Subgroups — #15833643

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#13113688	C₂	1	8	yes	maximal, complemented
4	#13113595	C₂	1	1	yes	maximal, characteristic
5	#13113292	C₂	1	2	yes	maximal
6	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
7	#11719175	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718777	C₄	1	4	yes	—
12	#11718763	—	2	2	no	—
13	#11718762	—	2	2	no	—
14	#11718585	C₂²	1	2	yes	—
15	#11718360	C₄	1	1	yes	characteristic
16	C₄×Dic₅₂	C₂²	1	4	yes	—
17	#11718343	C₄	1	1	yes	characteristic
18	#11718342	C₄	1	2	yes	—
19	#11718340	—	2	2	no	—
20	#11718340	C₂²	1	4	yes	—
21	C₈×Dic₂₆	—	2	2	no	—
22	C₈×Dic₂₆	C₂²	1	4	yes	—
23	C₁₃×#322	C₂²	1	4	yes	—
24	#11718050	C₂²	1	4	yes	—
25	#11718048	C₂²	1	8	yes	—
26	C₂²×C₈×C₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	#88180	—	2	2	no	—
32	#88178	—	2	2	no	—
33	#88178	C₂×C₄	1	4	yes	—
34	C₂×#6099	—	2	2	no	—
35	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
36	#88124	—	2	2	no	—
37	#88123	C₂×C₄	1	2	yes	—
38	#88123	C₂×C₄	1	2	yes	—
39	C₄×Dic₂₆	—	2	2	no	—
40	C₄×Dic₂₆	C₂³	1	4	yes	—
41	C₄×C₈×C₁₃	—	2	1	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	2	yes	—
45	C₄×C₈×C₁₃	C₂³	1	2	yes	—
46	C₄×C₈×C₁₃	C₂³	1	2	yes	—
47	C₄×C₈×C₁₃	C₂³	1	2	yes	—
48	#88058	C₂×C₄	1	4	yes	—
49	#88056	C₂×C₄	1	8	yes	—
50	#88055	—	2	4	no	—
51	#88054	—	2	4	no	—
52	#88054	—	2	4	no	—
53	C₈×Dic₁₃	—	2	4	no	—
54	C₄⋊#1055	—	2	2	no	—
55	C₄⋊#1055	C₂³	1	4	yes	—
56	#17977	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₂×Dic₂₆	—	2	2	no	—
61	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	Q₁₆	1	4	yes	complemented
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₃	#41	1	1	yes	characteristic
74	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²⋊Dic₁₃	—	2	4	no	—
76	C₄⋊Dic₁₃	—	2	2	no	—
77	C₄⋊Dic₁₃	—	2	2	no	—
78	C₄⋊Dic₁₃	—	2	2	no	—
79	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
80	#6101	—	2	4	no	—
81	#6101	—	2	4	no	—
82	C₄×Dic₁₃	—	2	4	no	—
83	#6099	—	4	4	no	—
84	C₂×#1055	—	2	4	no	—
85	Dic₅₂	—	4	4	no	—
86	#3170	—	13	1	no	—
87	#2960	—	13	2	no	—
88	#1584	—	13	8	no	—
89	#1567	—	13	1	no	—
90	#1493	—	13	2	no	—
91	C₂×#322	D₁₃	1	1	yes	characteristic
92	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₁₃	#135	1	2	yes	—
100	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
103	C₂×Dic₁₃	—	2	4	no	—
104	C₂×Dic₁₃	—	2	4	no	—
105	Dic₂₆	—	2	8	no	—
106	Dic₂₆	—	4	4	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	—	2	2	no	—
110	C₈×C₁₃	—	4	2	no	—
111	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
112	#1055	—	4	4	no	—
113	#533	—	13	2	no	—
114	#500	—	13	1	no	—
115	#494	—	13	1	no	—
116	#487	—	13	2	no	—
117	#467	—	13	4	no	—
118	#446	—	13	4	no	—
119	#446	—	26	2	no	—
120	C₈×Q₈	—	13	4	no	—
121	C₈×Q₈	—	26	2	no	—
122	C₄×Q₁₆	—	13	4	no	—
123	#424	—	26	2	no	—
124	#423	—	13	2	no	—
125	#405	—	26	2	no	—
126	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#335	—	13	4	no	—
130	Q₈⋊C₈	—	13	8	no	—
131	#322	C₂×D₁₃	1	4	yes	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
135	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
139	C₄×C₁₃	#444	1	2	yes	—
140	C₄×C₁₃	#568	1	2	yes	—
141	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
142	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
