Subgroups — C₂×#13116624 #16015953

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	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	#13116624	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718097	C₂	1	2	yes	maximal, complemented
5	C₂×#11718096	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×#88180	—	2	1	no	—
9	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₄	1	8	yes	complemented
11	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
13	#11718767	—	2	2	no	—
14	#11718767	C₂²	1	4	yes	—
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×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88057	—	2	1	no	—
22	#11718097	C₂²	1	8	yes	complemented
23	#11718096	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	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
33	#88180	—	2	4	no	—
34	#88180	—	4	2	no	—
35	C₂×C₄⋊Dic₁₃	—	2	2	no	—
36	C₂×C₄⋊Dic₁₃	—	2	4	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	—	4	1	no	—
40	C₂×C₄⋊Dic₁₃	—	4	2	no	—
41	C₂×C₄×Dic₁₃	—	4	1	no	—
42	C₁₃×M₃₂	C₂×C₄	1	16	yes	complemented
43	C₂×C₁₃×C₁₆	C₂×C₄	1	8	yes	complemented
44	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
45	#88058	—	2	4	no	—
46	#88058	—	2	2	no	—
47	#88058	—	2	1	no	—
48	#88058	—	2	1	no	—
49	#88058	C₂³	1	4	yes	—
50	#88058	C₂³	1	2	yes	—
51	#88058	C₂³	1	2	yes	—
52	#88057	—	2	4	no	—
53	C₂×#2159	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	D₈	1	2	yes	—
58	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×Dic₁₃	—	4	1	no	—
62	C₂²×Dic₁₃	—	4	1	no	—
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	C₂²⋊Dic₁₃	—	4	4	no	—
76	C₄⋊Dic₁₃	—	2	4	no	—
77	C₄⋊Dic₁₃	—	2	2	no	—
78	C₄⋊Dic₁₃	—	2	2	no	—
79	C₄⋊Dic₁₃	—	4	4	no	—
80	C₄⋊Dic₁₃	—	4	2	no	—
81	C₄⋊Dic₁₃	—	4	1	no	—
82	C₄⋊Dic₁₃	—	4	1	no	—
83	C₄×Dic₁₃	—	4	4	no	—
84	C₁₃×C₁₆	C₂²×C₄	1	8	yes	complemented
85	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
86	C₂×#427	—	13	1	no	—
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	#107	1	8	yes	—
97	C₂×C₄×C₁₃	#107	1	4	yes	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	C₂×Q₁₆	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×Dic₁₃	—	4	2	no	—
106	C₂×Dic₁₃	—	4	2	no	—
107	C₂×Dic₁₃	—	4	4	no	—
108	C₂×Dic₁₃	—	8	8	no	—
109	C₂×Dic₁₃	—	8	2	no	—
110	C₂×Dic₁₃	—	8	4	no	—
111	C₂×Dic₁₃	—	8	2	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	—	26	1	no	—
118	C₂²×C₄⋊C₄	—	26	1	no	—
119	C₂×M₃₂	Dic₁₃	1	8	yes	complemented
120	C₂×M₃₂	C₂×D₁₃	1	4	yes	—
121	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
122	#427	—	13	4	no	—
123	#427	—	26	2	no	—
124	C₂×#107	—	13	2	no	—
125	C₂×#107	—	13	4	no	—
126	C₂×#107	—	13	1	no	—
127	C₂×#107	—	13	1	no	—
128	C₂×#107	—	26	1	no	—
129	C₂×#107	—	26	2	no	—
130	C₂×#106	—	26	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	C₂×#107	1	4	yes	—
142	C₄×C₁₃	C₂×#107	1	2	yes	—
143	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
145	Dic₁₃	—	8	2	no	—
146	Dic₁₃	—	8	4	no	—
147	Dic₁₃	—	8	2	no	—
148	C₂³×C₄	—	52	1	no	—
149	C₂³×C₄	D₅₂	1	1	yes	characteristic
150	C₂²×C₈	Dic₂₆	1	4	yes	—
151	C₂²×C₈	D₅₂	1	1	yes	characteristic
152	C₂²×C₈	D₅₂	1	2	yes	—
153	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
156	#117	—	26	4	no	—
157	#117	—	52	2	no	—
158	C₂×C₄⋊C₄	—	26	4	no	—
159	C₂×C₄⋊C₄	—	26	2	no	—
160	C₂×C₄⋊C₄	—	26	1	no	—
161	C₂×C₄⋊C₄	—	26	1	no	—
162	C₂×C₄⋊C₄	—	52	2	no	—
163	C₂×C₄⋊C₄	—	52	1	no	—
164	C₂×C₂²⋊C₄	—	52	1	no	—
165	C₂×C₄²	—	52	1	no	—
166	M₃₂	C₂×Dic₁₃	1	16	yes	complemented
167	C₂×C₁₆	C₂×Dic₁₃	1	8	yes	complemented
168	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
169	#107	—	13	2	no	—
170	#107	—	13	4	no	—
171	#107	—	13	2	no	—
