Subgroups — #15833452

Order 3328.

614 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	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13123770	C₂	1	2	yes	maximal
3	#13113734	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₅₂	C₂	1	4	yes	maximal, complemented
5	#13113595	C₂	1	1	yes	maximal, characteristic
6	#13113292	C₂	1	2	yes	maximal
7	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
8	#11719175	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718777	C₄	1	4	yes	—
13	#11718763	—	2	2	no	—
14	#11718762	—	2	2	no	—
15	#11718585	C₂²	1	2	yes	—
16	#11718360	C₄	1	1	yes	characteristic
17	C₄×Dic₅₂	C₂²	1	4	yes	—
18	#11718343	C₄	1	1	yes	characteristic
19	#11718342	C₄	1	2	yes	—
20	#11718340	—	2	2	no	—
21	#11718340	C₂²	1	4	yes	—
22	C₈×Dic₂₆	—	2	2	no	—
23	C₈×Dic₂₆	C₂²	1	4	yes	—
24	C₈²×C₁₃	C₂²	1	2	yes	—
25	C₈²×C₁₃	C₂²	1	2	yes	—
26	#11718050	C₂²	1	4	yes	—
27	#11718048	C₂²	1	8	yes	—
28	C₂²×C₈×C₁₃	—	2	1	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	#88180	—	2	2	no	—
34	#88178	—	2	2	no	—
35	#88178	C₂×C₄	1	4	yes	—
36	C₂×#6099	—	2	2	no	—
37	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
38	#88124	—	2	2	no	—
39	#88123	C₂×C₄	1	2	yes	—
40	#88123	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₆	—	2	2	no	—
42	C₄×Dic₂₆	C₂³	1	4	yes	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#88058	C₂×C₄	1	4	yes	—
53	#88056	C₂×C₄	1	8	yes	—
54	#88055	—	2	4	no	—
55	#88054	—	2	4	no	—
56	#88054	—	2	4	no	—
57	C₈×Dic₁₃	—	2	4	no	—
58	C₄⋊#1055	—	2	2	no	—
59	C₄⋊#1055	C₂³	1	4	yes	—
60	#17958	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	#41	1	1	yes	characteristic
80	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	—	2	4	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
86	#6101	—	2	4	no	—
87	#6101	—	2	4	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	#6099	—	4	4	no	—
90	C₂×#1055	—	2	4	no	—
91	Dic₅₂	—	2	8	no	—
92	#3170	—	13	1	no	—
93	#2960	—	13	2	no	—
94	#1591	—	13	4	no	—
95	C₈×Q₁₆	—	13	4	no	—
96	#1567	—	13	1	no	—
97	#1493	—	13	2	no	—
98	C₂×C₈²	D₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	#135	1	2	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	2	4	no	—
112	Dic₂₆	—	2	8	no	—
113	Dic₂₆	—	4	4	no	—
114	C₈×C₁₃	—	2	4	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	#131	1	4	yes	—
119	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
120	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
121	#1055	—	4	4	no	—
122	#533	—	13	2	no	—
123	#500	—	13	1	no	—
124	#494	—	13	1	no	—
125	#487	—	13	2	no	—
126	#467	—	13	4	no	—
127	#446	—	13	4	no	—
128	#446	—	26	2	no	—
129	C₈×Q₈	—	13	4	no	—
130	C₈×Q₈	—	26	2	no	—
131	C₄×Q₁₆	—	13	4	no	—
132	#424	—	26	2	no	—
133	#423	—	13	2	no	—
134	#405	—	26	2	no	—
135	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	#335	—	13	4	no	—
139	Q₈⋊C₈	—	13	8	no	—
140	C₈²	C₂×D₁₃	1	2	yes	—
141	C₈²	C₂×D₁₃	1	2	yes	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
145	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
149	C₄×C₁₃	#443	1	2	yes	—
150	C₄×C₁₃	C₂×#131	1	2	yes	—
151	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
153	Dic₁₃	—	4	4	no	—
154	Dic₁₃	—	4	4	no	—
155	C₂×Q₁₆	—	13	4	no	—
156	C₂×M₁₆	—	26	2	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₄×D₁₃	1	2	yes	—
