Subgroups — C₂×#13116702 #16016210

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88040	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
4	#13116702	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718096	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718095	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718101	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₅₂	—	2	1	no	—
12	C₂×C₄×Dic₂₆	—	2	1	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88061	—	2	1	no	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	C₂×#88056	—	2	1	no	—
20	C₂×#88040	C₄	1	8	yes	complemented
21	C₂×#88040	C₂²	1	4	yes	—
22	C₄×Dic₅₂	—	2	2	no	—
23	C₄×Dic₅₂	C₂²	1	4	yes	complemented
24	C₄×D₁₀₄	—	2	2	no	—
25	C₄×D₁₀₄	C₂²	1	4	yes	—
26	C₁₃×#346	C₂²	1	4	yes	complemented
27	#11718101	C₂²	1	4	yes	—
28	#11718096	C₂²	1	4	yes	complemented
29	#11718095	C₂²	1	4	yes	complemented
30	C₂²×D₅₂	—	2	1	no	—
31	C₂²×C₄×D₁₃	—	4	1	no	—
32	C₂²×Dic₂₆	—	2	1	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×#6103	—	4	1	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×#6101	—	4	1	no	—
40	C₂×C₄×Dic₁₃	—	4	1	no	—
41	C₂×Dic₅₂	—	2	1	no	—
42	C₂×Dic₅₂	—	2	2	no	—
43	C₂×Dic₅₂	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₅₂	C₂³	1	2	yes	—
45	C₂×D₁₀₄	—	2	1	no	—
46	C₂×D₁₀₄	—	2	2	no	—
47	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
48	C₂×D₁₀₄	C₂³	1	2	yes	—
49	C₄×D₅₂	—	2	4	no	—
50	C₄×D₅₂	—	4	2	no	—
51	C₄×Dic₂₆	—	2	4	no	—
52	C₄×Dic₂₆	—	4	2	no	—
53	C₂×C₁₃×C₁₆	—	2	2	no	—
54	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₄×C₈×C₁₃	D₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	#88061	—	2	4	no	—
60	#88058	—	2	1	no	—
61	#88058	—	2	1	no	—
62	#88058	C₂³	1	2	yes	—
63	#88058	C₂³	1	2	yes	—
64	#88056	—	2	4	no	—
65	#88040	C₂×C₄	1	16	yes	complemented
66	C₂×#2175	—	13	1	no	maximal, Sylow
67	C₂³×D₁₃	—	4	1	no	—
68	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	4	1	no	—
72	C₂²×Dic₁₃	—	4	1	no	—
73	C₂²×Dic₁₃	—	4	1	no	—
74	C₂×D₅₂	—	2	4	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	2	no	—
78	C₂×C₄×D₁₃	—	4	4	no	—
79	C₂×C₄×D₁₃	—	8	4	no	—
80	C₂×Dic₂₆	—	2	4	no	—
81	C₂×Dic₂₆	—	2	2	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₁₃	D₈	1	2	yes	—
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	#6103	—	4	4	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	—	4	1	no	—
96	C₄⋊Dic₁₃	—	4	1	no	—
97	#6101	—	4	4	no	—
98	C₄×Dic₁₃	—	4	4	no	—
99	Dic₅₂	—	2	4	no	—
100	Dic₅₂	—	2	2	no	—
101	Dic₅₂	C₂²×C₄	1	4	yes	complemented
102	D₁₀₄	—	2	4	no	—
103	D₁₀₄	—	2	2	no	—
104	D₁₀₄	C₂²×C₄	1	4	yes	—
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	C₂²×C₄	1	4	yes	complemented
107	C₂²×SD₃₂	—	13	1	no	—
108	C₂×C₄×Q₁₆	—	13	1	no	—
109	C₂×C₄×D₈	—	13	1	no	—
110	#2175	—	13	8	no	—
111	C₂×#366	—	13	1	no	—
112	C₂×#358	—	13	1	no	—
113	C₂×#357	—	13	1	no	—
114	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₁₃	—	4	2	no	—
117	C₂²×D₁₃	—	4	4	no	—
118	C₂²×D₁₃	—	8	2	no	—
119	C₂²×D₁₃	—	8	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#135	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃	—	4	2	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	2	no	—
