Subgroups — C₂×#13117075 #16018228

Order 3328.

634 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718357	C₂	1	1	yes	maximal, characteristic
2	#13117075	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718084	C₂	1	2	yes	maximal, complemented
4	C₂×#11718083	C₂	1	2	yes	maximal
5	C₂×#11718053	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
9	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
10	C₂²×#6095	—	2	1	no	—
11	C₂×#88129	—	2	1	no	—
12	C₁₃×#500	—	2	2	no	—
13	C₁₃×#500	C₂²	1	4	yes	complemented
14	C₂×C₁₃×#107	—	2	1	no	—
15	C₂×C₁₃×#107	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	2	yes	—
18	#11718357	—	2	2	no	—
19	#11718357	C₂²	1	4	yes	—
20	#11718084	C₂²	1	8	yes	complemented
21	#11718083	C₂²	1	8	yes	—
22	#11718053	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂²×D₅₂	—	2	1	no	—
25	C₂²×Dic₂₆	—	2	1	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×C₄⋊Q₈	—	2	4	no	—
29	C₁₃×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×#6103	—	4	1	no	—
34	C₂×#6103	—	4	1	no	—
35	C₂×Dic₅₂	—	2	4	no	—
36	C₂×Dic₅₂	—	2	1	no	—
37	C₂×Dic₅₂	C₂³	1	2	yes	—
38	C₂×D₁₀₄	—	2	4	no	—
39	C₂×D₁₀₄	—	2	1	no	—
40	C₂×D₁₀₄	C₂³	1	2	yes	—
41	C₂×#6095	—	2	4	no	—
42	C₂×#6095	—	4	4	no	—
43	#88129	—	2	4	no	—
44	#88129	—	4	2	no	—
45	C₁₃×#107	—	2	4	no	—
46	C₁₃×#107	—	2	2	no	—
47	C₁₃×#107	C₂³	1	4	yes	—
48	C₄×C₈×C₁₃	D₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂×#6090	—	2	4	no	—
51	C₂×#6090	C₂³	1	4	yes	—
52	C₂×#2243	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	4	no	—
55	C₂×Q₈×C₁₃	—	8	4	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	4	1	no	—
61	C₂×D₅₂	—	2	2	no	—
62	C₂×D₅₂	—	4	2	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×D₅₂	—	4	1	no	—
65	C₂×Dic₂₆	—	2	2	no	—
66	C₂×Dic₂₆	—	4	2	no	—
67	C₂×Dic₂₆	—	4	2	no	—
68	C₂×Dic₂₆	—	4	1	no	—
69	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	#41	1	4	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	4	no	—
77	C₄²×C₁₃	D₈	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	#6103	—	4	4	no	—
80	#6103	—	4	4	no	—
81	Dic₅₂	—	2	4	no	—
82	Dic₅₂	—	4	2	no	—
83	D₁₀₄	—	2	4	no	—
84	D₁₀₄	—	4	2	no	—
85	#6095	—	4	8	no	—
86	#6090	—	2	4	no	—
87	C₂×#500	D₁₃	1	1	yes	characteristic, complemented
88	C₂×#495	—	13	1	no	—
89	#2243	—	13	8	no	—
90	C₂×#358	—	13	2	no	—
91	C₂×#357	—	13	2	no	—
92	C₂×#346	—	13	1	no	—
93	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₁₃	—	4	2	no	—
95	C₂²×D₁₃	—	8	2	no	—
96	C₂²×D₁₃	—	8	2	no	—
97	C₂²×D₁₃	—	8	2	no	—
98	Q₈×C₁₃	—	8	8	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	8	4	no	—
103	C₂×C₄×C₁₃	#124	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	—	8	2	no	—
111	D₅₂	—	4	2	no	—
112	D₅₂	—	4	2	no	—
113	D₅₂	—	8	2	no	—
114	Dic₂₆	—	4	2	no	—
115	Dic₂₆	—	4	2	no	—
116	Dic₂₆	—	8	2	no	—
117	C₈×C₁₃	#124	1	2	yes	—
118	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
119	C₈×C₁₃	C₂×#41	1	2	yes	—
120	C₂²×Q₁₆	—	13	1	no	—
121	C₂²×SD₁₆	—	26	1	no	—
122	C₂²×D₈	—	13	1	no	—
123	C₂×C₄⋊Q₈	—	2	1	no	—
124	C₂×#124	—	26	1	no	—
125	C₂²×C₁₆	—	13	2	no	—
126	#500	—	2	2	no	—
127	#500	C₂×D₁₃	1	4	yes	complemented
128	#495	—	13	4	no	—
129	#495	—	26	2	no	—
130	C₂×#107	—	2	1	no	—
131	C₂×#107	C₂×D₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	#358	—	13	8	no	—
134	#357	—	13	8	no	—
135	#346	—	13	4	no	—
136	C₂²×C₁₃	#486	1	2	yes	—
137	C₂²×C₁₃	#509	1	2	yes	—
138	C₂²×C₁₃	#510	1	2	yes	—
139	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₁₃	—	8	2	no	—
141	C₂×D₁₃	—	8	2	no	—
142	C₂×D₁₃	—	8	2	no	—
143	C₂×D₁₃	—	16	4	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	8	4	no	—
147	C₄×C₁₃	#486	1	2	yes	—
148	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
149	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