159	C₂²×C₈	D₅₂	1	1	yes	characteristic
160	C₂²×C₈	D₅₂	1	1	yes	characteristic
161	C₄⋊Q₈	—	13	2	no	—
162	C₄⋊Q₈	—	13	2	no	—
163	#125	—	26	2	no	—
164	C₂²⋊Q₈	—	13	4	no	—
165	C₂²⋊Q₈	—	26	2	no	—
166	C₄×Q₈	—	13	4	no	—
167	C₄×Q₈	—	26	2	no	—
168	#117	—	26	2	no	—
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	#107	—	13	4	no	—
171	C₄⋊C₈	—	13	4	no	—
172	C₄⋊C₈	—	26	4	no	—
173	C₄⋊C₈	—	26	4	no	—
174	C₄⋊C₈	—	26	2	no	—
175	#103	—	13	8	no	—
176	#97	—	26	4	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	26	4	no	—
179	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
180	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₄×C₈	D₅₂	1	1	yes	characteristic
182	C₄×C₈	D₅₂	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₁₃	1	2	yes	—
184	C₄×C₈	C₂²×D₁₃	1	2	yes	—
185	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	#2931	1	1	yes	characteristic
189	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
191	C₂×Q₈	—	13	4	no	—
192	C₂×Q₈	—	26	2	no	—
193	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
195	C₂²×C₄	#6127	1	1	yes	characteristic
196	Q₁₆	—	26	8	no	—
197	M₁₆	—	52	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	Dic₅₂	1	4	yes	complemented
204	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
206	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
207	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
208	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
209	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
210	C₄⋊C₄	—	13	4	no	—
211	C₄⋊C₄	—	26	2	no	—
212	C₄⋊C₄	—	26	2	no	—
213	C₄⋊C₄	—	26	4	no	—
214	C₄⋊C₄	—	26	2	no	—
215	C₄⋊C₄	—	26	4	no	—
216	C₂²⋊C₄	—	26	4	no	—
217	C₄²	—	26	4	no	—
218	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₄²	C₂×D₅₂	1	1	yes	characteristic
220	C₄²	#6127	1	1	yes	characteristic
221	C₄²	C₂³×D₁₃	1	1	yes	characteristic
222	C₁₃	#17958	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₄×D₅₂	1	1	yes	characteristic
224	Q₈	—	26	8	no	—
225	Q₈	—	52	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	26	4	no	—
229	C₂×C₄	—	26	4	no	—
230	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
231	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
232	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
233	C₂×C₄	#88158	1	2	yes	—
234	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
237	C₂×C₄	C₂×#6127	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	52	4	no	—
243	C₈	#88155	1	4	yes	—
244	C₈	C₂×Dic₅₂	1	2	yes	—
245	C₈	C₂×Dic₅₂	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₅₂	1	2	yes	—
249	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	52	4	no	—
253	C₄	—	52	4	no	—
254	C₄	C₄×Dic₅₂	1	2	yes	—
255	C₄	#11718361	1	2	yes	—
256	C₄	C₂×#88155	1	2	yes	—
257	C₄	C₂×#88158	1	1	yes	characteristic
258	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#13123549	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
262	C₂	C₂×#11718361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#17958
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #13113734	C₂	semidirect
4. C₈×Dic₅₂	C₂	semidirect
70. C₂×C₈×C₁₃	Q₁₆	semidirect
203. C₂×C₈	Dic₅₂	semidirect
222. C₁₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13123770	2
#13113734	2
C₈×Dic₅₂	2
#13113595	2
#13113292	2
C₂×C₈²×C₁₃	2
#17958	13