Subgroups — #15777985

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718145	C₂	1	2	yes	maximal
2	C₂×#11718058	C₂	1	2	yes	maximal, complemented
3	#13112918	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₄×C₁₃×M₁₆	—	2	4	no	—
9	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
10	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88052	C₂²	1	4	yes	—
14	#11718145	C₄	1	8	yes	complemented
15	#11718058	—	2	4	no	—
16	C₂³×C₄×C₁₃	—	2	1	no	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	2	no	—
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	—	2	1	no	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₁₃×#97	—	2	2	no	—
34	C₁₃×#97	D₄	1	4	yes	—
35	C₄×C₈×C₁₃	—	2	2	no	—
36	C₄×C₈×C₁₃	—	2	2	no	—
37	#88052	—	2	4	no	—
38	#88052	C₂×C₄	1	8	yes	complemented
39	C₂×#6090	—	2	4	no	—
40	#12607	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₁₃×M₁₆	—	2	8	no	—
55	C₁₃×M₁₆	—	2	8	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	C₄²	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	#41	1	2	yes	—
68	C₄²×C₁₃	—	2	4	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₃	M₁₆	1	2	yes	—
73	#6090	—	2	8	no	—
74	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
75	C₂×#349	—	13	2	no	—
76	C₂×#347	—	13	2	no	—
77	#1376	—	13	2	no	—
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	4	2	no	—
81	C₂³×C₁₃	#97	1	2	yes	—
82	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	8	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	#97	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	C₄×D₄	1	4	yes	—
105	C₂²×M₁₆	Dic₁₃	1	1	yes	characteristic
106	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
107	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
108	C₂×M₃₂	—	13	4	no	—
109	C₄×M₁₆	—	2	4	no	—
110	C₂×#97	Dic₁₃	1	1	yes	characteristic
111	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
112	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
114	#349	—	13	8	no	—
115	#347	—	26	4	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	4	2	no	—
119	C₂²×C₁₃	—	4	2	no	—
120	C₂²×C₁₃	—	4	2	no	—
121	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
122	C₂²×C₁₃	#407	1	1	yes	characteristic
123	C₂²×C₁₃	#432	1	1	yes	characteristic
124	C₄×C₁₃	—	2	4	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
129	C₄×C₁₃	#407	1	1	yes	characteristic
130	C₄×C₁₃	#432	1	1	yes	characteristic
131	C₂³×C₄	—	2	1	no	—
132	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
139	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
140	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	—	2	2	no	—
146	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
148	M₃₂	—	13	8	no	—
149	M₃₂	—	26	4	no	—
150	C₂×C₁₆	—	26	4	no	—
151	#97	—	2	2	no	—
152	#97	C₂²⋊D₁₃	1	4	yes	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₂×C₁₃	—	2	1	no	—
156	C₂×C₁₃	—	2	1	no	—
157	C₂×C₁₃	—	4	2	no	—
158	C₂×C₁₃	#1749	1	1	yes	characteristic
159	C₂×C₁₃	#2165	1	2	yes	—
160	C₂⁴	#6099	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	#6099	1	2	yes	—
170	C₂²×C₄	#6099	1	1	yes	characteristic
171	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	M₁₆	—	2	8	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	#6099	1	2	yes	—
182	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
183	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
184	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
185	C₂×C₈	#6127	1	2	yes	—
186	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	#6099	1	2	yes	—
191	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
192	C₁₆	—	26	8	no	—
193	C₁₃	#12607	1	1	yes	characteristic, complemented, Sylow
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#88043	1	2	yes	—
198	C₂³	C₂×#6099	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	#88043	1	2	yes	—
211	C₂×C₄	C₂×#6099	1	1	yes	characteristic
212	C₂×C₄	C₂×#6099	1	1	yes	characteristic
213	C₂×C₄	C₂×#6099	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
215	C₂×C₄	#88180	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	C₄×C₂²⋊D₁₃	1	4	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂×#88043	1	1	yes	characteristic
227	C₂²	#11718587	1	1	yes	characteristic
228	C₂²	#11718868	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₄×#6099	1	1	yes	characteristic
233	C₄	#11718587	1	1	yes	characteristic
234	C₄	#11718868	1	1	yes	characteristic
235	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	4	2	no	—
239	C₂	#13114598	1	1	yes	characteristic
240	C₂	#13116648	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#12607
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11718058	C₂	semidirect
3. #13112918	C₂	semidirect
14. #11718145	C₄	semidirect
38. #88052	C₂×C₄	semidirect
193. C₁₃	#12607	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718145	2
C₂×#11718058	2
#13112918	2
C₂×C₄×C₁₃×M₁₆	2
#12607	13