131	C₂×Dic₁₃	—	4	2	no	—
132	C₂×Dic₁₃	—	8	2	no	—
133	C₂×Dic₁₃	—	8	2	no	—
134	D₅₂	—	2	4	no	—
135	D₅₂	—	4	2	no	—
136	D₅₂	—	4	4	no	—
137	C₄×D₁₃	—	8	8	no	—
138	Dic₂₆	—	2	4	no	—
139	Dic₂₆	—	4	2	no	—
140	Dic₂₆	—	4	4	no	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	C₄×D₄	1	2	yes	—
143	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₂²×Q₁₆	—	13	1	no	—
146	C₂²×D₈	—	13	1	no	—
147	C₂×C₄×Q₈	—	26	1	no	—
148	C₂×C₄×D₄	—	26	1	no	—
149	C₂×SD₃₂	—	13	8	no	—
150	C₂×SD₃₂	—	13	4	no	—
151	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
152	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
153	C₄×Q₁₆	—	13	4	no	—
154	C₄×Q₁₆	—	26	2	no	—
155	C₄×D₈	—	13	4	no	—
156	C₄×D₈	—	26	2	no	—
157	C₂×#107	—	13	1	no	—
158	C₂×#107	—	13	1	no	—
159	C₂×#103	—	26	1	no	—
160	C₂×#102	—	26	1	no	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	#366	—	13	4	no	—
163	#358	—	13	4	no	—
164	#357	—	13	4	no	—
165	#346	C₂×D₁₃	1	4	yes	complemented
166	C₂²×C₁₃	C₄×D₈	1	2	yes	—
167	C₂²×C₁₃	#509	1	2	yes	—
168	C₂²×C₁₃	#510	1	2	yes	—
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₁₃	—	4	4	no	—
171	C₂×D₁₃	—	8	2	no	—
172	C₂×D₁₃	—	8	2	no	—
173	C₂×D₁₃	—	8	8	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	C₄×D₈	1	2	yes	—
177	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
179	Dic₁₃	—	4	4	no	—
180	Dic₁₃	—	8	2	no	—
181	Dic₁₃	—	8	2	no	—
182	C₂²×Q₈	—	26	1	no	—
183	C₂²×D₄	—	26	1	no	—
184	C₂³×C₄	—	52	1	no	—
185	C₂×Q₁₆	—	13	4	no	—
186	C₂×Q₁₆	—	13	2	no	—
187	C₂×Q₁₆	—	26	1	no	—
188	C₂×Q₁₆	—	26	2	no	—
189	C₂×D₈	—	13	4	no	—
190	C₂×D₈	—	13	2	no	—
191	C₂×D₈	—	26	1	no	—
192	C₂×D₈	—	26	2	no	—
193	C₂²×C₈	D₅₂	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×Q₈	—	26	4	no	—
196	C₄×Q₈	—	52	2	no	—
197	C₄×D₄	—	26	4	no	—
198	C₄×D₄	—	52	2	no	—
199	C₂×C₄⋊C₄	—	26	1	no	—
200	C₂×C₄⋊C₄	—	26	1	no	—
201	C₂×C₄⋊C₄	—	52	1	no	—
202	C₂×C₂²⋊C₄	—	52	1	no	—
203	C₂×C₄²	—	52	1	no	—
204	C₂×C₄²	D₅₂	1	1	yes	characteristic
205	SD₃₂	—	13	16	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
208	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
209	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
210	#107	—	13	2	no	—
211	#107	—	13	2	no	—
212	#107	—	26	1	no	—
213	#107	—	26	1	no	—
214	#103	—	26	4	no	—
215	#102	—	26	4	no	—
216	C₄×C₈	D₅₂	1	2	yes	—
217	C₄×C₈	C₂²×D₁₃	1	2	yes	—
218	C₂×C₁₃	#2175	1	4	yes	complemented
219	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
220	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
222	D₁₃	—	8	4	no	—
223	C₂⁴	—	52	1	no	—
224	C₂×Q₈	—	26	4	no	—
225	C₂×Q₈	—	26	2	no	—
226	C₂×Q₈	—	52	1	no	—
227	C₂×Q₈	—	52	2	no	—
228	C₂×D₄	—	26	4	no	—
229	C₂×D₄	—	26	2	no	—
230	C₂×D₄	—	52	1	no	—
231	C₂×D₄	—	52	2	no	—
232	C₂²×C₄	—	52	1	no	—
233	C₂²×C₄	—	52	1	no	—
234	C₂²×C₄	—	52	1	no	—
235	C₂²×C₄	—	52	4	no	—
236	C₂²×C₄	—	104	4	no	—
237	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
238	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
240	Q₁₆	—	13	4	no	—
241	Q₁₆	—	26	4	no	—
242	Q₁₆	—	26	2	no	—
243	D₈	—	13	4	no	—
244	D₈	—	26	4	no	—
