Subgroups — C₂×#13114743 #15861021

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88062	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
4	#13114743	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718150	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	—	2	2	no	—
13	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₂×#88062	C₄	1	16	yes	complemented
17	C₂×#88062	C₂²	1	8	yes	—
18	C₄×#6099	—	2	2	no	—
19	C₄×#6099	C₂²	1	4	yes	—
20	C₂×#88043	—	2	1	no	—
21	C₂×#88043	C₄	1	4	yes	—
22	C₂×#88043	C₄	1	4	yes	—
23	C₂×#88043	C₂²	1	4	yes	—
24	C₂×C₄×#1055	—	2	1	no	—
25	#11718150	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	4	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	4	no	—
42	C₂×#6099	C₂×C₄	1	8	yes	—
43	C₂×#6099	C₂³	1	4	yes	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	—	2	4	no	—
46	C₂²×#1055	C₂×C₄	1	2	yes	—
47	C₂²×#1055	C₂×C₄	1	2	yes	—
48	C₂²×#1055	C₂×C₄	1	8	yes	—
49	C₂²×#1055	C₂×C₄	1	2	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	C₁₃×#97	—	2	4	no	—
54	C₄×C₈×C₁₃	—	2	4	no	—
55	#88062	C₂×C₄	1	32	yes	complemented
56	#88043	—	2	4	no	—
57	#88043	C₂×C₄	1	16	yes	—
58	C₄×#1055	—	2	4	no	—
59	C₂×#1778	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×M₁₆	—	2	8	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₁₃	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	C₂×D₄	1	2	yes	—
84	C₄²×C₁₃	C₂×D₄	1	2	yes	—
85	#6099	—	2	8	no	—
86	#6099	—	2	4	no	—
87	#6099	C₂²×C₄	1	8	yes	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	8	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	C₄²	1	8	yes	—
94	C₂×#1055	C₄²	1	8	yes	—
95	C₂×#1055	C₂²×C₄	1	4	yes	—
96	C₂×#1055	C₂²×C₄	1	4	yes	—
97	C₂×#1055	C₂²×C₄	1	8	yes	—
98	C₂²×#108	—	13	2	no	—
99	C₂×C₄×M₁₆	—	13	1	no	—
100	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
101	C₂²×#97	—	13	1	no	—
102	#1778	—	13	8	no	—
103	C₂×#341	—	13	2	no	—
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
108	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#117	1	2	yes	—
120	C₂×C₄×C₁₃	#117	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
123	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₈×C₁₃	—	2	8	no	—
129	#1055	—	2	4	no	—
130	#1055	—	2	8	no	—
131	#1055	C₂×C₄²	1	8	yes	—
132	C₂²×M₁₆	—	13	2	no	—
133	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
134	C₂³×C₈	—	13	1	no	—
135	C₂³×C₈	—	13	1	no	—
136	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#108	—	13	16	no	—
138	C₂×#108	—	13	8	no	—
139	C₄×M₁₆	—	2	2	no	—
140	C₄×M₁₆	—	13	4	no	—
141	C₄×M₁₆	—	26	2	no	—
142	C₄×M₁₆	C₂×D₁₃	1	4	yes	—
143	C₂×#97	—	2	1	no	—
144	C₂×#97	—	13	4	no	—
145	C₂×#97	—	13	4	no	—
146	C₂×#97	—	13	4	no	—
147	C₂×#97	—	26	1	no	—
148	C₂×C₄×C₈	—	2	1	no	—
149	C₂×C₄×C₈	—	26	1	no	—
150	#341	—	13	8	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	8	no	—
154	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
155	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
156	C₂²×C₁₃	#430	1	4	yes	—
157	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
165	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
166	C₂³×C₄	Dic₂₆	1	2	yes	—
167	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	13	8	no	—
171	C₂×M₁₆	—	13	4	no	—
172	C₂×M₁₆	—	26	4	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	C₄×D₁₃	1	8	yes	—
175	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	13	8	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	26	4	no	—