172	#107	—	26	4	no	—
173	#107	—	26	2	no	—
174	#107	—	26	1	no	—
175	#107	—	26	1	no	—
176	#106	—	26	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	D₁₀₄	1	1	yes	characteristic
183	C₂²×C₄	—	52	4	no	—
184	C₂²×C₄	—	52	1	no	—
185	C₂²×C₄	—	52	1	no	—
186	C₂²×C₄	—	52	2	no	—
187	C₂²×C₄	—	104	4	no	—
188	C₂²×C₄	D₁₀₄	1	2	yes	—
189	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
190	C₂²×C₄	Dic₅₂	1	4	yes	—
191	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
192	C₂²×C₄	C₂×D₅₂	1	2	yes	—
193	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
194	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
195	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
196	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
197	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
198	C₂×C₈	C₂×D₅₂	1	2	yes	—
199	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
200	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
201	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
202	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	26	2	no	—
204	C₄⋊C₄	—	26	2	no	—
205	C₄⋊C₄	—	26	4	no	—
206	C₄⋊C₄	—	52	4	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	—	52	1	no	—
209	C₄⋊C₄	—	52	1	no	—
210	C₂²⋊C₄	—	52	4	no	—
211	C₄²	—	52	4	no	—
212	C₁₆	C₂²×Dic₁₃	1	8	yes	complemented
213	C₁₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#88058	1	4	yes	—
217	C₂³	C₂×D₁₀₄	1	2	yes	—
218	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
219	C₂×C₄	—	52	2	no	—
220	C₂×C₄	—	52	2	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	—	104	8	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	4	no	—
225	C₂×C₄	—	104	2	no	—
226	C₂×C₄	#88058	1	8	yes	—
227	C₂×C₄	#88058	1	4	yes	—
228	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
229	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
231	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
234	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
235	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
236	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
237	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
238	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	#11718569	1	2	yes	—
243	C₂²	#11718570	1	2	yes	—
244	C₂²	C₂×#88058	1	4	yes	—
245	C₂²	C₂×#88058	1	2	yes	—
246	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
247	C₄	—	104	2	no	—
248	C₄	—	104	4	no	—
249	C₄	—	104	2	no	—
250	C₄	C₂×#88058	1	4	yes	—
251	C₄	C₂×#88058	1	2	yes	—
252	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
253	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#13116624	1	4	yes	direct product factor
256	C₂	C₂²×#88058	1	1	yes	characteristic
257	C₂	C₂×#11718569	1	1	yes	characteristic
258	C₂	C₂×#11718570	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂×#2159
LCS	G ▷ 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
3. #13116624	C₂	direct (N × H)
4. C₂×#11718097	C₂	semidirect
5. C₂×#11718096	C₂	semidirect
10. C₂×C₁₃×M₃₂	C₄	semidirect
22. #11718097	C₂²	semidirect
23. #11718096	C₂²	semidirect
42. C₁₃×M₃₂	C₂×C₄	semidirect
43. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
84. C₁₃×C₁₆	C₂²×C₄	semidirect
119. C₂×M₃₂	Dic₁₃	semidirect
166. M₃₂	C₂×Dic₁₃	semidirect
167. C₂×C₁₆	C₂×Dic₁₃	semidirect
178. C₂×C₁₃	#2159	semidirect
212. C₁₆	C₂²×Dic₁₃	semidirect
213. C₁₃	C₂×#2159	semidirect
255. C₂	#13116624	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11718767	2
C₂²×#88058	2
#13116624	2
C₂×#11718097	2
C₂×#11718096	2
C₂²×C₁₃×M₃₂	2
C₂×#2159	13

Subgroups — C2×#13116624 #16015953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116624 #16015953