Subgroups — #15867646

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115777	C₂	1	2	yes	maximal
2	#13115560	C₂	1	2	yes	maximal
3	#13115395	C₂	1	4	yes	maximal
4	#13114693	C₂	1	2	yes	maximal
5	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
6	#13114588	C₂	1	2	yes	maximal
7	C₄×#88043	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2295	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#391	C₄	1	2	yes	—
11	C₁₃×#389	C₄	1	2	yes	—
12	C₁₃×#387	C₄	1	4	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₁₃×#376	C₂²	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	C₂²	1	4	yes	—
18	C₂×C₄⋊#1055	C₂²	1	2	yes	—
19	C₂×#88043	C₂²	1	1	yes	characteristic
20	C₂×#88043	C₂²	1	2	yes	—
21	C₂×#88043	C₂²	1	1	yes	characteristic
22	C₂×C₄×#1055	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	#11718116	C₂²	1	2	yes	—
25	#11718116	C₂²	1	4	yes	—
26	#11718116	C₂²	1	4	yes	—
27	#11718116	C₂²	1	4	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	—	2	4	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×#1055	C₂³	1	4	yes	—
42	C₂²×#1055	C₂³	1	2	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₁₃×#95	C₂×C₄	1	2	yes	—
45	C₁₃×#95	C₂×C₄	1	4	yes	—
46	C₁₃×#95	C₂×C₄	1	4	yes	—
47	C₁₃×#95	C₂×C₄	1	4	yes	—
48	C₄⋊#1055	—	2	4	no	—
49	C₄⋊#1055	—	2	8	no	—
50	#88043	—	2	4	no	—
51	#88043	—	2	2	no	—
52	#88043	Q₈	1	4	yes	complemented
53	C₄×#1055	—	2	4	no	—
54	C₄×#1055	—	2	4	no	—
55	#21344	—	13	1	no	maximal, Sylow
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×Q₈×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	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	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	2	8	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₄²×C₁₃	—	2	4	no	—
71	C₄²×C₁₃	—	2	4	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	—	4	2	no	—
75	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₃	#41	1	4	yes	—
77	C₄²×C₁₃	#41	1	2	yes	—
78	C₂×#1055	—	2	2	no	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	C₂×Q₈	1	4	yes	—
86	C₂×#1055	#41	1	4	yes	—
87	#2295	D₁₃	1	1	yes	characteristic
88	#1985	—	13	2	no	—
89	#1940	—	13	2	no	—
90	#1916	—	13	4	no	—
91	#1768	—	13	2	no	—
92	C₄×C₄⋊C₈	—	13	1	no	—
93	#1746	—	13	2	no	—
94	C₄×#97	—	13	1	no	—
95	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	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	#117	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	2	yes	—
112	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₁₃	#131	1	4	yes	—
114	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
118	#1055	—	2	4	no	—
119	#1055	—	2	4	no	—
120	#1055	—	4	4	no	—
121	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	13	4	no	—
123	C₂×C₄⋊C₈	—	13	2	no	—
124	C₂×#97	—	13	1	no	—
125	C₂×#97	—	13	2	no	—
126	C₂×#97	—	13	1	no	—
127	C₂×C₄×C₈	—	13	2	no	—
128	C₂×C₄×C₈	—	13	2	no	—
129	#391	Dic₁₃	1	2	yes	—
130	#389	Dic₁₃	1	2	yes	—
131	#387	Dic₁₃	1	4	yes	—
132	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
133	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
134	#376	C₂×D₁₃	1	2	yes	—
135	C₄³	C₂×D₁₃	1	1	yes	characteristic
136	#336	—	13	2	no	—
137	#336	—	13	4	no	—
138	#336	—	13	4	no	—
139	#336	—	13	4	no	—
140	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×#131	1	2	yes	—
144	C₂²×C₁₃	#568	1	2	yes	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	4	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	#529	1	2	yes	—
153	C₄×C₁₃	#533	1	2	yes	—
154	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
155	C₂²×C₈	—	13	2	no	—
156	C₂²×C₈	—	13	2	no	—
157	C₂²×C₈	—	13	4	no	—
158	C₄×Q₈	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
162	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
163	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
164	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
165	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
166	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	C₄⋊C₈	—	26	8	no	—
171	C₄⋊C₈	—	26	4	no	—
172	#97	—	13	4	no	—
173	#97	—	26	4	no	—
174	#97	—	26	2	no	—
175	C₄×C₈	—	26	4	no	—
176	C₄×C₈	—	26	4	no	—
177	#95	C₂×Dic₁₃	1	4	yes	—
178	#95	C₂×Dic₁₃	1	4	yes	—
179	#95	C₂×Dic₁₃	1	2	yes	—
180	#95	C₂×Dic₁₃	1	4	yes	—
181	C₂×C₁₃	#2295	1	1	yes	characteristic
182	C₂×C₁₃	#2921	1	1	yes	characteristic
183	C₂×C₁₃	#2922	1	1	yes	characteristic
184	C₂×C₁₃	#2962	1	2	yes	—
185	C₂×C₁₃	#2968	1	2	yes	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
189	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
190	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
191	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
195	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
196	C₂×C₈	—	13	4	no	—
197	C₂×C₈	—	13	4	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	—	26	2	no	—
200	C₂×C₈	—	26	2	no	—
201	C₂×C₈	—	26	2	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	26	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	4	2	no	—
214	C₄²	#6127	1	4	yes	—
215	C₄²	Q₈×D₁₃	1	2	yes	—
216	C₄²	#6131	1	2	yes	—
217	C₁₃	#21344	1	1	yes	characteristic, complemented, Sylow
218	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	#88180	1	2	yes	—
234	C₂×C₄	#88180	1	2	yes	—
235	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
236	C₂×C₄	#88202	1	4	yes	—
237	C₂×C₄	C₂×#6127	1	2	yes	—
238	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#6131	1	1	yes	characteristic
240	C₂×C₄	#41×D₁₃	1	2	yes	—
241	C₈	—	26	4	no	—
242	C₈	—	26	4	no	—
243	C₈	—	52	4	no	—
244	C₂²	C₂×#88180	1	1	yes	characteristic
245	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
246	C₂²	C₂×#88202	1	2	yes	—
247	C₂²	#11719524	1	2	yes	—
248	C₂²	#41×Dic₁₃	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	4	no	—
256	C₄	#11719283	1	2	yes	—
257	C₄	#11719285	1	2	yes	—
258	C₂	#13117314	1	1	yes	characteristic
259	C₂	#13123474	1	1	yes	characteristic
260	C₂	#13123480	1	1	yes	characteristic
261	C₂	#13123785	1	2	yes	—
262	C₂	#13123832	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₁₃×#2295	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2295	D₁₃
hypocenter	C₁₃	#21344
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
52. #88043	Q₈	semidirect
217. C₁₃	#21344	semidirect

Maximal subgroups

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

subgroup	index
#13115777	2
#13115560	2
#13115395	2
#13114693	2
C₄×C₄⋊#1055	2
#13114588	2
C₄×#88043	2
C₁₃×#2295	2
#21344	13