Subgroups — C₂×#13117097 #16018318

Order 3328.

854 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₂²×#88040	C₂	1	2	yes	maximal, complemented
3	C₂²×D₂₀₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718357	C₂	1	2	yes	maximal, complemented
5	#13117097	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₅₂	C₂²	1	2	yes	—
8	C₂²×D₁₀₄	C₂²	1	2	yes	—
9	C₂²×#6095	—	2	2	no	—
10	C₂×#88129	—	2	2	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₄	—	2	4	no	—
15	C₂×Dic₁₀₄	C₂²	1	4	yes	complemented
16	C₂×#88040	—	2	8	no	—
17	C₂×#88040	C₂²	1	8	yes	complemented
18	C₂×D₂₀₈	—	2	4	no	—
19	C₂×D₂₀₈	C₂²	1	4	yes	—
20	#11718357	—	2	4	no	—
21	#11718357	C₂²	1	8	yes	complemented
22	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
23	C₂²×D₅₂	—	2	2	no	—
24	C₂²×Dic₂₆	—	2	2	no	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×#6103	—	4	4	no	—
29	C₂×Dic₅₂	—	2	4	no	—
30	C₂×Dic₅₂	—	2	4	no	—
31	C₂×Dic₅₂	—	2	2	no	—
32	C₂×Dic₅₂	C₂³	1	4	yes	—
33	C₂×D₁₀₄	—	2	4	no	—
34	C₂×D₁₀₄	—	2	4	no	—
35	C₂×D₁₀₄	—	2	2	no	—
36	C₂×D₁₀₄	C₂³	1	4	yes	—
37	C₂×#6095	—	2	8	no	—
38	C₂×#6095	—	4	8	no	—
39	#88129	—	2	8	no	—
40	#88129	—	4	4	no	—
41	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
42	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂³	1	2	yes	—
47	Dic₁₀₄	—	2	8	no	—
48	#88040	—	2	16	no	—
49	D₂₀₈	—	2	8	no	—
50	C₂×#2249	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	D₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	4	2	no	—
55	C₂×D₅₂	—	2	4	no	—
56	C₂×D₅₂	—	4	4	no	—
57	C₂×D₅₂	—	4	4	no	—
58	C₂×D₅₂	—	4	2	no	—
59	C₂×Dic₂₆	—	2	4	no	—
60	C₂×Dic₂₆	—	4	4	no	—
61	C₂×Dic₂₆	—	4	4	no	—
62	C₂×Dic₂₆	—	4	2	no	—
63	C₂×C₈×C₁₃	D₈	1	4	yes	—
64	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₄²×C₁₃	D₈	1	2	yes	—
70	C₄²×C₁₃	C₂×D₄	1	2	yes	—
71	#6103	—	4	16	no	—
72	Dic₅₂	—	2	4	no	—
73	Dic₅₂	—	2	4	no	—
74	Dic₅₂	—	4	4	no	—
75	D₁₀₄	—	2	4	no	—
76	D₁₀₄	—	2	4	no	—
77	D₁₀₄	—	4	4	no	—
78	#6095	—	4	16	no	—
79	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
80	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
81	C₂²×Q₃₂	—	13	1	no	—
82	C₂²×SD₃₂	—	13	2	no	—
83	C₂²×D₁₆	—	13	1	no	—
84	C₂×#495	—	13	2	no	—
85	#2249	—	13	8	no	—
86	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
87	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₁₃	—	4	4	no	—
89	C₂²×D₁₃	—	8	4	no	—
90	C₂²×D₁₃	—	8	8	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	#127	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₁₃	—	4	4	no	—
98	C₂×Dic₁₃	—	8	4	no	—
99	D₅₂	—	4	4	no	—
100	D₅₂	—	4	4	no	—
101	D₅₂	—	8	4	no	—
102	Dic₂₆	—	4	4	no	—
103	Dic₂₆	—	4	4	no	—
104	Dic₂₆	—	8	4	no	—
105	C₈×C₁₃	#127	1	2	yes	—
106	C₈×C₁₃	C₂×D₈	1	4	yes	—
107	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂²×Q₁₆	—	13	2	no	—
110	C₂²×SD₁₆	—	26	2	no	—
111	C₂²×D₈	—	13	2	no	—
112	C₂×#124	—	26	2	no	—
113	C₂×Q₃₂	—	13	4	no	—
114	C₂×Q₃₂	—	26	4	no	—
115	C₂×SD₃₂	—	13	8	no	—
116	C₂×SD₃₂	—	26	8	no	—
117	C₂×D₁₆	—	13	4	no	—
118	C₂×D₁₆	—	26	4	no	—
119	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
121	#495	—	13	8	no	—
122	#495	—	26	4	no	—
123	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
124	C₄×C₁₆	C₂×D₁₃	1	4	yes	complemented
125	C₂²×C₁₃	#493	1	2	yes	—
126	C₂²×C₁₃	#508	1	4	yes	—
127	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₁₃	—	8	4	no	—
129	C₂×D₁₃	—	8	8	no	—
130	C₂×D₁₃	—	16	8	no	—
131	C₄×C₁₃	—	2	4	no	—
132	C₄×C₁₃	#493	1	2	yes	—
133	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
134	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
135	Dic₁₃	—	8	4	no	—
136	C₂²×Q₈	—	26	2	no	—
137	C₂²×D₄	—	26	2	no	—
138	C₂×Q₁₆	—	13	4	no	—
139	C₂×Q₁₆	—	26	4	no	—