143	Dic₁₃	—	4	4	no	—
144	Dic₁₃	—	4	4	no	—
145	C₂×Q₁₆	—	13	4	no	—
146	C₂×M₁₆	—	26	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	C₄×D₁₃	1	2	yes	—
149	C₂²×C₈	D₅₂	1	1	yes	characteristic
150	C₂²×C₈	D₅₂	1	1	yes	characteristic
151	C₄⋊Q₈	—	13	2	no	—
152	C₄⋊Q₈	—	13	2	no	—
153	#125	—	26	2	no	—
154	C₂²⋊Q₈	—	13	4	no	—
155	C₂²⋊Q₈	—	26	2	no	—
156	C₄×Q₈	—	13	4	no	—
157	C₄×Q₈	—	26	2	no	—
158	#117	—	26	2	no	—
159	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
160	#107	—	13	4	no	—
161	C₄⋊C₈	—	13	4	no	—
162	C₄⋊C₈	—	26	4	no	—
163	C₄⋊C₈	—	26	4	no	—
164	C₄⋊C₈	—	26	2	no	—
165	#103	—	13	8	no	—
166	#97	—	26	4	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	26	4	no	—
169	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
170	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
171	C₄×C₈	D₅₂	1	2	yes	—
172	C₄×C₈	C₂²×D₁₃	1	2	yes	—
173	C₄×C₈	C₂²×D₁₃	1	2	yes	—
174	C₄×C₈	C₂²×D₁₃	1	2	yes	—
175	C₂×C₁₃	—	2	2	no	—
176	C₂×C₁₃	#2932	1	1	yes	characteristic
177	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
179	C₂×Q₈	—	13	4	no	—
180	C₂×Q₈	—	26	2	no	—
181	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
183	C₂²×C₄	#6127	1	1	yes	characteristic
184	Q₁₆	—	52	4	no	—
185	M₁₆	—	52	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	26	4	no	—
191	C₂×C₈	Dic₅₂	1	4	yes	complemented
192	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
194	C₂×C₈	C₂×D₅₂	1	2	yes	—
195	C₂×C₈	C₂×D₅₂	1	2	yes	—
196	C₄⋊C₄	—	13	4	no	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	2	no	—
199	C₄⋊C₄	—	26	4	no	—
200	C₄⋊C₄	—	26	2	no	—
201	C₄⋊C₄	—	26	4	no	—
202	C₂²⋊C₄	—	26	4	no	—
203	C₄²	—	26	4	no	—
204	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₄²	C₂×D₅₂	1	1	yes	characteristic
206	C₄²	#6127	1	1	yes	characteristic
207	C₄²	C₂³×D₁₃	1	1	yes	characteristic
208	C₁₃	#17977	1	1	yes	characteristic, complemented, Sylow
209	C₂³	C₄×D₅₂	1	1	yes	characteristic
210	Q₈	—	26	8	no	—
211	Q₈	—	52	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	26	4	no	—
215	C₂×C₄	—	26	4	no	—
216	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
217	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
218	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
219	C₂×C₄	#88158	1	2	yes	—
220	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
221	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
223	C₂×C₄	C₂×#6127	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	52	4	no	—
229	C₈	C₂×Dic₅₂	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Dic₅₂	1	2	yes	—
233	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	52	4	no	—
237	C₄	—	52	4	no	—
238	C₄	C₄×Dic₅₂	1	2	yes	—
239	C₄	#11718378	1	2	yes	—
240	C₄	C₂×#88158	1	1	yes	characteristic
241	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
242	C₄	#11719450	1	2	yes	—
243	C₂	—	2	2	no	—
244	C₂	#13123555	1	1	yes	characteristic
245	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
246	C₂	C₂×#11718378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#17977
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #13113688	C₂	semidirect
66. C₂×C₈×C₁₃	Q₁₆	semidirect
191. C₂×C₈	Dic₅₂	semidirect
208. C₁₃	#17977	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
#13113688	2
#13113595	2
#13113292	2
C₂×C₁₃×#322	2
#17977	13