Subgroups — C₂×#13117045 #16017698

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718777	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	#13117045	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718097	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718096	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718095	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718095	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	—	2	1	no	—
11	C₂×#88178	—	2	1	no	—
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	#11718777	—	2	2	no	—
16	#11718777	C₂²	1	4	yes	—
17	#11718767	—	2	2	no	—
18	#11718767	C₂²	1	4	yes	—
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88058	C₂²	1	1	yes	characteristic
22	C₂×#88057	—	2	1	no	—
23	C₂×#88056	—	2	1	no	—
24	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	#11718097	C₂²	1	4	yes	complemented
26	#11718096	C₂²	1	4	yes	complemented
27	#11718095	C₂²	1	4	yes	complemented
28	#11718095	C₂²	1	4	yes	complemented
29	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×Dic₂₆	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
38	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
39	#88180	—	2	4	no	—
40	#88180	—	4	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	1	no	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×C₄⋊Dic₁₃	—	2	1	no	—
44	#88178	—	2	4	no	—
45	#88178	—	4	2	no	—
46	C₂×#6101	—	4	1	no	—
47	C₂×C₄×Dic₁₃	—	4	1	no	—
48	C₂×Dic₅₂	—	2	2	no	—
49	C₂×Dic₅₂	—	2	2	no	—
50	C₂×Dic₅₂	—	2	1	no	—
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	C₂×C₁₃×C₁₆	—	2	2	no	—
53	C₂×C₁₃×C₁₆	—	2	2	no	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	#88058	—	2	1	no	—
57	#88058	—	2	1	no	—
58	#88058	—	2	1	no	—
59	#88058	C₂³	1	2	yes	—
60	#88058	C₂³	1	2	yes	—
61	#88058	C₂³	1	2	yes	—
62	#88057	—	2	4	no	—
63	#88056	—	2	4	no	—
64	C₂×#2237	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	D₈	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	4	1	no	—
73	C₂²×Dic₁₃	—	4	1	no	—
74	C₂²×Dic₁₃	—	4	1	no	—
75	C₂²×Dic₁₃	—	4	1	no	—
76	C₂×Dic₂₆	—	2	2	no	—
77	C₂×Dic₂₆	—	4	2	no	—
78	C₂×Dic₂₆	—	4	2	no	—
79	C₂×Dic₂₆	—	4	1	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	—	4	4	no	—
90	C₂²⋊Dic₁₃	—	4	4	no	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	4	1	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	2	no	—
100	Dic₅₂	—	2	2	no	—
101	Dic₅₂	—	4	2	no	—
102	C₁₃×C₁₆	—	2	2	no	—
103	C₁₃×C₁₆	—	2	2	no	—
104	C₂×#467	—	13	1	no	—
105	C₂×#427	—	13	1	no	—
106	#2237	—	13	8	no	—
107	C₂×#367	—	13	1	no	—
108	C₂×#366	—	13	1	no	—
109	C₂×#358	—	13	1	no	—
110	C₂×#358	—	13	1	no	—
111	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	C₂×D₈	1	2	yes	—
116	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	#123	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#137	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	4	2	no	—
128	C₂×Dic₁₃	—	4	2	no	—
129	C₂×Dic₁₃	—	4	2	no	—
130	C₂×Dic₁₃	—	8	2	no	—
131	C₂×Dic₁₃	—	8	2	no	—
132	C₂×Dic₁₃	—	8	2	no	—
133	C₂×Dic₁₃	—	8	2	no	—
134	Dic₂₆	—	4	2	no	—
135	Dic₂₆	—	4	2	no	—
136	Dic₂₆	—	8	2	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	#123	1	2	yes	—
139	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₁₆	—	13	1	no	—
142	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₂²⋊Q₈	—	26	1	no	—
144	C₂×#117	—	26	1	no	—
145	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
146	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
147	#467	—	13	4	no	—
148	#467	—	26	2	no	—
149	#427	—	13	4	no	—
150	#427	—	26	2	no	—
151	C₂×#107	—	13	1	no	—
152	C₂×#107	—	13	1	no	—
153	C₂×#107	—	13	1	no	—
154	C₂×#106	—	26	1	no	—
155	C₂×#103	—	26	1	no	—
156	#367	—	13	4	no	—
157	#366	—	13	4	no	—
158	#358	—	13	4	no	—
159	#358	—	13	4	no	—
160	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	4	4	no	—
165	C₂²×C₁₃	#480	1	2	yes	—
166	C₂²×C₁₃	#508	1	2	yes	—
167	C₂²×C₁₃	#510	1	2	yes	—
168	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	#480	1	2	yes	—
171	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
173	Dic₁₃	—	8	2	no	—
174	Dic₁₃	—	8	2	no	—
175	Dic₁₃	—	8	2	no	—
