Subgroups — C₂×#13116625 #16015954

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	#13116625	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718082	C₂	1	2	yes	maximal, complemented
5	C₂×#11718081	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	—	2	1	no	—
9	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
10	C₁₃×#427	—	2	2	no	—
11	C₁₃×#427	C₂²	1	4	yes	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	—	2	1	no	—
15	C₂×#88058	—	2	1	no	—
16	C₂×#88058	—	2	2	no	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88058	C₂²	1	4	yes	—
19	C₂×#88052	C₄	1	8	yes	complemented
20	C₂×#88052	C₂²	1	4	yes	—
21	C₂²×#6090	C₂²	1	2	yes	—
22	#11718082	C₂²	1	8	yes	complemented
23	#11718081	C₂²	1	8	yes	complemented
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	4	1	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₁₃×#117	—	2	4	no	—
33	C₁₃×#117	—	4	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₁₃	—	4	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	—	2	4	no	—
40	C₂×C₄⋊Dic₁₃	—	4	1	no	—
41	C₂×C₄⋊Dic₁₃	—	4	2	no	—
42	C₁₃×#107	—	2	1	no	—
43	C₁₃×#107	—	2	1	no	—
44	C₁₃×#107	C₂³	1	2	yes	—
45	C₁₃×#107	C₂³	1	2	yes	—
46	C₁₃×#106	—	2	4	no	—
47	#88058	—	2	4	no	—
48	#88058	—	2	2	no	—
49	#88058	C₂³	1	4	yes	—
50	#88052	C₂×C₄	1	16	yes	complemented
51	C₂×#6090	C₂×C₄	1	8	yes	complemented
52	C₂×#6090	C₂³	1	4	yes	—
53	C₂×#2159	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	D₈	1	2	yes	—
60	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×Dic₁₃	—	4	4	no	—
64	C₂²×Dic₁₃	—	4	2	no	—
65	C₂²×Dic₁₃	—	8	4	no	—
66	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	4	1	no	—
78	C₁₃×C₄⋊C₄	—	4	1	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₄²×C₁₃	—	4	4	no	—
81	C₄⋊Dic₁₃	—	2	4	no	—
82	C₄⋊Dic₁₃	—	4	4	no	—
83	C₄⋊Dic₁₃	—	4	2	no	—
84	#6090	C₂²×C₄	1	8	yes	complemented
85	C₂²×M₃₂	—	13	1	no	—
86	C₂×#427	D₁₃	1	1	yes	characteristic
87	C₂²×#107	—	13	1	no	—
88	#2159	—	13	8	no	—
89	C₂×#367	—	13	2	no	—
90	C₂×#366	—	13	2	no	—
91	C₂³×C₁₃	—	2	2	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	#107	1	4	yes	—
94	C₂³×C₁₃	C₂×D₈	1	2	yes	—
95	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	8	2	no	—
99	C₂×C₄×C₁₃	—	8	2	no	—
100	C₂×C₄×C₁₃	#107	1	8	yes	—
101	C₂×C₄×C₁₃	#107	1	4	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
107	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	—	8	8	no	—
111	C₂×Dic₁₃	—	8	4	no	—
112	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₂×#117	—	2	1	no	—
118	C₂²×C₄⋊C₄	—	26	1	no	—
119	C₂×M₃₂	—	13	8	no	—
120	C₂×M₃₂	—	13	4	no	—
121	C₂²×C₁₆	—	13	2	no	—
122	#427	—	2	2	no	—
123	#427	C₂×D₁₃	1	4	yes	—
124	C₂×#107	—	13	2	no	—
125	C₂×#107	—	13	4	no	—
126	C₂×#107	—	26	1	no	—
127	C₂×#107	—	26	2	no	—
128	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
129	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
130	C₂×#106	—	2	1	no	—
131	#367	—	13	8	no	—
132	#366	—	13	8	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	8	no	—
136	C₂²×C₁₃	C₂×#107	1	4	yes	—
137	C₂²×C₁₃	C₂×#107	1	2	yes	—
138	C₂²×C₁₃	#509	1	2	yes	—
139	C₂²×C₁₃	#510	1	2	yes	—
140	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₁₃	—	8	2	no	—
142	C₄×C₁₃	—	8	2	no	—
143	C₄×C₁₃	C₂×#107	1	4	yes	—
144	C₄×C₁₃	C₂×#107	1	2	yes	—
145	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
147	Dic₁₃	—	8	4	no	—
148	C₂³×C₄	—	52	1	no	—
149	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂²×C₈	Dic₂₆	1	4	yes	—
151	C₂²×C₈	C₄×D₁₃	1	4	yes	—
