Subgroups — #16069003

Order 3328.

1990 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123223	C₂	1	4	yes	maximal, complemented
2	#13123199	C₂	1	4	yes	maximal, complemented
3	#13123146	C₂	1	8	yes	maximal
4	C₄×#88124	C₂	1	8	yes	maximal, complemented
5	#13117319	C₂	1	4	yes	maximal
6	#13117243	C₂	1	2	yes	maximal
7	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	4	yes	—
12	C₂×#88124	C₂²	1	8	yes	—
13	C₄³×C₁₃	C₂²	1	2	yes	—
14	C₄³×C₁₃	C₂²	1	2	yes	—
15	C₄³×C₁₃	C₂²	1	4	yes	—
16	#11718873	C₂²	1	8	yes	—
17	#11718871	C₂²	1	16	yes	—
18	#11718597	C₂²	1	4	yes	—
19	#11718597	C₂²	1	8	yes	—
20	#11718596	C₂²	1	4	yes	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	8	yes	—
22	#11718594	C₂²	1	32	yes	—
23	C₄×#6101	C₂²	1	16	yes	—
24	#11718589	C₂²	1	16	yes	—
25	#11718589	C₂²	1	16	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	—	2	8	no	—
30	C₂×C₄²×C₁₃	—	2	16	no	—
31	C₂×C₄²×C₁₃	Q₈	1	8	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₁₃	C₂³	1	8	yes	—
41	#88180	—	2	16	no	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
44	C₂×#6101	C₂³	1	32	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	8	yes	—
46	#88124	—	2	32	no	—
47	#88071	C₂³	1	32	yes	—
48	#88071	C₂³	1	16	yes	—
49	#36640	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	8	no	—
52	C₂²×C₄×C₁₃	—	2	16	no	—
53	C₂²×C₄×C₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
55	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₁₃	#41	1	8	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
63	C₄²×C₁₃	—	2	16	no	—
64	C₄²×C₁₃	—	2	8	no	—
65	C₄²×C₁₃	—	2	8	no	—
66	C₄²×C₁₃	—	2	8	no	—
67	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
68	C₄²×C₁₃	C₂×Q₈	1	8	yes	—
69	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
70	C₄²×C₁₃	#41	1	16	yes	—
71	C₂²⋊Dic₁₃	—	2	16	no	—
72	C₄⋊Dic₁₃	—	2	32	no	—
73	C₄⋊Dic₁₃	—	2	16	no	—
74	#6101	—	2	64	no	—
75	C₄×Dic₁₃	—	2	16	no	—
76	#2864	—	13	4	no	—
77	#2859	—	13	4	no	—
78	#2851	—	13	8	no	—
79	C₄×#125	—	13	8	no	—
80	#2296	—	13	4	no	—
81	#2285	—	13	2	no	—
82	C₂×C₄³	D₁₃	1	1	yes	characteristic
83	C₂³×C₁₃	—	2	4	no	—
84	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
85	C₂³×C₁₃	C₂×#41	1	4	yes	—
86	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
87	C₂×C₄×C₁₃	—	2	32	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	8	no	—
90	C₂×C₄×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	2	16	no	—
92	C₂×C₄×C₁₃	#125	1	16	yes	complemented
93	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
94	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
97	C₂×C₄×C₁₃	C₂×#41	1	16	yes	—
98	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
100	C₂×C₄×C₁₃	C₂×#41	1	8	yes	—
101	C₂×Dic₁₃	—	2	16	no	—
102	C₂×Dic₁₃	—	2	32	no	—
103	C₂×#125	—	13	8	no	—
104	C₂×#117	—	13	4	no	—
105	C₂²×C₄²	C₂×D₁₃	1	4	yes	—
106	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
107	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
108	#389	—	13	16	no	—
109	#389	—	13	16	no	—
110	#387	—	13	32	no	—
111	#385	—	13	16	no	—
112	#384	—	13	4	no	—
113	#382	—	13	8	no	—
114	#382	—	13	4	no	—
115	#380	—	13	8	no	—
116	C₄×C₄⋊C₄	—	13	16	no	—
117	C₄×C₄⋊C₄	—	13	8	no	—
118	C₄³	C₂×D₁₃	1	4	yes	—
119	C₄³	C₂×D₁₃	1	2	yes	—
120	C₄³	C₂×D₁₃	1	2	yes	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	8	no	—
123	C₂²×C₁₃	C₂×#125	1	4	yes	—
124	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
125	C₂²×C₁₃	C₂²×#41	1	2	yes	—
126	C₂²×C₁₃	C₂²×#41	1	4	yes	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	2	16	no	—