140	C₂×Q₁₆	—	26	4	no	—
141	C₂×Q₁₆	—	26	2	no	—
142	C₂×SD₁₆	—	26	8	no	—
143	C₂×SD₁₆	—	52	8	no	—
144	C₂×D₈	—	13	4	no	—
145	C₂×D₈	—	26	4	no	—
146	C₂×D₈	—	26	4	no	—
147	C₂×D₈	—	26	2	no	—
148	C₂²×C₈	D₅₂	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
150	#124	—	26	8	no	—
151	#124	—	52	4	no	—
152	C₂×C₂²⋊C₄	—	52	4	no	—
153	C₂×C₄²	D₅₂	1	1	yes	characteristic
154	Q₃₂	—	26	8	no	—
155	SD₃₂	—	26	16	no	—
156	D₁₆	—	26	8	no	—
157	C₂×C₁₆	D₅₂	1	4	yes	—
158	C₂×C₁₆	D₅₂	1	4	yes	—
159	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
160	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
161	C₄×C₈	D₅₂	1	2	yes	—
162	C₄×C₈	C₂²×D₁₃	1	2	yes	—
163	C₂×C₁₃	#2249	1	4	yes	complemented
164	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
165	C₂×C₁₃	C₂×#508	1	2	yes	—
166	D₁₃	—	16	4	no	—
167	C₂⁴	—	52	2	no	—
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	C₂×D₄	—	26	4	no	—
173	C₂×D₄	—	52	4	no	—
174	C₂×D₄	—	52	4	no	—
175	C₂×D₄	—	52	2	no	—
176	C₂²×C₄	—	52	2	no	—
177	C₂²×C₄	D₁₀₄	1	2	yes	—
178	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
179	Q₁₆	—	26	4	no	—
180	Q₁₆	—	26	4	no	—
181	Q₁₆	—	52	4	no	—
182	SD₁₆	—	52	16	no	—
183	D₈	—	26	4	no	—
184	D₈	—	26	4	no	—
185	D₈	—	52	4	no	—
186	C₂×C₈	D₁₀₄	1	4	yes	—
187	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
188	C₂×C₈	C₂×D₅₂	1	2	yes	—
189	C₂×C₈	C₂×D₅₂	1	2	yes	—
190	C₂×C₈	C₂×D₅₂	1	2	yes	—
191	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
192	C₂²⋊C₄	—	52	16	no	—
193	C₄²	D₁₀₄	1	2	yes	—
194	C₄²	C₂×D₅₂	1	2	yes	—
195	C₁₆	C₂×D₅₂	1	4	yes	—
196	C₁₆	C₂×D₅₂	1	4	yes	—
197	C₁₃	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	52	4	no	—
199	C₂³	—	104	8	no	—
200	C₂³	—	104	4	no	—
201	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
202	Q₈	—	52	4	no	—
203	Q₈	—	52	4	no	—
204	Q₈	—	104	4	no	—
205	D₄	—	52	4	no	—
206	D₄	—	52	4	no	—
207	D₄	—	104	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	52	4	no	—
210	C₂×C₄	—	104	4	no	—
211	C₂×C₄	#88128	1	2	yes	—
212	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
213	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
214	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
216	C₈	#88128	1	2	yes	—
217	C₈	C₂×D₁₀₄	1	4	yes	—
218	C₈	C₂²×D₅₂	1	1	yes	characteristic
219	C₈	C₂²×D₅₂	1	1	yes	characteristic
220	C₂²	—	104	8	no	—
221	C₂²	—	104	4	no	—
222	C₂²	—	208	8	no	—
223	C₂²	#11718356	1	2	yes	—
224	C₂²	#11718565	1	4	yes	—
225	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	104	4	no	—
228	C₄	#11718356	1	2	yes	—
229	C₄	C₂×#88128	1	1	yes	characteristic
230	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
231	C₂	—	208	4	no	—
232	C₂	#13117097	1	4	yes	direct product factor
233	C₂	C₂×#11718356	1	1	yes	characteristic
234	C₂	C₂×#11718565	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₂₀₈	C₂
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2249
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
1. C₂²×Dic₁₀₄	C₂	semidirect
2. C₂²×#88040	C₂	semidirect
4. C₂×#11718357	C₂	semidirect
5. #13117097	C₂	direct (N × H)
6. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
15. C₂×Dic₁₀₄	C₂²	semidirect
17. C₂×#88040	C₂²	semidirect
21. #11718357	C₂²	semidirect
22. C₄×C₁₃×C₁₆	C₂²	semidirect
86. C₂×C₄×C₁₆	D₁₃	semidirect
124. C₄×C₁₆	C₂×D₁₃	semidirect
163. C₂×C₁₃	#2249	semidirect
197. C₁₃	C₂×#2249	semidirect
232. C₂	#13117097	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
C₂²×#88040	2
C₂²×D₂₀₈	2
C₂×#11718357	2
#13117097	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2249	13

Subgroups — C2×#13117097 #16018318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117097 #16018318