Subgroups — #15867645

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	#13117313	C₂	1	1	yes	maximal, characteristic
2	#13115776	C₂	1	2	yes	maximal
3	#13115559	C₂	1	2	yes	maximal
4	#13115393	C₂	1	4	yes	maximal
5	#13114696	C₂	1	2	yes	maximal
6	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
7	#13114589	C₂	1	2	yes	maximal
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#97	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88054	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	#11718594	C₄	1	4	yes	—
18	#11718593	C₂²	1	2	yes	—
19	C₄×#6101	C₂²	1	2	yes	—
20	#11718590	C₄	1	2	yes	—
21	#11718589	C₄	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	#11718142	C₂²	1	4	yes	—
25	#11718142	C₂²	1	4	yes	—
26	#11718142	C₂²	1	4	yes	—
27	#11718116	C₂²	1	2	yes	—
28	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂×#6101	C₂×C₄	1	2	yes	—
39	C₂×#6101	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
42	C₂²×#1055	C₂³	1	4	yes	—
43	C₄×Dic₂₆	—	2	4	no	—
44	C₁₃×#97	—	2	4	no	—
45	C₁₃×#97	—	2	2	no	—
46	C₁₃×#97	Q₈	1	4	yes	complemented
47	C₄×C₈×C₁₃	—	2	4	no	—
48	#88071	C₂×C₄	1	4	yes	—
49	#88071	C₂×C₄	1	4	yes	—
50	#88071	C₂×C₄	1	4	yes	—
51	#88071	C₂×C₄	1	2	yes	—
52	#88054	—	2	8	no	—
53	C₄⋊#1055	—	2	4	no	—
54	C₄×#1055	—	2	4	no	—
55	#21344	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	—	4	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₁₃	#41	1	4	yes	—
72	C₄²×C₁₃	—	2	4	no	—
73	C₄²×C₁₃	—	2	2	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	C₄⋊Dic₁₃	—	2	2	no	—
79	C₄⋊Dic₁₃	—	4	2	no	—
80	#6101	—	2	4	no	—
81	#6101	—	2	8	no	—
82	C₄×Dic₁₃	—	2	4	no	—
83	C₄×Dic₁₃	—	2	4	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	4	no	—
87	#2295	—	13	1	no	—
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	D₁₃	1	1	yes	characteristic
95	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	2	2	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	#117	1	2	yes	—
103	C₂×C₄×C₁₃	#117	1	2	yes	—
104	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₁₃	#131	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	2	2	no	—
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	2	4	no	—
115	C₂×Dic₁₃	—	2	4	no	—
116	C₂×Dic₁₃	—	2	4	no	—
117	Dic₂₆	—	4	4	no	—
118	C₈×C₁₃	—	2	4	no	—
119	C₈×C₁₃	—	2	4	no	—
120	#1055	—	4	4	no	—
121	C₂×C₄×Q₈	—	13	1	no	—
122	C₂×C₄⋊C₈	—	13	4	no	—
123	C₂×C₄⋊C₈	—	13	2	no	—
124	C₂×#97	C₂×D₁₃	1	2	yes	—
125	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
126	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄×C₈	—	13	2	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
129	#391	—	13	2	no	—
130	#389	—	13	2	no	—
131	#387	—	13	4	no	—
132	C₄×C₄⋊C₄	—	13	2	no	—
133	C₄×C₄⋊C₄	—	13	1	no	—
134	#376	—	13	2	no	—
135	C₄³	C₂×D₁₃	1	1	yes	characteristic
136	#336	—	13	4	no	—
137	#336	—	13	4	no	—
138	#336	—	13	4	no	—
139	#336	—	13	2	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	#529	1	2	yes	—
150	C₄×C₁₃	#533	1	2	yes	—
151	Dic₁₃	—	4	2	no	—
152	Dic₁₃	—	4	2	no	—
153	Dic₁₃	—	4	4	no	—
154	C₂²×Q₈	—	13	1	no	—
155	C₂²×C₈	—	13	4	no	—
156	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
157	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
158	C₄×Q₈	—	26	4	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	13	1	no	—
161	C₂×C₄⋊C₄	—	13	4	no	—
162	C₂×C₄⋊C₄	—	13	2	no	—
163	C₂×C₄²	—	13	2	no	—
164	C₂×C₄²	—	13	2	no	—
165	C₂×C₄²	C₄×D₁₃	1	2	yes	—
166	C₂×C₄²	C₄×D₁₃	1	2	yes	—
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
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	—	2	2	no	—
173	#97	—	2	4	no	—
174	#97	Dic₂₆	1	4	yes	complemented
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	26	4	no	—
177	#95	—	13	4	no	—
178	#95	—	13	4	no	—
179	#95	—	13	2	no	—
180	#95	—	13	4	no	—
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	—	26	2	no	—
187	C₂×Q₈	—	52	2	no	—
188	C₂²×C₄	—	13	4	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
192	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
195	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	26	4	no	—
201	C₂×C₈	—	26	4	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
204	C₂×C₈	#6127	1	4	yes	—
205	C₄⋊C₄	—	26	4	no	—
206	C₄⋊C₄	—	26	2	no	—
207	C₄⋊C₄	—	26	8	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	26	4	no	—
213	C₄²	—	26	4	no	—
214	C₄²	C₂×Dic₂₆	1	2	yes	—
215	C₄²	#6127	1	2	yes	—
216	C₄²	#6127	1	4	yes	—
217	C₁₃	#21344	1	1	yes	characteristic, complemented, Sylow
218	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
219	Q₈	—	52	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	—	26	2	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	4	no	—
229	C₂×C₄	—	26	4	no	—
230	C₂×C₄	—	26	4	no	—
231	C₂×C₄	—	26	4	no	—
232	C₂×C₄	—	26	4	no	—
233	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
234	C₂×C₄	#88126	1	2	yes	—
235	C₂×C₄	#88126	1	2	yes	—
236	C₂×C₄	#88161	1	4	yes	—
237	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
238	C₂×C₄	C₂×#6127	1	2	yes	—
239	C₂×C₄	C₂×#6127	1	2	yes	—
240	C₂×C₄	C₂×#6127	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	—	2	4	no	—
243	C₈	—	52	4	no	—
244	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
245	C₂²	C₂×#88126	1	1	yes	characteristic
246	C₂²	C₄×#6127	1	1	yes	characteristic
247	C₂²	C₂×#88161	1	2	yes	—
248	C₂²	#11719450	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	52	2	no	—
254	C₄	—	52	2	no	—
255	C₄	—	52	4	no	—
256	C₄	#11719175	1	2	yes	—
257	C₄	#11719184	1	2	yes	—
258	C₂	#13117313	1	1	yes	characteristic
259	C₂	#13123472	1	1	yes	characteristic
260	C₂	#13123481	1	1	yes	characteristic
261	C₂	#13123783	1	2	yes	—
262	C₂	#13123830	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#97	C₂
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#21344
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
46. C₁₃×#97	Q₈	semidirect
174. #97	Dic₂₆	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
#13117313	2
#13115776	2
#13115559	2
#13115393	2
#13114696	2
C₄×C₄⋊#1055	2
#13114589	2
C₄×C₁₃×#97	2
#21344	13