129	C₄×C₁₃	C₂×#125	1	4	yes	—
130	C₄×C₁₃	C₂×#125	1	4	yes	—
131	C₄×C₁₃	#533	1	8	yes	—
132	Dic₁₃	—	4	16	no	—
133	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
134	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	#125	—	26	32	no	—
137	#117	—	26	16	no	—
138	C₂×C₄⋊C₄	—	13	8	no	—
139	C₂×C₄⋊C₄	—	13	32	no	—
140	C₂×C₄⋊C₄	—	13	16	no	—
141	C₂×C₂²⋊C₄	—	13	8	no	—
142	C₂×C₄²	—	2	8	no	—
143	C₂×C₄²	—	2	16	no	—
144	C₂×C₄²	—	13	8	no	—
145	C₂×C₄²	Dic₂₆	1	8	yes	—
146	C₂×C₄²	C₂²×D₁₃	1	8	yes	—
147	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
148	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
149	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
150	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
151	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
153	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
154	#95	—	13	16	no	—
155	#95	—	13	32	no	—
156	C₂×C₁₃	—	2	4	no	—
157	C₂×C₁₃	C₂²×#125	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#533	1	2	yes	—
159	C₂×C₁₃	#3444	1	4	yes	—
160	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	16	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	13	16	no	—
165	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
166	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
167	C₂²×C₄	#6127	1	8	yes	—
168	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
169	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
170	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
172	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
173	C₄⋊C₄	—	26	16	no	—
174	C₄⋊C₄	—	26	64	no	—
175	C₄⋊C₄	—	26	32	no	—
176	C₂²⋊C₄	—	26	16	no	—
177	C₄²	—	2	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	26	16	no	—
182	C₄²	C₂×Dic₂₆	1	4	yes	—
183	C₄²	C₂×Dic₂₆	1	8	yes	—
184	C₄²	C₂×Dic₂₆	1	4	yes	—
185	C₄²	#6127	1	16	yes	—
186	C₁₃	#36640	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	2	4	no	—
188	C₂³	C₂²×Dic₂₆	1	2	yes	—
189	C₂³	C₂×#6127	1	4	yes	—
190	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
191	C₂×C₄	—	2	32	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	26	16	no	—
197	C₂×C₄	—	26	32	no	—
198	C₂×C₄	#88124	1	16	yes	complemented
199	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
200	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
201	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
202	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
203	C₂×C₄	C₂×#6127	1	16	yes	—
204	C₂×C₄	C₂×#6127	1	4	yes	—
205	C₂×C₄	C₂×#6127	1	4	yes	—
206	C₂×C₄	C₂×#6127	1	8	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	C₂×#88124	1	4	yes	—
210	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
211	C₂²	C₂²×#6127	1	2	yes	—
212	C₂²	C₂²×#6127	1	4	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	16	no	—
215	C₄	—	52	16	no	—
216	C₄	C₂×#88124	1	4	yes	—
217	C₄	C₂×#88124	1	4	yes	—
218	C₄	#11719175	1	8	yes	—
219	C₂	—	2	4	no	—
220	C₂	C₂²×#88124	1	1	yes	characteristic
221	C₂	C₂×#11719175	1	2	yes	—
222	C₂	#13128132	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	#36640
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. #13123223	C₂	semidirect
2. #13123199	C₂	semidirect
4. C₄×#88124	C₂	semidirect
92. C₂×C₄×C₁₃	#125	semidirect
186. C₁₃	#36640	semidirect
198. C₂×C₄	#88124	semidirect

Maximal subgroups

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

subgroup	index
#13123223	2
#13123199	2
#13123146	2
C₄×#88124	2
#13117319	2
#13117243	2
C₂×C₄³×C₁₃	2
#36640	13