150	Dic₁₃	—	8	2	no	—
151	C₂²×Q₈	—	4	1	no	—
152	C₂²×Q₈	—	26	1	no	—
153	C₂²×D₄	—	26	1	no	—
154	C₂×Q₁₆	—	13	2	no	—
155	C₂×Q₁₆	—	26	4	no	—
156	C₂×Q₁₆	—	26	1	no	—
157	C₂×SD₁₆	—	26	4	no	—
158	C₂×SD₁₆	—	52	4	no	—
159	C₂×D₈	—	13	2	no	—
160	C₂×D₈	—	26	4	no	—
161	C₂×D₈	—	26	1	no	—
162	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	4	no	—
165	C₄⋊Q₈	—	4	2	no	—
166	#124	—	26	4	no	—
167	#124	—	52	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₂²⋊C₄	—	52	1	no	—
171	C₂×C₂²⋊C₄	—	52	1	no	—
172	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
173	C₂×C₁₆	—	13	4	no	—
174	C₂×C₁₆	—	26	4	no	—
175	#107	—	2	4	no	—
176	#107	—	2	2	no	—
177	#107	C₂²×D₁₃	1	4	yes	—
178	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
179	C₄×C₈	C₂²×D₁₃	1	2	yes	—
180	C₂×C₁₃	#2243	1	4	yes	complemented
181	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
182	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
183	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
184	D₁₃	—	16	2	no	—
185	C₂⁴	—	52	1	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	8	4	no	—
188	C₂×Q₈	—	26	2	no	—
189	C₂×Q₈	—	52	1	no	—
190	C₂×Q₈	—	52	2	no	—
191	C₂×Q₈	—	52	2	no	—
192	C₂×D₄	—	26	2	no	—
193	C₂×D₄	—	52	1	no	—
194	C₂×D₄	—	52	2	no	—
195	C₂×D₄	—	52	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	52	1	no	—
198	C₂²×C₄	#6104	1	1	yes	characteristic
199	C₂²×C₄	#6104	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	Q₁₆	—	26	4	no	—
202	Q₁₆	—	52	2	no	—
203	SD₁₆	—	52	8	no	—
204	D₈	—	26	4	no	—
205	D₈	—	52	2	no	—
206	C₂×C₈	#6106	1	4	yes	—
207	C₂×C₈	#6131	1	4	yes	—
208	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	52	4	no	—
215	C₂²⋊C₄	—	52	4	no	—
216	C₄²	#6104	1	2	yes	—
217	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
218	C₁₆	—	26	4	no	—
219	C₁₃	C₂×#2243	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	52	2	no	—
221	C₂³	—	104	2	no	—
222	C₂³	—	104	2	no	—
223	C₂³	—	104	2	no	—
224	C₂³	C₂×#6104	1	1	yes	characteristic
225	Q₈	—	8	8	no	—
226	Q₈	—	52	2	no	—
227	Q₈	—	52	2	no	—
228	Q₈	—	104	2	no	—
229	D₄	—	52	2	no	—
230	D₄	—	52	2	no	—
231	D₄	—	104	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	104	2	no	—
238	C₂×C₄	C₂×#6104	1	2	yes	—
239	C₂×C₄	C₂×#6104	1	2	yes	—
240	C₂×C₄	C₂×#6104	1	1	yes	characteristic
241	C₂×C₄	C₂×#6106	1	2	yes	—
242	C₂×C₄	#88201	1	2	yes	—
243	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
244	C₈	C₂×#6106	1	4	yes	—
245	C₈	#88201	1	2	yes	—
246	C₈	C₂×#6131	1	2	yes	—
247	C₂²	—	104	2	no	—
248	C₂²	—	104	2	no	—
249	C₂²	—	104	2	no	—
250	C₂²	—	208	4	no	—
251	C₂²	#11718749	1	2	yes	—
252	C₂²	#11718808	1	2	yes	—
253	C₂²	#11718840	1	2	yes	—
254	C₂²	C₂²×#6104	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	8	4	no	—
258	C₄	—	104	2	no	—
259	C₄	#11718749	1	2	yes	—
260	C₄	C₂²×#6106	1	1	yes	characteristic
261	C₄	C₂×#88201	1	1	yes	characteristic
262	C₂	—	208	2	no	—
263	C₂	#13117075	1	4	yes	direct product factor
264	C₂	C₂×#11718749	1	1	yes	characteristic
265	C₂	C₂×#11718808	1	1	yes	characteristic
266	C₂	C₂×#11718840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	C₂×#2243
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #13117075	C₂	direct (N × H)
3. C₂×#11718084	C₂	semidirect
5. C₂×#11718053	C₂	semidirect
6. C₂×C₁₃×#500	C₂	semidirect
13. C₁₃×#500	C₂²	semidirect
20. #11718084	C₂²	semidirect
22. #11718053	C₂²	semidirect
87. C₂×#500	D₁₃	semidirect
127. #500	C₂×D₁₃	semidirect
180. C₂×C₁₃	#2243	semidirect
219. C₁₃	C₂×#2243	semidirect
263. C₂	#13117075	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×#11718357	2
#13117075	2
C₂×#11718084	2
C₂×#11718083	2
C₂×#11718053	2
C₂×C₁₃×#500	2
C₂×#2243	13

Subgroups — C2×#13117075 #16018228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117075 #16018228