
70	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊Dic₁₃	—	4	8	no	—
72	C₄⋊Dic₁₃	—	2	4	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	#6101	—	4	8	no	—
75	Dic₅₂	—	2	4	no	—
76	Dic₅₂	—	2	4	no	—
77	Dic₅₂	—	4	4	no	—
78	C₁₃×C₁₆	—	2	2	no	—
79	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
80	C₂²×Q₃₂	—	13	1	no	—
81	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
82	C₂×#467	—	13	2	no	—
83	#2219	—	13	8	no	—
84	C₂×#367	—	13	1	no	—
85	C₂×#358	—	13	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	4	no	—
88	C₂³×C₁₃	C₂×D₈	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	#121	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#135	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₁₃	—	4	4	no	—
100	C₂×Dic₁₃	—	4	4	no	—
101	C₂×Dic₁₃	—	8	4	no	—
102	C₂×Dic₁₃	—	8	4	no	—
103	Dic₂₆	—	4	4	no	—
104	Dic₂₆	—	4	4	no	—
105	Dic₂₆	—	8	4	no	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	#121	1	2	yes	—
108	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂²×Q₁₆	—	13	2	no	—
111	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
112	C₂×C₂²⋊Q₈	—	26	2	no	—
113	C₂×Q₃₂	—	13	4	no	—
114	C₂×Q₃₂	—	26	4	no	—
115	C₂×M₃₂	—	2	4	no	—
116	C₂×M₃₂	C₂×D₁₃	1	4	yes	—
117	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
119	#467	—	13	8	no	—
120	#467	—	26	4	no	—
121	C₂×#107	—	13	2	no	—
122	C₂×#103	—	26	2	no	—
123	#367	—	13	4	no	—
124	#358	—	13	8	no	—
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	4	4	no	—
128	C₂²×C₁₃	#466	1	2	yes	—
129	C₂²×C₁₃	#510	1	4	yes	—
130	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	#466	1	2	yes	—
133	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
134	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
135	Dic₁₃	—	8	4	no	—
136	Dic₁₃	—	8	4	no	—
137	C₂²×Q₈	—	26	2	no	—
138	C₂³×C₄	D₅₂	1	1	yes	characteristic
139	C₂×Q₁₆	—	13	4	no	—
140	C₂×Q₁₆	—	26	2	no	—
141	C₂×Q₁₆	—	26	4	no	—
142	C₂×Q₁₆	—	26	4	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	D₅₂	1	1	yes	characteristic
146	C₂²×C₈	D₅₂	1	2	yes	—
147	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
148	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
149	C₂²⋊Q₈	—	26	8	no	—
150	C₂²⋊Q₈	—	52	4	no	—
151	C₂×C₄⋊C₄	—	26	2	no	—
152	C₂×C₄⋊C₄	—	52	2	no	—
153	C₂×C₂²⋊C₄	—	52	2	no	—
154	Q₃₂	—	26	8	no	—
155	M₃₂	—	2	8	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	—
158	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
159	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
160	#107	—	13	4	no	—
161	#107	—	26	2	no	—
162	#103	—	26	8	no	—
163	C₂×C₁₃	—	4	2	no	—
164	C₂×C₁₃	#2219	1	4	yes	complemented
165	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
166	C₂×C₁₃	C₂×#510	1	2	yes	—
167	C₂⁴	D₁₀₄	1	1	yes	characteristic
168	C₂×Q₈	—	26	4	no	—
169	C₂×Q₈	—	52	2	no	—
170	C₂×Q₈	—	52	4	no	—
171	C₂×Q₈	—	52	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	52	2	no	—
175	C₂²×C₄	—	52	2	no	—
176	C₂²×C₄	D₁₀₄	1	2	yes	—
177	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₅₂	1	2	yes	—
179	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
180	Q₁₆	—	26	4	no	—
181	Q₁₆	—	26	4	no	—
182	Q₁₆	—	52	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂×D₅₂	1	2	yes	—
188	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
189	C₂×C₈	#6127	1	2	yes	—
190	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
191	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	26	4	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₄⋊C₄	—	52	8	no	—
195	C₂²⋊C₄	—	52	8	no	—
196	C₁₆	—	2	2	no	—
197	C₁₆	C₂×C₂²⋊D₁₃	1	4	yes	—
198	C₁₃	C₂×#2219	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₂×D₁₀₄	1	2	yes	—
202	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
203	Q₈	—	52	4	no	—
204	Q₈	—	52	4	no	—
205	Q₈	—	104	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	52	4	no	—
211	C₂×C₄	—	52	4	no	—
212	C₂×C₄	—	104	4	no	—
213	C₂×C₄	—	104	4	no	—
214	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
215	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
216	C₂×C₄	#88158	1	2	yes	—
217	C₂×C₄	#88184	1	2	yes	—
218	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	#88184	1	2	yes	—
221	C₈	C₂×#6127	1	1	yes	characteristic
222	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#11718570	1	4	yes	—
227	C₂²	#11718776	1	2	yes	—
228	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	104	4	no	—
231	C₄	—	104	4	no	—
232	C₄	#11718776	1	2	yes	—
233	C₄	C₂×#88158	1	1	yes	characteristic
234	C₄	C₂×#88184	1	1	yes	characteristic
235	C₂	—	4	2	no	—
236	C₂	#13116944	1	4	yes	direct product factor
237	C₂	C₂×#11718776	1	1	yes	characteristic
238	C₂	C₂×#11718570	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
prime radical	C₂⁴×C₁₃	D₈
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂×#2219
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×Dic₁₀₄	C₂	semidirect
3. #13116944	C₂	direct (N × H)
4. C₂×#11718097	C₂	semidirect
5. C₂×#11718095	C₂	semidirect
19. C₂×Dic₁₀₄	C₂²	semidirect
20. #11718097	C₂²	semidirect
21. #11718095	C₂²	semidirect
164. C₂×C₁₃	#2219	semidirect
198. C₁₃	C₂×#2219	semidirect
236. C₂	#13116944	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×#11718777	2
#13116944	2
C₂×#11718097	2
C₂×#11718095	2
C₂²×C₁₃×M₃₂	2
C₂×#2219	13

Subgroups — C2×#13116944 #16017371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116944 #16017371