Subgroups — #15768655

Order 3328.

718 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718144	C₂	1	2	yes	maximal
2	C₂×#11718144	C₂	1	2	yes	maximal
3	#13114530	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6099	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	—	2	4	no	—
10	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
11	#11718144	—	2	4	no	—
12	#11718144	C₄	1	8	yes	complemented
13	#11718116	—	2	4	no	—
14	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	4	yes	—
24	C₂×C₄²×C₁₃	—	2	2	no	—
25	C₂×C₄²×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	D₄	1	2	yes	—
28	C₂×#6099	—	2	4	no	—
29	C₂×#6099	—	2	4	no	—
30	C₂×#6099	—	2	4	no	—
31	C₂×#6099	—	2	4	no	—
32	C₂×#6099	—	2	4	no	—
33	C₂×#6099	C₂×C₄	1	8	yes	—
34	C₂²×#1055	—	2	4	no	—
35	C₄×C₈×C₁₃	—	2	8	no	—
36	#10822	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	—	2	8	no	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	#41	1	2	yes	—
46	C₂²×C₄×C₁₃	#41	1	2	yes	—
47	C₂²×C₄×C₁₃	#41	1	2	yes	—
48	C₂²×C₄×C₁₃	#41	1	2	yes	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	8	no	—
54	C₂×C₈×C₁₃	—	2	8	no	—
55	C₂×C₈×C₁₃	C₄²	1	4	yes	—
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	4	no	—
60	#6099	—	2	8	no	—
61	#6099	—	2	8	no	—
62	#6099	—	4	4	no	—
63	#6099	—	4	4	no	—
64	C₂×#1055	—	2	8	no	—
65	C₂×#1055	—	4	4	no	—
66	C₂×#1055	—	4	4	no	—
67	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
68	C₂×#341	—	13	2	no	—
69	C₂×#341	—	13	2	no	—
70	#1728	—	13	2	no	—
71	C₂³×C₁₃	—	2	4	no	—
72	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
74	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
75	C₂³×C₁₃	#125	1	2	yes	—
76	C₂×C₄×C₁₃	—	2	16	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	#108	1	8	yes	—
80	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#117	1	4	yes	—
85	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₁₃	#121	1	2	yes	—
90	C₂×C₄×C₁₃	#123	1	4	yes	—
91	C₂×C₄×C₁₃	#124	1	2	yes	—
92	C₂×C₄×C₁₃	#126	1	4	yes	—
93	C₈×C₁₃	—	2	4	no	—
94	C₈×C₁₃	—	2	4	no	—
95	#1055	—	4	8	no	—
96	C₂²×M₁₆	—	13	4	no	—
97	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
98	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
99	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
100	C₂×C₄×C₈	—	2	4	no	—
101	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
102	#341	—	13	8	no	—
103	#341	—	26	4	no	—
104	#336	—	26	4	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	4	no	—
107	C₂²×C₁₃	—	2	4	no	—
108	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₁₃	#380	1	1	yes	characteristic
110	C₂²×C₁₃	#382	1	1	yes	characteristic
111	C₂²×C₁₃	#387	1	4	yes	—
112	C₂²×C₁₃	C₂×#108	1	4	yes	—
113	C₄×C₁₃	—	2	8	no	—
114	C₄×C₁₃	#376	1	1	yes	characteristic
115	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	#381	1	1	yes	characteristic
117	C₄×C₁₃	#383	1	1	yes	characteristic
118	C₄×C₁₃	#388	1	4	yes	—
119	C₂³×C₄	Dic₂₆	1	2	yes	—
120	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
121	C₂×M₁₆	—	13	8	no	—
122	C₂×M₁₆	—	26	4	no	—
123	C₂×M₁₆	—	26	4	no	—
124	C₂×M₁₆	—	26	4	no	—
125	C₂×M₁₆	—	26	4	no	—
126	C₂×M₁₆	—	26	4	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	26	4	no	—
132	C₂²×C₈	C₄×D₁₃	1	4	yes	—
133	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
134	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
135	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	D₅₂	1	2	yes	—
139	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
140	C₄×C₈	—	2	8	no	—
141	C₂×C₁₃	—	2	4	no	—
142	C₂×C₁₃	#1434	1	1	yes	characteristic
143	C₂×C₁₃	C₄×#108	1	2	yes	—
144	C₂×C₁₃	#1810	1	2	yes	—
145	C₂×C₁₃	#1837	1	2	yes	—
146	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
150	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
151	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
152	C₂²×C₄	#6127	1	2	yes	—
153	C₂²×C₄	#6127	1	2	yes	—
154	C₂²×C₄	#6127	1	2	yes	—
155	C₂²×C₄	#6127	1	2	yes	—
156	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
158	M₁₆	—	26	8	no	—
159	M₁₆	—	26	8	no	—
160	M₁₆	—	52	4	no	—
161	M₁₆	—	52	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	26	8	no	—
169	C₂×C₈	—	52	4	no	—
170	C₂×C₈	—	52	4	no	—
171	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
172	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₁₃	#10822	1	1	yes	characteristic, complemented, Sylow
177	C₂³	—	2	4	no	—
178	C₂³	C₄×Dic₂₆	1	2	yes	—
179	C₂³	#88124	1	2	yes	—
180	C₂³	#88178	1	2	yes	—
181	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	#88059	1	8	yes	—
186	C₂×C₄	#88126	1	4	yes	—
187	C₂×C₄	C₄×D₅₂	1	2	yes	—
188	C₂×C₄	#88129	1	2	yes	—
189	C₂×C₄	#88130	1	4	yes	—
190	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
191	C₂×C₄	#88180	1	1	yes	characteristic
192	C₂×C₄	#88180	1	1	yes	characteristic
193	C₂×C₄	#88180	1	1	yes	characteristic
194	C₂×C₄	#88180	1	1	yes	characteristic
195	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
196	C₂×C₄	#88183	1	4	yes	—
197	C₂×C₄	#88184	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₈	—	2	4	no	—
201	C₈	—	52	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#11718594	1	4	yes	—
206	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
207	C₂²	#11718597	1	1	yes	characteristic
208	C₂²	C₂×#88059	1	4	yes	—
209	C₂²	#11718873	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	#11718593	1	1	yes	characteristic
212	C₄	#11718598	1	1	yes	characteristic
213	C₄	#11718601	1	4	yes	—
214	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
215	C₄	#11718872	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	#13113063	1	1	yes	characteristic
218	C₂	C₄×#88059	1	2	yes	—
219	C₂	#13114874	1	2	yes	—
220	C₂	#13114974	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#10822
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
12. #11718144	C₄	semidirect
176. C₁₃	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718144	2
#13114530	2
C₂²×C₄×C₈×C₁₃	2
#10822	13