Subgroups — #15768706

Order 3328.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	2	yes	maximal
2	C₂×#11718116	C₂	1	4	yes	maximal
3	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	4	yes	—
9	C₄³×C₁₃	—	2	4	no	—
10	C₂×C₄⋊#1055	—	2	4	no	—
11	C₂×C₄⋊#1055	—	2	4	no	—
12	C₂×C₄⋊#1055	C₄	1	8	yes	complemented
13	#11718116	—	2	8	no	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	—	2	4	no	—
19	C₂×C₄²×C₁₃	—	2	4	no	—
20	C₂×C₄²×C₁₃	—	2	8	no	—
21	C₂×C₄²×C₁₃	—	2	4	no	—
22	C₂×C₄²×C₁₃	—	2	4	no	—
23	C₂×C₄²×C₁₃	—	2	4	no	—
24	C₂×C₄²×C₁₃	C₈	1	8	yes	complemented
25	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃	D₄	1	4	yes	—
27	C₂×C₄²×C₁₃	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃	D₄	1	4	yes	—
29	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
30	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
31	C₂²×#1055	—	2	4	no	—
32	C₂²×#1055	—	2	4	no	—
33	C₂²×#1055	—	2	8	no	—
34	C₂²×#1055	—	2	8	no	—
35	C₂²×#1055	C₂×C₄	1	8	yes	—
36	C₄⋊#1055	—	2	16	no	—
37	#10832	—	13	1	no	maximal, Sylow
38	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	—	2	4	no	—
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
54	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	4	yes	—
62	C₄²×C₁₃	—	2	16	no	—
63	C₄²×C₁₃	—	2	16	no	—
64	C₄²×C₁₃	—	2	8	no	—
65	C₄²×C₁₃	—	2	8	no	—
66	C₄²×C₁₃	C₂²⋊C₄	1	8	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
68	C₂×#1055	—	2	8	no	—
69	C₂×#1055	—	2	16	no	—
70	C₂×#1055	—	4	8	no	—
71	C₂×#1055	—	4	4	no	—
72	C₂×#1055	—	4	4	no	—
73	C₂²×C₄⋊C₈	—	13	2	no	—
74	C₂×C₄³	D₁₃	1	1	yes	characteristic
75	C₂×#336	—	13	4	no	—
76	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
77	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₁₃	#117	1	2	yes	—
81	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
83	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
84	C₂×C₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	8	no	—
89	C₂×C₄×C₁₃	—	2	8	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	8	no	—
92	C₂×C₄×C₁₃	—	2	8	no	—
93	C₂×C₄×C₁₃	#95	1	4	yes	—
94	C₂×C₄×C₁₃	#95	1	4	yes	—
95	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
96	C₂×C₄×C₁₃	#97	1	4	yes	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	8	yes	—
98	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
99	C₂×C₄×C₁₃	C₄⋊C₈	1	16	yes	complemented
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
104	C₂×C₄×C₁₃	#121	1	4	yes	—
105	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₁₃	#124	1	2	yes	—
107	C₂×C₄×C₁₃	#125	1	2	yes	—
108	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
111	#1055	—	4	8	no	—
112	C₂³×C₈	—	13	4	no	—
113	C₂²×C₄²	Dic₁₃	1	2	yes	—
114	C₂²×C₄²	Dic₁₃	1	2	yes	—
115	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
116	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	13	8	no	—
118	C₂×C₄⋊C₈	—	26	4	no	—
119	C₂×C₄⋊C₈	—	26	4	no	—
120	C₄³	—	2	4	no	—
121	#336	—	26	8	no	—
122	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
123	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
124	C₂²×C₁₃	#380	1	2	yes	—
125	C₂²×C₁₃	#382	1	1	yes	characteristic
126	C₂²×C₁₃	#384	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
129	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
130	C₂²×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
131	C₂²×C₁₃	#407	1	2	yes	—
132	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
133	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
134	C₂²×C₁₃	#423	1	4	yes	—
135	C₂²×C₁₃	#423	1	2	yes	—
136	C₂²×C₁₃	#431	1	4	yes	—
137	C₂²×C₁₃	#432	1	4	yes	—
138	C₄×C₁₃	—	2	16	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	#336	1	8	yes	complemented
141	C₄×C₁₃	#389	1	4	yes	—
142	C₄×C₁₃	#390	1	2	yes	—
143	C₄×C₁₃	#391	1	2	yes	—
144	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
145	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
147	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
148	C₂²×C₈	—	13	8	no	—
149	C₂²×C₈	—	26	8	no	—
150	C₂²×C₈	—	26	4	no	—
151	C₂²×C₈	—	26	8	no	—