152	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
156	#117	—	2	4	no	—
157	#117	—	4	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	26	4	no	—
161	C₂×C₄⋊C₄	—	26	2	no	—
162	C₂×C₄⋊C₄	—	52	2	no	—
163	C₂×C₄⋊C₄	—	52	1	no	—
164	C₂×C₂²⋊C₄	—	4	1	no	—
165	C₂×C₄²	—	4	1	no	—
166	M₃₂	—	13	16	no	—
167	C₂×C₁₆	—	13	8	no	—
168	C₂×C₁₆	—	13	4	no	—
169	#107	—	2	1	no	—
170	#107	—	2	1	no	—
171	#107	—	13	4	no	—
172	#107	—	26	4	no	—
173	#107	—	26	2	no	—
174	#107	C₂²×D₁₃	1	2	yes	—
175	#107	C₂²×D₁₃	1	2	yes	—
176	#106	—	2	4	no	—
177	C₂×C₁₃	—	2	4	no	—
178	C₂×C₁₃	#2159	1	4	yes	complemented
179	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
181	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
182	C₂⁴	#6104	1	1	yes	characteristic
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	52	4	no	—
186	C₂²×C₄	—	52	2	no	—
187	C₂²×C₄	—	104	4	no	—
188	C₂²×C₄	#6101	1	4	yes	—
189	C₂²×C₄	#6104	1	2	yes	—
190	C₂²×C₄	#6104	1	1	yes	characteristic
191	C₂²×C₄	#6107	1	4	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂×C₈	#6101	1	4	yes	—
195	C₂×C₈	#6101	1	8	yes	—
196	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
197	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
198	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
199	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
200	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	26	4	no	—
208	C₄⋊C₄	—	52	4	no	—
209	C₄⋊C₄	—	52	2	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	4	4	no	—
212	C₁₆	—	13	8	no	—
213	C₁₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#88047	1	4	yes	—
217	C₂³	C₂×#6104	1	2	yes	—
218	C₂³	C₂×#6104	1	1	yes	characteristic
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	52	4	no	—
224	C₂×C₄	—	104	8	no	—
225	C₂×C₄	—	104	4	no	—
226	C₂×C₄	#88047	1	8	yes	—
227	C₂×C₄	#88047	1	4	yes	—
228	C₂×C₄	C₂×#6101	1	4	yes	—
229	C₂×C₄	C₂×#6101	1	2	yes	—
230	C₂×C₄	C₂×#6104	1	2	yes	—
231	C₂×C₄	C₂×#6104	1	1	yes	characteristic
232	C₂×C₄	C₂×#6107	1	4	yes	—
233	C₂×C₄	C₂×#6107	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₈	C₂×#6101	1	2	yes	—
236	C₈	C₂×#6101	1	4	yes	—
237	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
238	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	C₂×#88047	1	4	yes	—
243	C₂²	C₂×#88047	1	2	yes	—
244	C₂²	#11718854	1	2	yes	—
245	C₂²	#11718856	1	2	yes	—
246	C₂²	C₂²×#6104	1	1	yes	characteristic
247	C₄	—	8	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	104	4	no	—
250	C₄	C₂×#88047	1	4	yes	—
251	C₄	C₂×#88047	1	2	yes	—
252	C₄	C₂²×#6101	1	1	yes	characteristic
253	C₄	C₂²×#6107	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#13116625	1	4	yes	direct product factor
256	C₂	C₂²×#88047	1	1	yes	characteristic
257	C₂	C₂×#11718854	1	1	yes	characteristic
258	C₂	C₂×#11718856	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	C₂×#2159
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #13116625	C₂	direct (N × H)
4. C₂×#11718082	C₂	semidirect
5. C₂×#11718081	C₂	semidirect
19. C₂×#88052	C₄	semidirect
22. #11718082	C₂²	semidirect
23. #11718081	C₂²	semidirect
50. #88052	C₂×C₄	semidirect
51. C₂×#6090	C₂×C₄	semidirect
84. #6090	C₂²×C₄	semidirect
178. C₂×C₁₃	#2159	semidirect
213. C₁₃	C₂×#2159	semidirect
255. C₂	#13116625	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×#88052	2
C₂²×#88058	2
#13116625	2
C₂×#11718082	2
C₂×#11718081	2
C₂×C₁₃×#427	2
C₂×#2159	13

Subgroups — C2×#13116625 #16015954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116625 #16015954