Subgroups — C₂×#13114317 #15845516

Order 3328.

850 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	C₂×#11718736	C₂	1	2	yes	maximal, complemented
2	C₂×#11718340	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114317	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718136	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×D₈	—	2	1	no	—
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂×C₁₃×#127	C₂²	1	2	yes	—
10	C₂²×#6105	—	2	2	no	—
11	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
12	C₁₃×#493	—	2	2	no	—
13	C₁₃×#493	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88072	—	2	2	no	—
16	C₂×#88055	—	2	1	no	—
17	C₂×#88054	—	2	1	no	—
18	#11718736	—	2	4	no	—
19	#11718736	C₂²	1	8	yes	complemented
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	#11718340	—	2	2	no	—
23	#11718340	C₂²	1	4	yes	complemented
24	#11718136	C₂²	1	8	yes	complemented
25	#11718049	C₂²	1	4	yes	complemented
26	C₂²×D₄×C₁₃	—	2	2	no	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	2	2	no	—
29	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
30	C₂×C₁₃×D₈	—	2	4	no	—
31	C₂×C₁₃×D₈	—	2	4	no	—
32	C₂×C₁₃×D₈	—	4	4	no	—
33	C₂×C₁₃×D₈	—	4	4	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₁₃×#127	—	2	2	no	—
37	C₁₃×#127	—	2	2	no	—
38	C₁₃×#127	—	2	2	no	—
39	C₁₃×#127	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×#6105	—	2	8	no	—
42	C₂×#6105	—	4	8	no	—
43	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
44	C₂×#6101	—	2	1	no	—
45	C₂×C₄×Dic₁₃	—	2	1	no	—
46	C₂²×#1055	—	2	2	no	—
47	C₂²×#1055	—	2	1	no	—
48	C₄×Dic₂₆	—	2	1	no	—
49	C₄×Dic₂₆	—	2	1	no	—
50	C₄×Dic₂₆	D₄	1	2	yes	—
51	C₄×Dic₂₆	C₂³	1	2	yes	—
52	C₄×C₈×C₁₃	D₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	#88072	—	2	8	no	—
55	#88055	—	2	4	no	—
56	#88054	—	2	4	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	C₂³	1	4	yes	—
60	C₄⋊#1055	C₂³	1	2	yes	—
61	C₂×#1688	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	2	4	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	4	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	—	4	4	no	—
70	C₂×D₄×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	4	4	no	—
72	C₂×D₄×C₁₃	—	4	4	no	—
73	C₂×D₄×C₁₃	—	8	4	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	2	1	no	—
78	C₂²×Dic₁₃	—	2	1	no	—
79	C₂×Dic₂₆	—	2	4	no	—
80	C₂×Dic₂₆	—	2	1	no	—
81	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
82	C₁₃×D₈	—	4	8	no	—
83	C₁₃×D₈	—	4	8	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
92	#6105	—	4	16	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
95	#6101	—	2	2	no	—
96	#6101	—	2	2	no	—
97	C₄×Dic₁₃	—	2	2	no	—
98	C₄×Dic₁₃	—	2	2	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂×#1055	—	4	4	no	—
103	C₂×#493	D₁₃	1	1	yes	characteristic
104	C₂×#460	—	13	2	no	—
105	C₂×#446	—	13	1	no	—
106	#1688	—	13	8	no	—
107	C₂×#334	—	13	1	no	—
108	C₂×#331	—	13	2	no	—
109	C₂³×C₁₃	—	4	4	no	—
110	C₂³×C₁₃	—	8	4	no	—
111	C₂³×C₁₃	—	8	4	no	—
112	C₂³×C₁₃	—	8	4	no	—
113	C₂³×C₁₃	#121	1	1	yes	characteristic
114	D₄×C₁₃	—	4	4	no	—
115	D₄×C₁₃	—	4	4	no	—
116	D₄×C₁₃	—	4	4	no	—
117	D₄×C₁₃	—	4	4	no	—
118	D₄×C₁₃	—	8	4	no	—
119	D₄×C₁₃	—	8	4	no	—
120	D₄×C₁₃	—	8	4	no	—
121	D₄×C₁₃	—	8	4	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
127	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	Dic₂₆	—	2	4	no	—
137	Dic₂₆	—	4	2	no	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	C₂×SD₁₆	1	4	yes	complemented
140	#1055	—	4	2	no	—
141	#1055	—	4	4	no	—
142	C₂²×SD₁₆	—	26	2	no	—
143	C₂²×D₈	—	2	1	no	—
144	C₂²×D₈	—	2	1	no	—
145	C₂×#127	C₂×D₁₃	1	2	yes	—
146	C₂×C₄×Q₈	—	13	1	no	—
147	#493	—	2	2	no	—
148	#493	C₂×D₁₃	1	4	yes	—
149	#460	—	13	8	no	—
150	#460	—	26	4	no	—
151	#446	—	13	4	no	—
152	#446	—	26	2	no	—
153	C₂×C₄⋊C₈	—	13	1	no	—
154	C₂×C₄⋊C₈	—	13	2	no	—
155	C₂×C₄⋊C₈	—	26	1	no	—
156	C₂×#102	—	26	2	no	—
157	C₂×#97	—	26	1	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	#334	—	13	4	no	—
160	#331	—	13	8	no	—
161	C₂²×C₁₃	—	8	4	no	—
162	C₂²×C₁₃	—	8	4	no	—
163	C₂²×C₁₃	—	8	4	no	—
164	C₂²×C₁₃	—	16	8	no	—
165	C₂²×C₁₃	#460	1	2	yes	—
166	C₂²×C₁₃	#469	1	2	yes	—
167	C₂²×C₁₃	#472	1	2	yes	—
168	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	#460	1	2	yes	—
172	C₄×C₁₃	#469	1	2	yes	—
173	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
174	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
