Subgroups — #15832765

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123919	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13113746	C₂	1	4	yes	maximal, complemented
4	#13113734	C₂	1	4	yes	maximal, complemented
5	#13113584	C₂	1	1	yes	maximal, characteristic
6	#13113232	C₂	1	2	yes	maximal
7	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88058	C₄	1	1	yes	characteristic
14	C₂×#88056	C₄	1	2	yes	—
15	#11718586	C₂²	1	2	yes	—
16	#11718492	—	2	4	no	—
17	#11718380	—	2	4	no	—
18	C₄×Dic₅₂	C₄	1	2	yes	—
19	C₄×Dic₅₂	C₄	1	2	yes	—
20	C₄×Dic₅₂	C₂²	1	2	yes	—
21	C₄×Dic₅₂	C₂²	1	2	yes	—
22	#11718340	C₂²	1	8	yes	—
23	C₁₃×#322	C₂²	1	2	yes	—
24	C₈²×C₁₃	C₂²	1	2	yes	—
25	#11718050	C₂²	1	2	yes	—
26	#11718049	C₂²	1	2	yes	—
27	#11718048	C₂²	1	8	yes	—
28	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
33	C₂×#6101	—	2	2	no	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×Dic₅₂	—	2	4	no	—
36	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
37	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
38	C₄×Dic₂₆	—	2	2	no	—
39	C₄×Dic₂₆	—	2	2	no	—
40	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₆	C₂³	1	4	yes	—
42	C₁₃×C₄⋊C₈	—	2	1	no	—
43	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
44	C₁₃×C₂²⋊C₈	—	2	1	no	—
45	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂²⋊#1055	—	2	2	no	—
53	#88058	C₂×C₄	1	2	yes	—
54	#88058	C₂×C₄	1	2	yes	—
55	#88056	C₂×C₄	1	4	yes	—
56	#88056	C₂×C₄	1	4	yes	—
57	#88055	—	2	8	no	—
58	#88054	—	2	8	no	—
59	C₄⋊#1055	C₂³	1	4	yes	—
60	C₄×#1055	—	2	2	no	—
61	#17897	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	2	no	—
66	C₂²×Dic₁₃	—	2	2	no	—
67	C₂×Dic₂₆	—	2	4	no	—
68	C₂×Dic₂₆	—	2	4	no	—
69	C₂×Dic₂₆	—	2	2	no	—
70	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
85	#6101	—	2	4	no	—
86	#6101	—	2	4	no	—
87	C₄×Dic₁₃	—	2	4	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	Dic₅₂	—	2	4	no	—
91	Dic₅₂	—	2	4	no	—
92	#2983	—	13	2	no	—
93	C₂×C₄×Q₁₆	—	13	1	no	—
94	#1593	—	13	4	no	—
95	#1591	—	13	4	no	—
96	#1564	—	13	1	no	—
97	#1479	—	13	2	no	—
98	#1450	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	—	4	1	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×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	#131	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	C₂×Dic₁₃	—	4	4	no	—
113	C₂×Dic₁₃	—	8	2	no	—
114	Dic₂₆	—	2	4	no	—
115	Dic₂₆	—	2	4	no	—
116	Dic₂₆	—	4	4	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	4	no	—
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
122	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
123	#1055	—	4	4	no	—
124	C₂²×Q₁₆	—	13	1	no	—
125	C₂×C₄×Q₈	—	13	2	no	—
126	#446	—	13	8	no	—
127	#446	—	26	4	no	—
128	C₄×Q₁₆	—	13	2	no	—
129	C₄×Q₁₆	—	13	2	no	—
130	C₄×Q₁₆	—	13	2	no	—
131	C₄×Q₁₆	—	13	2	no	—
132	#432	—	26	4	no	—
133	#431	—	13	2	no	—
134	#431	C₂×D₁₃	1	1	yes	characteristic
135	#431	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#107	—	13	1	no	—
137	C₂×#103	—	13	2	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#335	—	13	2	no	—
140	#334	—	13	2	no	—
141	Q₈⋊C₈	—	13	8	no	—
142	#322	C₂×D₁₃	1	2	yes	—
143	C₈²	C₂×D₁₃	1	2	yes	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	#436	1	2	yes	—
147	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	#436	1	2	yes	—
