Subgroups — #15881822

Order 3328.

876 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13115554	C₂	1	3	yes	maximal
4	#13115552	C₂	1	3	yes	maximal
5	#13114952	C₂	1	3	yes	maximal
6	C₄×#88054	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₃	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
14	C₂×#88054	C₂²	1	3	yes	—
15	C₂×#88054	C₂²	1	3	yes	—
16	Q₈×#1055	—	2	4	no	—
17	C₄⋊C₄×Dic₁₃	C₄	1	3	yes	—
18	C₄⋊C₄×Dic₁₃	C₄	1	3	yes	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
20	C₄²×Dic₁₃	C₂²	1	3	yes	—
21	C₂×C₄⋊#1055	C₂²	1	3	yes	—
22	C₂×C₄×#1055	C₂²	1	3	yes	—
23	#11718142	C₂²	1	1	yes	characteristic
24	#11718142	C₂²	1	3	yes	—
25	#11718142	C₂²	1	6	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	D₄	1	4	yes	—
30	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
33	Q₈×Dic₁₃	—	2	4	no	—
34	Q₈×Dic₁₃	C₈	1	8	yes	complemented
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
36	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
42	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	3	yes	—
44	C₂²×#1055	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	—	2	6	no	—
46	C₄×C₈×C₁₃	—	2	6	no	—
47	#88071	C₂×C₄	1	6	yes	—
48	#88054	—	2	6	no	—
49	#88054	—	2	6	no	—
50	C₄⋊#1055	—	2	6	no	—
51	C₄×#1055	—	2	6	no	—
52	#22734	—	13	1	no	maximal, Sylow
53	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
54	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
55	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
60	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
61	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	3	no	—
66	C₂×C₈×C₁₃	—	2	3	no	—
67	C₂×C₈×C₁₃	—	2	3	no	—
68	C₂×C₈×C₁₃	#41	1	2	yes	—
69	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
70	C₄²×C₁₃	C₂×D₄	1	6	yes	—
71	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
72	C₄⋊Dic₁₃	—	2	6	no	—
73	C₄⋊Dic₁₃	C₂×C₈	1	12	yes	—
74	C₄×Dic₁₃	—	2	6	no	—
75	C₄×Dic₁₃	—	2	3	no	—
76	C₄×Dic₁₃	—	2	3	no	—
77	C₄×Dic₁₃	—	2	1	no	—
78	C₄×Dic₁₃	—	2	1	no	—
79	C₄×Dic₁₃	—	2	6	no	—
80	C₄×Dic₁₃	—	4	3	no	—
81	C₄×Dic₁₃	C₂×C₈	1	12	yes	—
82	C₂×#1055	—	2	1	no	—
83	C₂×#1055	—	2	1	no	—
84	C₂×#1055	—	2	3	no	—
85	C₂×#1055	—	2	3	no	—
86	C₂×#1055	—	2	3	no	—
87	C₂×#1055	#41	1	2	yes	—
88	C₂×C₈×Q₈	—	13	1	no	—
89	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
90	C₄²×Q₈	—	13	1	no	—
91	#1939	—	13	3	no	—
92	#1939	—	13	3	no	—
93	#1831	—	13	3	no	—
94	C₄×C₄⋊C₈	—	13	3	no	—
95	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₁₃	C₄⋊C₈	1	16	yes	complemented
97	C₂×C₄×C₁₃	—	2	3	no	—
98	C₂×C₄×C₁₃	—	2	3	no	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
101	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
102	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	3	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	2	1	no	—
113	C₂×Dic₁₃	—	2	3	no	—
114	C₂×Dic₁₃	—	2	1	no	—
115	C₂×Dic₁₃	—	2	3	no	—
116	C₂×Dic₁₃	—	2	3	no	—
117	C₂×Dic₁₃	—	2	3	no	—
118	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₁₃	C₂²×C₈	1	6	yes	—
120	C₂×Dic₁₃	#131	1	2	yes	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	—	4	3	no	—
123	#1055	—	2	2	no	—
124	#1055	—	4	3	no	—
125	C₂×C₄×Q₈	—	13	1	no	—
126	C₂×C₄×Q₈	—	13	1	no	—
127	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
128	C₈×Q₈	—	2	4	no	—
129	C₈×Q₈	—	26	4	no	—
130	C₂×C₄⋊C₈	—	13	3	no	—
131	C₂×C₄⋊C₈	—	13	3	no	—
132	C₂×C₄⋊C₈	—	13	3	no	—
133	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
134	C₂×C₄×C₈	—	13	3	no	—
135	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
136	C₄×C₄⋊C₄	—	13	3	no	—
137	C₄×C₄⋊C₄	—	13	3	no	—
138	C₄×C₄⋊C₄	—	13	3	no	—
139	C₄³	—	13	3	no	—
140	#336	—	13	3	no	—
141	#336	—	13	6	no	—
142	#336	—	13	1	no	—
143	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
145	C₂²×C₁₃	#519	1	1	yes	characteristic
146	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
147	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
149	C₂²×C₁₃	#568	1	1	yes	characteristic
150	C₄×C₁₃	—	2	6	no	—
151	C₄×C₁₃	C₂×C₄⋊C₈	1	12	yes	—
152	C₄×C₁₃	#548	1	1	yes	characteristic
153	C₄×C₁₃	#548	1	1	yes	characteristic
154	C₄×C₁₃	#557	1	1	yes	characteristic
