Subgroups — #15863154

Order 3328.

778 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	#13123400	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
3	#13114976	C₂	1	4	yes	maximal, complemented
4	#13114973	C₂	1	4	yes	maximal, complemented
5	#13114562	C₂	1	2	yes	maximal
6	C₂×#11718150	C₂	1	2	yes	maximal
7	C₁₃×#2905	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#424	—	2	4	no	—
14	C₁₃×#423	C₂²	1	4	yes	—
15	#11718660	—	2	4	no	—
16	#11718658	C₄	1	8	yes	—
17	#11718585	C₂²	1	4	yes	—
18	C₂×#88043	C₂²	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	2	yes	—
20	#11718150	C₂²	1	4	yes	—
21	#11718150	C₂²	1	8	yes	—
22	#11718150	C₂²	1	4	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₁₃×#117	C₂×C₄	1	4	yes	—
30	C₁₃×#117	D₄	1	4	yes	—
31	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	2	no	—
36	C₂×#6099	—	2	2	no	—
37	C₂×#6099	—	2	4	no	—
38	C₂×#6099	—	2	4	no	—
39	C₂×#6099	C₂×C₄	1	4	yes	—
40	C₂×#6099	C₂³	1	4	yes	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	C₂×C₄	1	4	yes	—
45	C₂²×#1055	C₂³	1	1	yes	characteristic
46	C₂²×#1055	C₂³	1	1	yes	characteristic
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	—	2	8	no	—
49	C₄⋊#1055	—	2	8	no	—
50	#88043	C₂×C₄	1	8	yes	—
51	C₄×#1055	C₂×C₄	1	8	yes	—
52	#20768	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×M₁₆	—	2	4	no	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₁₃×M₁₆	—	4	4	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	—	4	4	no	—
68	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
69	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
70	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
71	C₄²×C₁₃	C₂×D₄	1	4	yes	—
72	#6099	—	2	8	no	—
73	#6099	—	2	4	no	—
74	#6099	—	2	4	no	—
75	#6099	—	4	4	no	—
76	C₂×#1055	—	2	8	no	—
77	C₂×#1055	—	2	1	no	—
78	C₂×#1055	—	2	1	no	—
79	C₂×#1055	—	2	1	no	—
80	C₂×#1055	—	2	1	no	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	—	2	2	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	4	4	no	—
85	C₂×#1055	C₂²×C₄	1	2	yes	—
86	C₂×#1055	C₂²×C₄	1	2	yes	—
87	C₂×#1055	C₂²×C₄	1	4	yes	—
88	#2905	—	13	1	no	—
89	#2905	D₁₃	1	1	yes	characteristic
90	C₂×#405	—	13	1	no	—
91	#1838	—	13	4	no	—
92	#1837	—	13	4	no	—
93	#1738	—	13	2	no	—
94	C₂×#341	—	13	2	no	—
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₁₃	#125	1	2	yes	—
98	C₂³×C₁₃	#125	1	2	yes	—
99	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₃	#117	1	4	yes	—
104	C₂×C₄×C₁₃	#117	1	4	yes	—
105	C₂×C₄×C₁₃	#124	1	4	yes	—
106	C₂×C₄×C₁₃	#124	1	4	yes	—
107	C₂×C₄×C₁₃	#125	1	2	yes	—
108	C₂×C₄×C₁₃	#125	1	2	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₁₃	—	4	4	no	—
116	#1055	—	2	2	no	—
117	#1055	—	2	2	no	—
118	#1055	—	2	4	no	—
119	#1055	—	4	4	no	—
120	C₂²×M₁₆	—	13	1	no	—
121	C₂²×M₁₆	—	13	2	no	—
122	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
123	C₂³×C₈	—	13	1	no	—
124	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
125	#424	—	2	4	no	—
126	#424	—	26	4	no	—
127	#423	—	13	4	no	—
128	#423	C₂×D₁₃	1	4	yes	—
129	#405	—	13	8	no	—
130	C₂×#97	—	13	2	no	—
131	C₂×C₄×C₈	—	13	2	no	—
132	#341	—	13	4	no	—
133	#341	—	13	8	no	—
134	#341	—	13	4	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	4	no	—
137	C₂²×C₁₃	—	4	2	no	—
138	C₂²×C₁₃	—	4	2	no	—
139	C₂²×C₁₃	#382	1	4	yes	—
140	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
143	C₄×C₁₃	—	2	8	no	—
144	C₄×C₁₃	#382	1	4	yes	—
145	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
148	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