185	C₂²×C₈	—	26	2	no	—
186	C₂×C₄²	Dic₂₆	1	2	yes	—
187	C₂×C₄²	C₄×D₁₃	1	2	yes	—
188	C₂×C₄²	C₄×D₁₃	1	2	yes	—
189	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
193	#108	—	13	32	no	—
194	#97	—	2	4	no	—
195	#97	—	13	16	no	—
196	#97	—	26	4	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	26	4	no	—
199	C₂×C₁₃	—	2	2	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	#1778	1	4	yes	complemented
202	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₁₃	C₂×#430	1	2	yes	—
204	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	#6101	1	2	yes	—
207	C₂²×C₄	#6101	1	2	yes	—
208	C₂²×C₄	#6101	1	4	yes	—
209	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
210	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	#6127	1	4	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
218	M₁₆	—	2	8	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	13	8	no	—
221	M₁₆	—	26	8	no	—
222	M₁₆	—	26	4	no	—
223	M₁₆	C₂×C₄×D₁₃	1	8	yes	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	13	8	no	—
227	C₂×C₈	—	13	8	no	—
228	C₂×C₈	—	13	4	no	—
229	C₂×C₈	—	13	8	no	—
230	C₂×C₈	—	13	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	8	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	4	no	—
236	C₄²	#6101	1	4	yes	—
237	C₄²	#6101	1	2	yes	—
238	C₄²	#6101	1	2	yes	—
239	C₄²	C₂×C₄×D₁₃	1	4	yes	—
240	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
241	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
242	C₁₃	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	2	4	no	—
244	C₂³	C₄×Dic₂₆	1	4	yes	—
245	C₂³	C₂×#6101	1	1	yes	characteristic
246	C₂³	C₂×#6101	1	1	yes	characteristic
247	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
253	C₂×C₄	C₄²×D₁₃	1	2	yes	—
254	C₂×C₄	C₄²×D₁₃	1	2	yes	—
255	C₂×C₄	#88126	1	2	yes	—
256	C₂×C₄	#88126	1	2	yes	—
257	C₂×C₄	C₂×#6101	1	4	yes	—
258	C₂×C₄	C₂×#6101	1	4	yes	—
259	C₂×C₄	C₂×#6101	1	1	yes	characteristic
260	C₂×C₄	C₂×#6101	1	1	yes	characteristic
261	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
262	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
265	C₂×C₄	C₂×#6127	1	2	yes	—
266	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
267	C₈	—	2	8	no	—
268	C₈	—	13	8	no	—
269	C₈	—	26	4	no	—
270	C₈	—	26	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	8	no	—
274	C₂²	C₄×#6101	1	2	yes	—
275	C₂²	C₄×#6101	1	2	yes	—
276	C₂²	#11718785	1	4	yes	—
277	C₂²	C₂×C₄×Dic₂₆	1	2	yes	—
278	C₂²	C₂²×#6101	1	1	yes	characteristic
279	C₄	—	2	4	no	—
280	C₄	—	2	4	no	—
281	C₄	C₄×#6101	1	2	yes	—
282	C₄	C₄×#6101	1	2	yes	—
283	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
284	C₄	C₂×#88126	1	1	yes	characteristic
285	C₄	C₂×C₄×C₂²⋊D₁₃	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	2	2	no	—
288	C₂	#13114743	1	4	yes	direct product factor
289	C₂	C₂×C₄×#6101	1	1	yes	characteristic
290	C₂	C₂×#11718785	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	C₂×#1778
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13114743	C₂	direct (N × H)
16. C₂×#88062	C₄	semidirect
55. #88062	C₂×C₄	semidirect
201. C₂×C₁₃	#1778	semidirect
242. C₁₃	C₂×#1778	semidirect
288. C₂	#13114743	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×#88062	2
C₂×C₄×#6099	2
C₂²×#88043	2
#13114743	2
C₂×#11718150	2
C₂×C₄×C₁₃×M₁₆	2
C₂×#1778	13

Subgroups — C2×#13114743 #15861021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114743 #15861021