152	C₂²×C₈	—	26	4	no	—
153	C₂×C₄²	—	2	8	no	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	—	2	4	no	—
156	C₂×C₄²	—	2	4	no	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	#1055	1	8	yes	complemented
160	C₂×C₄²	Dic₂₆	1	4	yes	—
161	C₂×C₄²	Dic₂₆	1	2	yes	—
162	C₂×C₄²	C₄×D₁₃	1	4	yes	—
163	C₂×C₄²	D₅₂	1	4	yes	—
164	C₂×C₄²	D₅₂	1	2	yes	—
165	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
166	C₄⋊C₈	—	26	16	no	—
167	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
169	C₂×C₁₃	#1728	1	1	yes	characteristic
170	C₂×C₁₃	C₄×C₄⋊C₈	1	2	yes	—
171	C₂×C₁₃	#1767	1	2	yes	—
172	C₂×C₁₃	#1797	1	4	yes	—
173	C₂×C₁₃	#1831	1	2	yes	—
174	C₂×C₁₃	#1842	1	2	yes	—
175	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂×#1055	1	8	yes	—
182	C₂²×C₄	C₂×#1055	1	4	yes	—
183	C₂²×C₄	#6099	1	8	yes	—
184	C₂²×C₄	#6099	1	4	yes	—
185	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
186	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
189	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
191	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
192	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
193	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
195	C₂²×C₄	C₂×D₅₂	1	2	yes	—
196	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
197	C₂²×C₄	#6127	1	4	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
199	C₂×C₈	—	26	16	no	—
200	C₂×C₈	—	26	8	no	—
201	C₂×C₈	—	52	4	no	—
202	C₂×C₈	—	52	4	no	—
203	C₂×C₈	—	52	8	no	—
204	C₄²	—	2	16	no	—
205	C₄²	—	2	16	no	—
206	C₄²	—	2	8	no	—
207	C₄²	—	2	8	no	—
208	C₄²	#6101	1	8	yes	—
209	C₄²	#6103	1	8	yes	—
210	C₁₃	#10832	1	1	yes	characteristic, complemented, Sylow
211	C₂³	C₂²×#1055	1	2	yes	—
212	C₂³	C₂×#6099	1	2	yes	—
213	C₂³	C₂×#6099	1	4	yes	—
214	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
217	C₂³	#88180	1	2	yes	—
218	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	C₄×#1055	1	4	yes	—
229	C₂×C₄	#88043	1	4	yes	—
230	C₂×C₄	C₄⋊#1055	1	8	yes	—
231	C₂×C₄	C₄⋊#1055	1	16	yes	complemented
232	C₂×C₄	C₂²⋊#1055	1	8	yes	—
233	C₂×C₄	#88071	1	4	yes	—
234	C₂×C₄	#88071	1	4	yes	—
235	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
236	C₂×C₄	#88123	1	2	yes	—
237	C₂×C₄	#88123	1	1	yes	characteristic
238	C₂×C₄	#88124	1	2	yes	—
239	C₂×C₄	C₄×D₅₂	1	4	yes	—
240	C₂×C₄	#88128	1	1	yes	characteristic
241	C₂×C₄	#88129	1	2	yes	—
242	C₂×C₄	C₂×#6101	1	2	yes	—
243	C₂×C₄	#88178	1	4	yes	—
244	C₂×C₄	C₂×#6103	1	2	yes	—
245	C₂×C₄	#88184	1	4	yes	—
246	C₈	—	52	8	no	—
247	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
248	C₂²	C₂×#88043	1	1	yes	characteristic
249	C₂²	C₄×#6099	1	2	yes	—
250	C₂²	C₂×C₄⋊#1055	1	4	yes	—
251	C₂²	C₂×C₄⋊#1055	1	2	yes	—
252	C₂²	#11718585	1	4	yes	—
253	C₂²	#11718585	1	2	yes	—
254	C₂²	#11718586	1	4	yes	—
255	C₂²	#11718587	1	4	yes	—
256	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
257	C₂²	#11718596	1	1	yes	characteristic
258	C₂²	#11718597	1	1	yes	characteristic
259	C₂²	C₂×C₂²⋊#1055	1	2	yes	—
260	C₂²	#11718868	1	2	yes	—
261	C₂²	C₂×#88071	1	1	yes	characteristic
262	C₂²	#11718873	1	2	yes	—
263	C₄	—	2	4	no	—
264	C₄	—	2	16	no	—
265	C₄	#11718116	1	8	yes	complemented
266	C₄	#11718589	1	4	yes	—
267	C₄	#11718590	1	2	yes	—
268	C₄	#11718600	1	2	yes	—
269	C₂	#13113066	1	1	yes	characteristic
270	C₂	C₂×#11718116	1	1	yes	characteristic
271	C₂	#13114530	1	1	yes	characteristic
272	C₂	C₄×C₄⋊#1055	1	2	yes	—
273	C₂	#13114689	1	2	yes	—
274	C₂	#13114816	1	4	yes	—
275	C₂	#13114953	1	2	yes	—
276	C₂	#13114988	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₄³×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	#10832
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
12. C₂×C₄⋊#1055	C₄	semidirect
24. C₂×C₄²×C₁₃	C₈	semidirect
99. C₂×C₄×C₁₃	C₄⋊C₈	semidirect
140. C₄×C₁₃	#336	semidirect
159. C₂×C₄²	#1055	semidirect
210. C₁₃	#10832	semidirect
231. C₂×C₄	C₄⋊#1055	semidirect
265. C₄	#11718116	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊#1055	2
C₂×#11718116	2
C₂×C₄³×C₁₃	2
#10832	13