149	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	13	4	no	—
154	C₂×M₁₆	—	13	4	no	—
155	C₂×M₁₆	—	26	2	no	—
156	C₂×M₁₆	—	26	2	no	—
157	C₂×M₁₆	—	26	2	no	—
158	C₂×M₁₆	—	26	4	no	—
159	C₂×M₁₆	—	26	4	no	—
160	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	13	2	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	—	26	2	no	—
168	C₂²×C₈	—	26	2	no	—
169	#117	C₄×D₁₃	1	4	yes	—
170	#117	C₂²⋊D₁₃	1	4	yes	—
171	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
172	C₂×C₂²⋊C₄	Dic₂₆	1	2	yes	—
173	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
174	C₄⋊C₈	—	2	8	no	—
175	C₄⋊C₈	—	26	8	no	—
176	#97	—	13	8	no	—
177	C₄×C₈	—	13	8	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	—	2	4	no	—
181	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
182	C₂×C₁₃	#2916	1	2	yes	—
183	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
185	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
187	C₂²×C₄	#6129	1	2	yes	—
188	C₂²×C₄	#6129	1	2	yes	—
189	C₂²×C₄	#6131	1	2	yes	—
190	C₂²×C₄	#6131	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
192	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
193	M₁₆	—	2	4	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	4	4	no	—
196	M₁₆	—	26	8	no	—
197	M₁₆	—	26	4	no	—
198	M₁₆	—	26	4	no	—
199	M₁₆	—	52	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	—	13	2	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	26	8	no	—
206	C₂×C₈	—	26	1	no	—
207	C₂×C₈	—	26	1	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	—	26	1	no	—
210	C₂×C₈	—	26	1	no	—
211	C₂×C₈	—	26	2	no	—
212	C₂×C₈	—	26	4	no	—
213	C₂×C₈	—	52	4	no	—
214	C₄⋊C₄	#6101	1	8	yes	—
215	C₂²⋊C₄	#6101	1	8	yes	—
216	C₄²	C₂×C₄×D₁₃	1	4	yes	—
217	C₄²	C₂×C₂²⋊D₁₃	1	4	yes	—
218	C₁₃	#20768	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	4	2	no	—
220	C₂³	#88144	1	2	yes	—
221	C₂³	#88144	1	2	yes	—
222	C₂³	C₂×#6101	1	2	yes	—
223	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	#88144	1	2	yes	—
226	C₂×C₄	#88144	1	2	yes	—
227	C₂×C₄	#88146	1	4	yes	—
228	C₂×C₄	#88146	1	4	yes	—
229	C₂×C₄	C₂×#6101	1	8	yes	—
230	C₂×C₄	C₂×#6101	1	2	yes	—
231	C₂×C₄	#88190	1	4	yes	—
232	C₂×C₄	#88201	1	4	yes	—
233	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#6129	1	1	yes	characteristic
235	C₂×C₄	C₂×#6129	1	1	yes	characteristic
236	C₂×C₄	C₂×#6131	1	1	yes	characteristic
237	C₂×C₄	C₂×#6131	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₈	—	4	4	no	—
240	C₈	—	26	2	no	—
241	C₈	—	26	2	no	—
242	C₈	—	26	4	no	—
243	C₈	—	52	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#11718638	1	4	yes	—
249	C₂²	C₂×#88144	1	1	yes	characteristic
250	C₂²	C₂×#88144	1	1	yes	characteristic
251	C₂²	C₂²×#6101	1	1	yes	characteristic
252	C₄	—	2	8	no	—
253	C₄	#11718638	1	4	yes	—
254	C₄	C₂×#88146	1	1	yes	characteristic
255	C₄	C₂×#88146	1	1	yes	characteristic
256	C₄	C₂×#88190	1	1	yes	characteristic
257	C₄	C₂×#88201	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	4	no	—
261	C₂	C₂×#11718638	1	1	yes	characteristic
262	C₂	#13123445	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2905	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2905	D₁₃
hypocenter	C₁₃	#20768
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2905

Complements

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

N	H ≅ G/N	type
3. #13114976	C₂	semidirect
4. #13114973	C₂	semidirect
218. C₁₃	#20768	semidirect

Maximal subgroups

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

subgroup	index
#13123400	2
C₂×#11718658	2
#13114976	2
#13114973	2
#13114562	2
C₂×#11718150	2
C₁₃×#2905	2
#20768	13