Subgroups — C₂×#13112718 #15752967

Order 3328.

830 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718597	C₂	1	1	yes	maximal, characteristic
3	#13112718	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	—	2	1	no	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	C₄	1	2	yes	—
8	C₂³×#1055	—	2	1	no	—
9	C₂×C₁₃×#97	—	2	1	no	—
10	C₂×C₁₃×#97	—	2	2	no	—
11	C₂×C₁₃×#97	C₄	1	4	yes	complemented
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×#88071	—	2	1	no	—
14	C₂×#88071	—	2	1	no	—
15	#11718597	—	2	1	no	—
16	#11718597	—	2	1	no	—
17	#11718597	C₄	1	2	yes	—
18	#11718597	C₂²	1	2	yes	—
19	C₂×#88043	—	2	2	no	—
20	C₂×#88043	—	2	1	no	—
21	C₂×#88043	C₄	1	4	yes	complemented
22	C₂×#88043	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	8	no	—
28	C₂²×C₈×C₁₃	—	2	4	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	—	4	4	no	—
40	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
41	C₂²×#1055	—	2	8	no	—
42	C₂²×#1055	—	2	4	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₁₃×#97	—	2	4	no	—
45	C₁₃×#97	—	2	2	no	—
46	C₁₃×#97	C₂×C₄	1	4	yes	complemented
47	#88071	—	2	2	no	—
48	#88071	—	2	2	no	—
49	#88071	—	2	2	no	—
50	#88071	—	2	2	no	—
51	#88043	—	2	4	no	—
52	#88043	—	2	2	no	—
53	#88043	C₂×C₄	1	4	yes	complemented
54	C₂×#1295	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	—	4	4	no	—
74	C₂²×Dic₁₃	—	4	4	no	—
75	C₂²×Dic₁₃	—	4	4	no	—
76	C₂×C₈×C₁₃	—	2	16	no	—
77	C₂×C₈×C₁₃	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	C₄²	1	2	yes	—
80	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
89	C₄⋊Dic₁₃	—	4	4	no	—
90	C₄⋊Dic₁₃	—	4	4	no	—
91	C₂×#1055	—	2	16	no	—
92	C₂×#1055	—	2	8	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂²×#97	—	13	1	no	—
95	C₂²×#97	D₁₃	1	1	yes	characteristic
96	C₂×#382	—	13	1	no	—
97	#1295	—	13	4	no	—
98	C₂³×C₁₃	#95	1	2	yes	—
99	C₂³×C₁₃	#100	1	2	yes	—
100	C₂³×C₁₃	#101	1	2	yes	—
101	C₂³×C₁₃	#104	1	4	yes	—
102	C₂³×C₁₃	#104	1	4	yes	—
103	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	#95	1	2	yes	—
119	C₂×C₄×C₁₃	#95	1	2	yes	—
120	C₂×C₄×C₁₃	#95	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	8	8	no	—
131	C₈×C₁₃	—	2	8	no	—
132	#1055	—	2	8	no	—
133	C₂³×C₈	—	2	1	no	—
134	C₂³×C₈	—	26	1	no	—
135	C₂²×C₄⋊C₄	—	13	2	no	—
136	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#97	—	2	1	no	—
138	C₂×#97	—	2	2	no	—
139	C₂×#97	—	13	4	no	—
140	C₂×#97	—	13	2	no	—
141	C₂×#97	—	26	1	no	—
142	C₂×#97	—	26	2	no	—
143	C₂×#97	Dic₁₃	1	4	yes	complemented
144	C₂×#97	C₂×D₁₃	1	2	yes	—
145	#382	—	13	2	no	—
146	#382	—	13	2	no	—
147	#382	—	26	1	no	—
148	#382	—	26	1	no	—
149	C₂×#95	—	26	1	no	—
150	C₂×#95	—	26	1	no	—
151	C₂²×C₁₃	#329	1	8	yes	—
152	C₂²×C₁₃	#330	1	8	yes	—
153	C₂²×C₁₃	#339	1	4	yes	—
154	C₂²×C₁₃	#339	1	4	yes	—
155	C₂²×C₁₃	#343	1	4	yes	—
156	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#104	1	2	yes	—
160	C₂²×C₁₃	C₂×#104	1	2	yes	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	2	1	no	—
170	Dic₁₃	—	8	4	no	—
171	C₂³×C₄	—	26	2	no	—
172	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
173	C₂³×C₄	D₅₂	1	1	yes	characteristic
174	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂²×C₈	—	2	8	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	26	8	no	—
179	C₂²×C₈	—	26	4	no	—
180	C₂²×C₈	—	26	2	no	—
181	C₂×C₄⋊C₄	—	13	4	no	—
182	C₂×C₄⋊C₄	—	26	2	no	—
183	C₂×C₄⋊C₄	—	52	4	no	—
184	C₂×C₄²	Dic₂₆	1	2	yes	—
185	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₁₃	1	2	yes	—
187	C₂×C₄²	D₅₂	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