245	D₈	—	26	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
248	C₂×C₈	C₂×D₅₂	1	2	yes	—
249	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
250	C₂×C₈	#6127	1	2	yes	—
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	26	2	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	52	1	no	—
255	C₄⋊C₄	—	52	1	no	—
256	C₄⋊C₄	—	52	4	no	—
257	C₂²⋊C₄	—	52	4	no	—
258	C₄²	—	52	4	no	—
259	C₄²	D₁₀₄	1	2	yes	—
260	C₄²	C₂×D₅₂	1	2	yes	—
261	C₁₆	—	2	2	no	—
262	C₁₆	C₂×C₄×D₁₃	1	4	yes	complemented
263	C₁₃	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	52	2	no	—
265	C₂³	—	52	4	no	—
266	C₂³	—	104	2	no	—
267	C₂³	—	104	2	no	—
268	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
269	Q₈	—	26	4	no	—
270	Q₈	—	52	2	no	—
271	Q₈	—	52	4	no	—
272	D₄	—	26	4	no	—
273	D₄	—	52	2	no	—
274	D₄	—	52	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	52	2	no	—
278	C₂×C₄	—	52	4	no	—
279	C₂×C₄	—	52	2	no	—
280	C₂×C₄	—	52	2	no	—
281	C₂×C₄	—	104	8	no	—
282	C₂×C₄	—	104	2	no	—
283	C₂×C₄	—	104	2	no	—
284	C₂×C₄	C₄×D₅₂	1	2	yes	—
285	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
286	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
287	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
288	C₂×C₄	#88158	1	2	yes	—
289	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×D₅₂	1	2	yes	—
292	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
293	C₈	C₂×#6127	1	1	yes	characteristic
294	C₂²	—	52	4	no	—
295	C₂²	—	104	2	no	—
296	C₂²	—	104	8	no	—
297	C₂²	—	104	2	no	—
298	C₂²	C₄×D₁₀₄	1	2	yes	—
299	C₂²	#11718569	1	2	yes	—
300	C₂²	#11718570	1	2	yes	—
301	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	52	4	no	—
305	C₄	—	104	2	no	—
306	C₄	—	104	2	no	—
307	C₄	C₄×D₁₀₄	1	2	yes	—
308	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
309	C₄	C₂×#88158	1	1	yes	characteristic
310	C₂	—	104	4	no	—
311	C₂	#13116702	1	4	yes	direct product factor
312	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
313	C₂	C₂×#11718569	1	1	yes	characteristic
314	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	C₂×C₁₃×#346	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	C₂×#2175
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₅₂	C₂	semidirect
4. #13116702	C₂	direct (N × H)
5. C₂×#11718096	C₂	semidirect
6. C₂×#11718095	C₂	semidirect
8. C₂×C₁₃×#346	C₂	semidirect
20. C₂×#88040	C₄	semidirect
23. C₄×Dic₅₂	C₂²	semidirect
26. C₁₃×#346	C₂²	semidirect
28. #11718096	C₂²	semidirect
29. #11718095	C₂²	semidirect
43. C₂×Dic₅₂	C₂×C₄	semidirect
54. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
65. #88040	C₂×C₄	semidirect
101. Dic₅₂	C₂²×C₄	semidirect
106. C₁₃×C₁₆	C₂²×C₄	semidirect
114. C₂×#346	D₁₃	semidirect
165. #346	C₂×D₁₃	semidirect
207. C₂×C₁₆	C₄×D₁₃	semidirect
218. C₂×C₁₃	#2175	semidirect
262. C₁₆	C₂×C₄×D₁₃	semidirect
263. C₁₃	C₂×#2175	semidirect
311. C₂	#13116702	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×#88040	2
C₂×C₄×Dic₅₂	2
C₂×C₄×D₁₀₄	2
#13116702	2
C₂×#11718096	2
C₂×#11718095	2
C₂×#11718101	2
C₂×C₁₃×#346	2
C₂×#2175	13

Subgroups — C2×#13116702 #16016210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116702 #16016210