Subgroups — C₂×#13114201 #15845154

Order 3328.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718668	C₂	1	2	yes	maximal
3	#13114201	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718050	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718074	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	—	2	1	no	—
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₂	—	2	1	no	—
10	C₂²×Dic₅₂	—	2	1	no	—
11	C₂×#88123	C₂²	1	2	yes	—
12	C₈×D₄×C₁₃	—	2	2	no	—
13	C₈×D₄×C₁₃	C₂²	1	4	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88072	—	2	2	no	—
18	#11718668	—	2	4	no	—
19	#11718668	C₂²	1	8	yes	—
20	C₂×#88047	—	2	2	no	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	#11718360	—	2	2	no	—
23	#11718360	C₂²	1	4	yes	complemented
24	#11718074	C₂²	1	8	yes	complemented
25	#11718050	C₂²	1	4	yes	complemented
26	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	—	2	1	no	—
28	C₂²×Dic₂₆	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂²×C₈×C₁₃	—	2	4	no	—
31	C₂²×C₈×C₁₃	—	4	4	no	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₄×D₄×C₁₃	—	2	1	no	—
35	C₄×D₄×C₁₃	—	2	1	no	—
36	C₄×D₄×C₁₃	D₄	1	2	yes	—
37	C₄×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₂×Dic₅₂	—	2	4	no	—
45	C₂×Dic₅₂	—	2	4	no	—
46	C₂×Dic₅₂	—	4	4	no	—
47	C₂×Dic₅₂	—	4	4	no	—
48	#88123	—	2	2	no	—
49	#88123	—	2	2	no	—
50	#88123	—	2	2	no	—
51	#88123	C₂³	1	4	yes	—
52	C₁₃×C₄⋊C₈	—	2	1	no	—
53	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
54	C₁₃×C₂²⋊C₈	—	2	4	no	—
55	C₄×C₈×C₁₃	D₄	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	#88072	—	2	8	no	—
58	#88047	—	2	8	no	—
59	C₄⋊#1055	—	2	2	no	—
60	C₄⋊#1055	C₂³	1	4	yes	—
61	C₂×#1669	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	Q₁₆	1	4	yes	complemented
66	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	4	2	no	—
76	C₂×Dic₂₆	—	2	4	no	—
77	C₂×Dic₂₆	—	4	4	no	—
78	C₂×Dic₂₆	—	4	4	no	—
79	C₂×Dic₂₆	—	4	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	—	4	8	no	—
84	C₂×C₈×C₁₃	D₈	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₁₃×C₄⋊C₄	—	2	1	no	—
88	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
89	C₁₃×C₂²⋊C₄	—	2	2	no	—
90	C₁₃×C₂²⋊C₄	—	2	2	no	—
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	4	4	no	—
101	Dic₅₂	—	4	8	no	—
102	Dic₅₂	—	4	8	no	—
103	C₂×#494	—	13	1	no	—
104	C₂×#474	—	13	2	no	—
105	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
106	#1669	—	13	8	no	—
107	C₂×#335	—	13	1	no	—
108	C₂×#332	—	13	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	—	2	4	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	#121	1	1	yes	characteristic
114	D₄×C₁₃	—	2	2	no	—
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	C₂×Q₁₆	1	4	yes	complemented
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	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×D₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₁₃	#135	1	2	yes	—
129	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	8	4	no	—
135	Dic₂₆	—	4	4	no	—
136	Dic₂₆	—	4	4	no	—
137	Dic₂₆	—	8	4	no	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	—	4	2	no	—
140	C₈×C₁₃	C₂×D₈	1	4	yes	—
141	#1055	—	4	4	no	—
142	C₂²×Q₁₆	—	26	1	no	—
143	C₂²×Q₁₆	—	26	1	no	—
144	C₂³×C₈	—	2	1	no	—
145	C₂×C₄⋊Q₈	—	13	2	no	—
146	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
147	#494	—	13	4	no	—
148	#494	—	26	2	no	—
149	#474	—	13	8	no	—
150	#474	—	26	4	no	—
151	C₈×D₄	—	2	2	no	—
152	C₈×D₄	C₂×D₁₃	1	4	yes	—
153	C₂×#107	—	26	2	no	—
154	C₂×C₄⋊C₈	—	13	2	no	—
155	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₂×#102	—	26	2	no	—
157	C₂×C₂²⋊C₈	—	2	1	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	#335	—	13	4	no	—
160	#332	—	13	8	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	4	8	no	—
163	C₂²×C₁₃	—	4	2	no	—
164	C₂²×C₁₃	—	4	2	no	—
165	C₂²×C₁₃	#459	1	2	yes	—
166	C₂²×C₁₃	#467	1	2	yes	—
167	C₂²×C₁₃	#473	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	—	4	2	no	—
172	C₄×C₁₃	#459	1	2	yes	—
173	C₄×C₁₃	#467	1	2	yes	—
174	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
175	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
176	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
177	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	Dic₁₃	—	8	4	no	—