152	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
153	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
154	C₄×C₁₃	#578	1	2	yes	—
155	Dic₁₃	—	4	4	no	—
156	Dic₁₃	—	4	4	no	—
157	C₂²×Q₈	—	13	2	no	—
158	C₂×Q₁₆	—	13	2	no	—
159	C₂×Q₁₆	—	13	2	no	—
160	C₂×Q₁₆	—	26	4	no	—
161	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
163	C₄×Q₈	—	13	4	no	—
164	C₄×Q₈	—	13	4	no	—
165	C₄×Q₈	—	26	2	no	—
166	C₄×Q₈	—	26	2	no	—
167	C₂×C₄⋊C₄	—	13	2	no	—
168	C₂×C₄⋊C₄	—	26	2	no	—
169	C₂×C₄²	—	26	2	no	—
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	#107	—	13	2	no	—
172	#107	—	13	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	13	4	no	—
175	C₄⋊C₈	—	26	8	no	—
176	C₄⋊C₈	D₅₂	1	2	yes	—
177	#103	—	13	4	no	—
178	#103	—	13	4	no	—
179	C₂²⋊C₈	—	2	1	no	—
180	C₂²⋊C₈	—	26	2	no	—
181	C₂²⋊C₈	D₅₂	1	2	yes	—
182	#97	—	26	8	no	—
183	C₄×C₈	—	26	2	no	—
184	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
185	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₁₃	1	2	yes	—
187	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₃	1	2	yes	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	13	4	no	—
195	C₂×Q₈	—	26	2	no	—
196	C₂×Q₈	—	26	4	no	—
197	C₂×Q₈	—	26	4	no	—
198	C₂²×C₄	—	26	2	no	—
199	C₂²×C₄	—	26	2	no	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
202	C₂²×C₄	#6127	1	1	yes	characteristic
203	Q₁₆	—	26	4	no	—
204	Q₁₆	—	26	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	#6094	1	4	yes	—
209	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₅₂	1	4	yes	—
214	C₄⋊C₄	—	13	4	no	—
215	C₄⋊C₄	—	26	2	no	—
216	C₄⋊C₄	—	26	4	no	—
217	C₄⋊C₄	—	26	4	no	—
218	C₄²	—	26	4	no	—
219	C₄²	—	26	4	no	—
220	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
221	C₄²	C₂×D₅₂	1	1	yes	characteristic
222	C₄²	#6127	1	1	yes	characteristic
223	C₄²	C₂³×D₁₃	1	1	yes	characteristic
224	C₁₃	#17897	1	1	yes	characteristic, complemented, Sylow
225	C₂³	C₄×D₅₂	1	1	yes	characteristic
226	Q₈	—	26	4	no	—
227	Q₈	—	26	4	no	—
228	Q₈	—	52	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	26	4	no	—
232	C₂×C₄	—	26	4	no	—
233	C₂×C₄	—	52	4	no	—
234	C₂×C₄	—	104	2	no	—
235	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
236	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
237	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
238	C₂×C₄	C₂×#6094	1	2	yes	—
239	C₂×C₄	#88155	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
242	C₂×C₄	C₂×#6127	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	52	4	no	—
248	C₈	C₂×#6094	1	2	yes	—
249	C₈	C₂×#6094	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#11718349	1	2	yes	—
253	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	52	4	no	—
258	C₄	—	52	4	no	—
259	C₄	#11718349	1	2	yes	—
260	C₄	C₂²×#6094	1	1	yes	characteristic
261	C₄	C₂×#88155	1	1	yes	characteristic
262	C₄	#11719458	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×#11718349	1	1	yes	characteristic
265	C₂	#13123705	1	1	yes	characteristic
266	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#17897
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #13113746	C₂	semidirect
4. #13113734	C₂	semidirect
224. C₁₃	#17897	semidirect

Maximal subgroups

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

subgroup	index
#13123919	2
C₂×C₄×Dic₅₂	2
#13113746	2
#13113734	2
#13113584	2
#13113232	2
C₁₃×#1450	2
#17897	13