155	C₄×C₁₃	#557	1	1	yes	characteristic
156	Dic₁₃	—	2	6	no	—
157	Dic₁₃	—	2	2	no	—
158	Dic₁₃	—	2	2	no	—
159	Dic₁₃	—	4	3	no	—
160	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	13	3	no	—
163	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
164	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	C₄×Q₈	—	13	8	no	—
166	C₄×Q₈	—	26	4	no	—
167	C₄×Q₈	Dic₂₆	1	4	yes	—
168	C₄×Q₈	C₂²⋊D₁₃	1	4	yes	—
169	C₂×C₄⋊C₄	—	13	3	no	—
170	C₂×C₄⋊C₄	—	13	3	no	—
171	C₂×C₄⋊C₄	C₄×D₁₃	1	3	yes	—
172	C₂×C₄²	—	13	3	no	—
173	C₂×C₄²	—	13	3	no	—
174	C₂×C₄²	—	13	3	no	—
175	C₂×C₄²	—	13	1	no	—
176	C₂×C₄²	—	13	3	no	—
177	C₂×C₄²	—	13	3	no	—
178	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
179	C₄⋊C₈	—	2	6	no	—
180	C₄⋊C₈	—	26	6	no	—
181	C₄⋊C₈	—	26	6	no	—
182	C₄⋊C₈	—	26	6	no	—
183	C₄×C₈	—	2	6	no	—
184	C₄×C₈	—	26	6	no	—
185	#95	—	13	6	no	—
186	C₂×C₁₃	#2351	1	1	yes	characteristic
187	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
188	C₂×C₁₃	#2906	1	1	yes	characteristic
189	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
190	C₂×C₁₃	#2971	1	1	yes	characteristic
191	C₂×C₁₃	#2982	1	1	yes	characteristic
192	C₂×C₁₃	#2984	1	1	yes	characteristic
193	C₂×Q₈	C₈×D₁₃	1	4	yes	—
194	C₂×Q₈	#6094	1	4	yes	—
195	C₂×Q₈	#6101	1	4	yes	—
196	C₂²×C₄	—	13	3	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	3	no	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
202	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	3	no	—
206	C₂×C₈	—	2	3	no	—
207	C₂×C₈	—	2	3	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	26	1	no	—
210	C₂×C₈	—	26	1	no	—
211	C₂×C₈	—	26	3	no	—
212	C₂×C₈	—	26	3	no	—
213	C₂×C₈	—	26	3	no	—
214	C₂×C₈	#6131	1	2	yes	—
215	C₄⋊C₄	—	13	12	no	—
216	C₄⋊C₄	—	26	6	no	—
217	C₄⋊C₄	#6101	1	12	yes	—
218	C₄²	—	13	12	no	—
219	C₄²	—	26	6	no	—
220	C₄²	—	26	6	no	—
221	C₄²	—	26	3	no	—
222	C₄²	—	26	1	no	—
223	C₄²	—	26	3	no	—
224	C₄²	—	26	1	no	—
225	C₄²	—	52	3	no	—
226	C₄²	C₂×Dic₂₆	1	6	yes	—
227	C₄²	C₂×C₂²⋊D₁₃	1	6	yes	—
228	C₁₃	#22734	1	1	yes	characteristic, complemented, Sylow
229	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
230	Q₈	#88054	1	16	yes	complemented
231	C₂×C₄	—	2	3	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	6	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	26	1	no	—
237	C₂×C₄	—	26	3	no	—
238	C₂×C₄	—	26	1	no	—
239	C₂×C₄	—	26	1	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	26	3	no	—
242	C₂×C₄	—	26	3	no	—
243	C₂×C₄	—	26	3	no	—
244	C₂×C₄	—	26	3	no	—
245	C₂×C₄	C₂×C₈×D₁₃	1	6	yes	—
246	C₂×C₄	C₂×#6094	1	6	yes	—
247	C₂×C₄	C₂×#6101	1	6	yes	—
248	C₂×C₄	C₂×#6101	1	6	yes	—
249	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
250	C₂×C₄	C₂×#6131	1	1	yes	characteristic
251	C₂×C₄	#88254	1	1	yes	characteristic
252	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
253	C₂×C₄	#88257	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	3	no	—
257	C₈	—	26	2	no	—
258	C₈	—	52	3	no	—
259	C₂²	#11719277	1	1	yes	characteristic
260	C₂²	C₄×#6131	1	1	yes	characteristic
261	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
262	C₂²	C₂²×#6094	1	1	yes	characteristic
263	C₂²	D₁₃×#131	1	1	yes	characteristic
264	C₂²	#11719455	1	1	yes	characteristic
265	C₂²	C₂²×#6101	1	1	yes	characteristic
266	C₄	—	2	6	no	—
267	C₄	—	26	2	no	—
268	C₄	—	26	2	no	—
269	C₄	—	26	6	no	—
270	C₄	—	52	3	no	—
271	C₄	C₂×#88054	1	12	yes	—
272	C₄	#11719274	1	1	yes	characteristic
273	C₄	#11719275	1	1	yes	characteristic
274	C₄	#11719522	1	1	yes	characteristic
275	C₄	#11719536	1	1	yes	characteristic
276	C₂	#13117754	1	1	yes	characteristic
277	C₂	C₂²×#88054	1	1	yes	characteristic
278	C₂	#13123402	1	1	yes	characteristic
279	C₂	C₈×#6131	1	1	yes	characteristic
280	C₂	#13123855	1	1	yes	characteristic
281	C₂	#13123915	1	1	yes	characteristic
282	C₂	#13123924	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#22734
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
34. Q₈×Dic₁₃	C₈	semidirect
96. Q₈×C₁₃	C₄⋊C₈	semidirect
228. C₁₃	#22734	semidirect
230. Q₈	#88054	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
C₄×Q₈×Dic₁₃	2
#13115554	2
#13115552	2
#13114952	2
C₄×#88054	2
C₂×C₈×Q₈×C₁₃	2
#22734	13