179	C₂²×Q₈	—	26	2	no	—
180	C₂²×D₄	D₅₂	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂×Q₁₆	—	26	4	no	—
183	C₂×Q₁₆	—	26	4	no	—
184	C₂×Q₁₆	—	52	4	no	—
185	C₂×Q₁₆	—	52	4	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	4	4	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₄⋊Q₈	—	13	4	no	—
193	C₄⋊Q₈	—	26	2	no	—
194	C₄⋊Q₈	—	26	2	no	—
195	C₄⋊Q₈	—	26	2	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	D₅₂	1	2	yes	—
199	C₄×D₄	C₂²×D₁₃	1	2	yes	—
200	C₂×C₄⋊C₄	—	26	2	no	—
201	C₂×C₄⋊C₄	—	26	2	no	—
202	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
205	#107	—	26	8	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	13	4	no	—
208	C₄⋊C₈	—	26	2	no	—
209	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
210	#102	—	26	8	no	—
211	C₂²⋊C₈	—	2	4	no	—
212	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
213	C₄×C₈	C₂²×D₁₃	1	2	yes	—
214	C₂×C₁₃	—	4	4	no	—
215	C₂×C₁₃	#1669	1	4	yes	complemented
216	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#467	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
219	C₂⁴	—	2	1	no	—
220	C₂×Q₈	—	26	4	no	—
221	C₂×Q₈	—	52	2	no	—
222	C₂×Q₈	—	52	4	no	—
223	C₂×Q₈	—	52	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	Dic₅₂	1	4	yes	complemented
227	C₂×D₄	C₂×D₅₂	1	2	yes	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	52	2	no	—
234	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
235	C₂²×C₄	#6127	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
237	Q₁₆	—	52	8	no	—
238	Q₁₆	—	52	8	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	—	26	4	no	—
244	C₂×C₈	—	52	4	no	—
245	C₂×C₈	#6104	1	4	yes	—
246	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	26	4	no	—
250	C₄⋊C₄	—	26	4	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₄⋊C₄	—	52	2	no	—
253	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₄²	C₂×D₅₂	1	1	yes	characteristic
257	C₄²	#6127	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₄²	C₂³×D₁₃	1	1	yes	characteristic
260	C₁₃	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	#88184	1	1	yes	characteristic
266	Q₈	—	52	4	no	—
267	Q₈	—	52	4	no	—
268	Q₈	—	104	4	no	—
269	D₄	—	2	2	no	—
270	D₄	—	2	4	no	—
271	D₄	C₂×Dic₅₂	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	52	4	no	—
279	C₂×C₄	—	104	4	no	—
280	C₂×C₄	#88158	1	2	yes	—
281	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
282	C₂×C₄	#88184	1	1	yes	characteristic
283	C₂×C₄	#88184	1	1	yes	characteristic
284	C₂×C₄	#88184	1	1	yes	characteristic
285	C₂×C₄	C₂×#6104	1	2	yes	—
286	C₂×C₄	#88203	1	2	yes	—
287	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
288	C₂×C₄	C₂×#6127	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	52	4	no	—
293	C₈	C₂×#6104	1	4	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#11718676	1	2	yes	—
299	C₂²	#11718777	1	2	yes	—
300	C₂²	#11718804	1	2	yes	—
301	C₂²	C₂×#88184	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	104	4	no	—
306	C₄	#11718676	1	2	yes	—
307	C₄	#11718777	1	2	yes	—
308	C₄	C₂×#88158	1	1	yes	characteristic
309	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
310	C₄	C₂²×#6104	1	1	yes	characteristic
311	C₄	C₂×#88203	1	1	yes	characteristic
312	C₂	—	4	4	no	—
313	C₂	#13114201	1	4	yes	direct product factor
314	C₂	C₂×#11718676	1	1	yes	characteristic
315	C₂	C₂×#11718777	1	1	yes	characteristic
316	C₂	C₂×#11718804	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂²×D₄×C₁₃	D₄
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	C₂×#1669
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11718360	C₂	semidirect
3. #13114201	C₂	direct (N × H)
4. C₂×#11718050	C₂	semidirect
5. C₂×#11718074	C₂	semidirect
23. #11718360	C₂²	semidirect
24. #11718074	C₂²	semidirect
25. #11718050	C₂²	semidirect
65. C₂×D₄×C₁₃	Q₁₆	semidirect
116. D₄×C₁₃	C₂×Q₁₆	semidirect
215. C₂×C₁₃	#1669	semidirect
226. C₂×D₄	Dic₅₂	semidirect
260. C₁₃	C₂×#1669	semidirect
271. D₄	C₂×Dic₅₂	semidirect
313. C₂	#13114201	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×#11718360	2
C₂×#11718668	2
#13114201	2
C₂×#11718050	2
C₂×#11718074	2
C₂×C₈×D₄×C₁₃	2
C₂×#1669	13

Subgroups — C2×#13114201 #15845154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114201 #15845154