189	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
191	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
192	#97	—	2	4	no	—
193	#97	—	2	2	no	—
194	#97	—	13	4	no	—
195	#97	—	26	4	no	—
196	#97	—	26	2	no	—
197	#97	C₂×Dic₁₃	1	4	yes	complemented
198	#95	—	26	2	no	—
199	#95	—	26	2	no	—
200	#95	—	26	2	no	—
201	#95	—	26	2	no	—
202	C₂×C₁₃	#1295	1	8	yes	complemented
203	C₂×C₁₃	C₂×#329	1	2	yes	—
204	C₂×C₁₃	C₂×#330	1	2	yes	—
205	C₂×C₁₃	C₂×#339	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#339	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#343	1	1	yes	characteristic
208	C₂⁴	#6103	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	26	4	no	—
214	C₂²×C₄	—	52	4	no	—
215	C₂²×C₄	—	52	4	no	—
216	C₂²×C₄	—	52	4	no	—
217	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
218	C₂²×C₄	#6101	1	2	yes	—
219	C₂²×C₄	#6101	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
221	C₂²×C₄	#6103	1	1	yes	characteristic
222	C₂²×C₄	#6103	1	1	yes	characteristic
223	C₂²×C₄	#6103	1	1	yes	characteristic
224	C₂²×C₄	#6103	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	2	16	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	26	16	no	—
233	C₂×C₈	—	26	8	no	—
234	C₂×C₈	—	26	4	no	—
235	C₄⋊C₄	—	52	4	no	—
236	C₄⋊C₄	—	52	4	no	—
237	C₄²	C₄×Dic₁₃	1	2	yes	—
238	C₄²	#6101	1	2	yes	—
239	C₄²	#6101	1	2	yes	—
240	C₄²	C₄⋊Dic₁₃	1	2	yes	—
241	C₄²	#6103	1	2	yes	—
242	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
243	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
244	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₁₃	C₂×#1295	1	1	yes	characteristic, complemented, Sylow
248	C₂³	#88045	1	4	yes	—
249	C₂³	#88063	1	2	yes	—
250	C₂³	#88064	1	2	yes	—
251	C₂³	#88065	1	4	yes	—
252	C₂³	#88071	1	2	yes	—
253	C₂³	C₂×#6103	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	52	4	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	104	8	no	—
272	C₂×C₄	#88071	1	2	yes	—
273	C₂×C₄	#88071	1	2	yes	—
274	C₂×C₄	#88071	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#6101	1	1	yes	characteristic
277	C₂×C₄	C₂×#6101	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#6103	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
281	C₈	—	2	8	no	—
282	C₈	—	26	8	no	—
283	C₂²	#11718056	1	8	yes	—
284	C₂²	#11718057	1	8	yes	—
285	C₂²	#11718115	1	4	yes	—
286	C₂²	#11718146	1	4	yes	—
287	C₂²	#11718147	1	4	yes	—
288	C₂²	C₂×#88045	1	2	yes	—
289	C₂²	C₂×#88063	1	1	yes	characteristic
290	C₂²	C₂×#88064	1	1	yes	characteristic
291	C₂²	C₂×#88065	1	2	yes	—
292	C₂²	C₂×#88071	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	104	4	no	—
303	C₂	#13112718	1	8	yes	direct product factor
304	C₂	C₂×#11718056	1	2	yes	—
305	C₂	C₂×#11718057	1	2	yes	—
306	C₂	C₂×#11718115	1	1	yes	characteristic
307	C₂	C₂×#11718147	1	1	yes	characteristic
308	C₂	C₂×#11718146	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	C₂×#1295
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #13112718	C₂	direct (N × H)
11. C₂×C₁₃×#97	C₄	semidirect
21. C₂×#88043	C₄	semidirect
46. C₁₃×#97	C₂×C₄	semidirect
53. #88043	C₂×C₄	semidirect
143. C₂×#97	Dic₁₃	semidirect
197. #97	C₂×Dic₁₃	semidirect
202. C₂×C₁₃	#1295	semidirect
247. C₁₃	C₂×#1295	semidirect
303. C₂	#13112718	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×#88043	2
C₂×#11718597	2
#13112718	2
C₂²×C₁₃×#97	2
C₂×#1295	13

Subgroups — C2×#13112718 #15752967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13112718 #15752967