Subgroups — #16018277

Order 3328.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic
2	#13117081	C₂	1	4	yes	maximal, complemented
3	#13117063	C₂	1	4	yes	maximal, complemented
4	#13116556	C₂	1	4	yes	maximal
5	C₄×#88052	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
10	#11719175	—	2	1	no	—
11	C₁₃×#494	—	2	1	no	—
12	C₁₃×#494	—	2	1	no	—
13	C₁₃×#494	C₂²	1	2	yes	—
14	C₁₃×#494	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718767	—	2	1	no	—
17	#11718767	C₂²	1	2	yes	—
18	C₂×#88052	C₂²	1	2	yes	—
19	#11718344	—	2	1	no	—
20	#11718344	C₂²	1	2	yes	—
21	#11718343	C₂²	1	2	yes	—
22	#11718341	—	2	1	no	—
23	#11718156	C₂²	1	16	yes	—
24	#11718053	C₂²	1	2	yes	—
25	C₄×#6090	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂×C₁₃×Q₁₆	—	2	2	no	—
28	C₂×C₁₃×Q₁₆	—	2	2	no	—
29	C₂×C₁₃×Q₁₆	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	—	2	8	no	—
31	C₂×C₁₃×Q₁₆	—	4	4	no	—
32	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	4	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	#88180	—	2	2	no	—
41	#88178	—	4	2	no	—
42	#88124	—	2	2	no	—
43	#88123	—	2	2	no	—
44	C₄×Dic₂₆	—	4	2	no	—
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	#88058	—	2	2	no	—
50	#88058	C₂³	1	4	yes	—
51	#88057	—	2	2	no	—
52	#88057	—	2	2	no	—
53	#88052	—	2	4	no	—
54	C₂×#6090	C₂³	1	4	yes	—
55	#33705	—	13	1	no	maximal, Sylow
56	C₂×Q₈×C₁₃	—	2	4	no	—
57	C₂×Q₈×C₁₃	—	4	2	no	—
58	C₂×Q₈×C₁₃	—	4	4	no	—
59	C₂×Q₈×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	D₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₂₆	—	4	2	no	—
64	C₁₃×Q₁₆	—	2	8	no	—
65	C₁₃×Q₁₆	—	4	4	no	—
66	C₁₃×Q₁₆	—	4	4	no	—
67	C₁₃×Q₁₆	—	4	4	no	—
68	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	4	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₄²×C₁₃	D₈	1	2	yes	—
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²⋊Dic₁₃	—	4	2	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	4	2	no	—
82	#6101	—	4	4	no	—
83	C₄×Dic₁₃	—	4	2	no	—
84	#6090	—	2	4	no	—
85	#3160	—	13	1	no	—
86	C₂×#494	D₁₃	1	1	yes	characteristic
87	#2245	—	13	4	no	—
88	#2241	—	13	4	no	—
89	#2141	—	13	4	no	—
90	C₄×M₃₂	—	13	1	no	—
91	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₁₃	—	4	4	no	—
93	Q₈×C₁₃	—	4	4	no	—
94	Q₈×C₁₃	—	8	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	8	4	no	—
98	C₂×C₄×C₁₃	#124	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₁₃	—	4	4	no	—
105	Dic₂₆	—	8	4	no	—
106	C₈×C₁₃	#124	1	2	yes	—
107	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
108	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
109	C₈×C₁₃	C₂×#41	1	2	yes	—
110	C₂²×Q₁₆	—	2	1	no	—
111	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
112	#533	—	26	1	no	—
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂×M₃₂	—	13	2	no	—
115	#500	—	13	2	no	—
116	#499	—	13	2	no	—
117	#499	—	26	1	no	—
118	#498	—	26	1	no	—
119	#494	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	C₂×D₁₃	1	2	yes	—
122	#494	C₂×D₁₃	1	2	yes	—
123	#427	—	13	2	no	—
124	#427	—	26	1	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	#358	—	13	16	no	—
127	#346	—	13	2	no	—
128	C₄×C₁₆	—	13	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	#486	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	8	4	no	—
134	C₄×C₁₃	#486	1	2	yes	—
135	C₄×C₁₃	#510	1	4	yes	—
136	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
137	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
138	Dic₁₃	—	8	4	no	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂×Q₁₆	—	2	8	no	—
141	C₂×Q₁₆	—	2	2	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	4	4	no	—
145	C₂×Q₁₆	C₂²×D₁₃	1	4	yes	—
146	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	C₄⋊Q₈	—	26	2	no	—
152	#125	—	26	2	no	—
153	C₂²⋊Q₈	—	52	2	no	—
154	C₄×Q₈	—	52	2	no	—
155	#117	—	26	2	no	—
156	C₂×C₄⋊C₄	—	4	2	no	—
157	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
158	M₃₂	—	26	4	no	—
159	C₂×C₁₆	—	13	4	no	—
160	#107	—	13	4	no	—
161	#107	—	26	2	no	—
162	#106	—	26	2	no	—
163	#106	—	26	2	no	—
164	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×C₁₃	—	2	2	no	—
169	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#510	1	2	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	52	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	#6104	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
179	Q₁₆	—	2	8	no	—
180	Q₁₆	—	4	4	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	C₂×C₈	#6105	1	4	yes	—
184	C₂×C₈	#6129	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	26	4	no	—
192	C₄⋊C₄	—	52	4	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₂²⋊C₄	—	52	2	no	—
195	C₄²	—	52	2	no	—
196	C₄²	#6104	1	2	yes	—
197	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
199	C₁₆	—	26	4	no	—
200	C₁₃	#33705	1	1	yes	characteristic, complemented, Sylow
201	C₂³	C₂×#6104	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	104	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	52	4	no	—
210	C₂×C₄	C₂×#6104	1	2	yes	—
211	C₂×C₄	C₂×#6104	1	1	yes	characteristic
212	C₂×C₄	C₂×#6104	1	2	yes	—
213	C₂×C₄	C₂×#6105	1	2	yes	—
214	C₂×C₄	#88190	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₈	C₂×#6105	1	2	yes	—
217	C₈	C₂×#6105	1	2	yes	—
218	C₈	#88190	1	2	yes	—
219	C₈	C₂×#6129	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#11718731	1	2	yes	—
222	C₂²	C₂²×#6104	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	104	4	no	—
226	C₄	#11718731	1	2	yes	—
227	C₄	#11718840	1	4	yes	—
228	C₄	C₂²×#6105	1	1	yes	characteristic
229	C₄	C₂×#88190	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#11718731	1	1	yes	characteristic
232	C₂	C₂×#11718840	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#33705
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #13117081	C₂	semidirect
3. #13117063	C₂	semidirect
200. C₁₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13117081	2
#13117063	2
#13116556	2
C₄×#88052	2
C₂×C₁₃×#494	2
#33705	13