175	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	Dic₁₃	—	4	2	no	—
177	Dic₁₃	—	4	2	no	—
178	C₂²×Q₈	—	13	1	no	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	—	2	2	no	—
182	C₂×SD₁₆	—	26	8	no	—
183	C₂×SD₁₆	—	52	8	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	4	4	no	—
187	C₂×D₈	—	4	4	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	26	1	no	—
190	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
192	#127	—	2	2	no	—
193	#127	—	2	2	no	—
194	#127	—	2	2	no	—
195	#127	C₂²×D₁₃	1	4	yes	—
196	C₄×Q₈	—	13	2	no	—
197	C₄×Q₈	—	13	2	no	—
198	C₄×Q₈	—	26	1	no	—
199	C₄×Q₈	—	26	1	no	—
200	C₂×C₄⋊C₄	—	13	1	no	—
201	C₂×C₄⋊C₄	—	26	1	no	—
202	C₂×C₄²	—	26	1	no	—
203	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊C₈	—	13	2	no	—
205	C₄⋊C₈	—	13	4	no	—
206	C₄⋊C₈	—	26	4	no	—
207	C₄⋊C₈	—	26	2	no	—
208	C₄⋊C₈	—	26	1	no	—
209	#102	—	26	8	no	—
210	#97	—	26	4	no	—
211	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
212	C₄×C₈	C₂²×D₁₃	1	2	yes	—
213	C₂×C₁₃	—	16	4	no	—
214	C₂×C₁₃	#1688	1	4	yes	complemented
215	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#469	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
218	C₂⁴	—	4	2	no	—
219	C₂×Q₈	—	13	2	no	—
220	C₂×Q₈	—	26	1	no	—
221	C₂×Q₈	—	26	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	8	4	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	—	26	1	no	—
235	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
236	C₂²×C₄	#6129	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
238	SD₁₆	—	52	16	no	—
239	D₈	—	4	8	no	—
240	D₈	—	4	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	26	4	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	52	4	no	—
245	C₂×C₈	—	52	2	no	—
246	C₂×C₈	#6105	1	4	yes	complemented
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
249	C₄⋊C₄	—	13	2	no	—
250	C₄⋊C₄	—	26	1	no	—
251	C₄⋊C₄	—	26	2	no	—
252	C₄⋊C₄	—	26	2	no	—
253	C₄²	—	26	2	no	—
254	C₄²	—	26	2	no	—
255	C₄²	D₄×D₁₃	1	1	yes	characteristic
256	C₄²	#6129	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₄²	C₂³×D₁₃	1	1	yes	characteristic
259	C₁₃	C₂×#1688	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	4	4	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	#88192	1	1	yes	characteristic
265	Q₈	—	26	4	no	—
266	Q₈	—	52	2	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	26	2	no	—
277	C₂×C₄	—	26	2	no	—
278	C₂×C₄	—	52	2	no	—
279	C₂×C₄	—	52	2	no	—
280	C₂×C₄	#88165	1	4	yes	—
281	C₂×C₄	#88186	1	2	yes	—
282	C₂×C₄	C₂×#6105	1	2	yes	—
283	C₂×C₄	#88192	1	1	yes	characteristic
284	C₂×C₄	#88192	1	1	yes	characteristic
285	C₂×C₄	#88192	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂×#6129	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	52	4	no	—
291	C₈	—	52	2	no	—
292	C₈	C₂×#6105	1	4	yes	complemented
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	8	no	—
297	C₂²	#11718736	1	2	yes	—
298	C₂²	#11718820	1	2	yes	—
299	C₂²	#11718821	1	2	yes	—
300	C₂²	C₂×#88192	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	52	2	no	—
304	C₄	—	52	2	no	—
305	C₄	#11718736	1	2	yes	—
306	C₄	#11718820	1	2	yes	—
307	C₄	C₂×#88165	1	2	yes	—
308	C₄	C₂×#88186	1	1	yes	characteristic
309	C₄	C₂²×#6105	1	1	yes	characteristic
310	C₂	—	16	4	no	—
311	C₂	#13114317	1	4	yes	direct product factor
312	C₂	C₂×#11718736	1	1	yes	characteristic
313	C₂	C₂×#11718820	1	1	yes	characteristic
314	C₂	C₂×#11718821	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#493	C₂
socle	C₂³×C₁₃	#121
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	C₂×#1688
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#11718736	C₂	semidirect
2. C₂×#11718340	C₂	semidirect
3. #13114317	C₂	direct (N × H)
4. C₂×#11718049	C₂	semidirect
5. C₂×#11718136	C₂	semidirect
19. #11718736	C₂²	semidirect
23. #11718340	C₂²	semidirect
24. #11718136	C₂²	semidirect
25. #11718049	C₂²	semidirect
85. C₂×C₈×C₁₃	SD₁₆	semidirect
139. C₈×C₁₃	C₂×SD₁₆	semidirect
214. C₂×C₁₃	#1688	semidirect
246. C₂×C₈	#6105	semidirect
259. C₁₃	C₂×#1688	semidirect
292. C₈	C₂×#6105	semidirect
311. C₂	#13114317	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×#11718736	2
C₂×#11718340	2
#13114317	2
C₂×#11718049	2
C₂×#11718136	2
C₂×C₁₃×#493	2
C₂×#1688	13

Subgroups — C2×#13114317 #15845516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114317 #15845516