176	Dic₁₃	—	8	2	no	—
177	C₂²×Q₈	—	26	1	no	—
178	C₂³×C₄	D₅₂	1	1	yes	characteristic
179	C₂×Q₁₆	—	13	2	no	—
180	C₂×Q₁₆	—	26	1	no	—
181	C₂×Q₁₆	—	26	2	no	—
182	C₂×Q₁₆	—	26	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	D₅₂	1	1	yes	characteristic
186	C₂²×C₈	D₅₂	1	2	yes	—
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂²⋊Q₈	—	26	4	no	—
190	C₂²⋊Q₈	—	52	2	no	—
191	#117	—	26	4	no	—
192	#117	—	52	2	no	—
193	C₂×C₄⋊C₄	—	26	1	no	—
194	C₂×C₄⋊C₄	—	26	1	no	—
195	C₂×C₄⋊C₄	—	26	1	no	—
196	C₂×C₄⋊C₄	—	52	1	no	—
197	C₂×C₂²⋊C₄	—	52	1	no	—
198	C₂×C₂²⋊C₄	—	52	1	no	—
199	C₂×C₄²	—	52	1	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
203	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
204	#107	—	13	2	no	—
205	#107	—	13	2	no	—
206	#107	—	13	2	no	—
207	#107	—	26	1	no	—
208	#107	—	26	1	no	—
209	#107	—	26	1	no	—
210	#106	—	26	4	no	—
211	#103	—	26	4	no	—
212	C₂×C₁₃	—	4	2	no	—
213	C₂×C₁₃	#2237	1	4	yes	complemented
214	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
217	C₂⁴	D₁₀₄	1	1	yes	characteristic
218	C₂×Q₈	—	26	2	no	—
219	C₂×Q₈	—	52	1	no	—
220	C₂×Q₈	—	52	2	no	—
221	C₂×Q₈	—	52	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	52	1	no	—
225	C₂²×C₄	—	52	1	no	—
226	C₂²×C₄	—	52	1	no	—
227	C₂²×C₄	—	52	1	no	—
228	C₂²×C₄	D₁₀₄	1	2	yes	—
229	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₅₂	1	2	yes	—
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	Q₁₆	—	26	2	no	—
233	Q₁₆	—	26	2	no	—
234	Q₁₆	—	52	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂×D₅₂	1	2	yes	—
240	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
241	C₂×C₈	#6129	1	2	yes	—
242	C₂×C₈	#6129	1	2	yes	—
243	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	26	2	no	—
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₄⋊C₄	—	52	1	no	—
248	C₄⋊C₄	—	52	1	no	—
249	C₄⋊C₄	—	52	1	no	—
250	C₄⋊C₄	—	52	4	no	—
251	C₂²⋊C₄	—	52	4	no	—
252	C₂²⋊C₄	—	52	4	no	—
253	C₄²	—	52	4	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₁₃	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×D₁₀₄	1	2	yes	—
261	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
262	Q₈	—	52	2	no	—
263	Q₈	—	52	2	no	—
264	Q₈	—	104	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	52	2	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	52	2	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	104	2	no	—
274	C₂×C₄	—	104	2	no	—
275	C₂×C₄	—	104	2	no	—
276	C₂×C₄	—	104	2	no	—
277	C₂×C₄	#88140	1	2	yes	—
278	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
279	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
280	C₂×C₄	#88163	1	2	yes	—
281	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#88140	1	2	yes	—
284	C₈	C₂×#6129	1	1	yes	characteristic
285	C₈	C₂×#6129	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	#11718397	1	2	yes	—
291	C₂²	#11718565	1	2	yes	—
292	C₂²	#11718570	1	2	yes	—
293	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	104	2	no	—
296	C₄	—	104	2	no	—
297	C₄	—	104	2	no	—
298	C₄	—	104	2	no	—
299	C₄	#11718397	1	2	yes	—
300	C₄	C₂×#88140	1	1	yes	characteristic
301	C₄	C₂×#88163	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#13117045	1	4	yes	direct product factor
304	C₂	C₂×#11718397	1	1	yes	characteristic
305	C₂	C₂×#11718565	1	1	yes	characteristic
306	C₂	C₂×#11718570	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
hypercenter	C₂×C₂²⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2237
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #13117045	C₂	direct (N × H)
4. C₂×#11718097	C₂	semidirect
5. C₂×#11718096	C₂	semidirect
6. C₂×#11718095	C₂	semidirect
7. C₂×#11718095	C₂	semidirect
25. #11718097	C₂²	semidirect
26. #11718096	C₂²	semidirect
27. #11718095	C₂²	semidirect
28. #11718095	C₂²	semidirect
213. C₂×C₁₃	#2237	semidirect
256. C₁₃	C₂×#2237	semidirect
303. C₂	#13117045	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×#11718777	2
C₂×#11718767	2
#13117045	2
C₂×#11718097	2
C₂×#11718096	2
C₂×#11718095	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2237	13

Subgroups — C2×#13117045 #16017698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117045 #16017698