Subgroups — #16018425

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125372	C₂	1	2	yes	maximal
2	#13117120	C₂	1	4	yes	maximal, complemented
3	#13117113	C₂	1	2	yes	maximal, complemented
4	#13117110	C₂	1	2	yes	maximal, complemented
5	#13116622	C₂	1	2	yes	maximal
6	#13116620	C₂	1	1	yes	maximal, characteristic
7	#13116432	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic
9	#11719205	—	2	2	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	#11718767	C₂²	1	2	yes	—
13	#11718762	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	#11718625	—	2	2	no	—
16	#11718539	C₂²	1	4	yes	—
17	#11718537	—	2	4	no	—
18	#11718535	C₂²	1	4	yes	—
19	C₁₃×#366	C₂²	1	2	yes	—
20	C₁₃×#366	C₂²	1	2	yes	—
21	#11718097	C₂²	1	2	yes	—
22	#11718096	C₂²	1	1	yes	characteristic
23	#11718096	C₂²	1	1	yes	characteristic
24	#11718094	C₂²	1	2	yes	—
25	C₁₆×Dic₁₃	C₂²	1	2	yes	—
26	#11718082	C₂²	1	4	yes	—
27	#11718081	C₂²	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	#88180	—	2	2	no	—
31	#88180	D₄	1	1	yes	characteristic
32	#88178	—	4	4	no	—
33	C₂×#6099	D₄	1	1	yes	characteristic
34	#88144	—	2	4	no	—
35	#88144	—	4	2	no	—
36	#88142	—	2	4	no	—
37	#88142	—	4	2	no	—
38	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
39	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₁₃×#107	—	2	2	no	—
42	C₁₃×#107	C₂³	1	4	yes	—
43	#88058	C₂³	1	2	yes	—
44	#88058	C₂³	1	2	yes	—
45	#88057	—	2	2	no	—
46	#88055	D₄	1	2	yes	—
47	C₈×Dic₁₃	C₂³	1	2	yes	—
48	#88052	Q₈	1	4	yes	complemented
49	C₂×#6090	C₂³	1	2	yes	—
50	#88048	—	2	4	no	—
51	#88047	—	2	4	no	—
52	#33729	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₂₆	—	4	4	no	—
56	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₂²⋊Dic₁₃	—	4	2	no	—
63	C₂²⋊Dic₁₃	D₈	1	2	yes	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄⋊Dic₁₃	—	2	2	no	—
66	C₄⋊Dic₁₃	—	4	4	no	—
67	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
68	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
69	#6101	—	4	4	no	—
70	#6101	—	4	4	no	—
71	C₄×Dic₁₃	—	4	2	no	—
72	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
73	#6099	Q₁₆	1	4	yes	—
74	C₂×#1055	C₂×D₄	1	2	yes	—
75	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
77	#6090	C₂×Q₈	1	4	yes	—
78	#3164	—	13	2	no	—
79	#2255	—	13	4	no	—
80	#2253	—	13	2	no	—
81	#2252	—	13	2	no	—
82	#2159	—	13	2	no	—
83	#2158	—	13	1	no	—
84	C₂×#366	D₁₃	1	1	yes	characteristic
85	#2108	—	13	1	no	—
86	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	16	2	no	—
89	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
96	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
97	Dic₂₆	—	16	4	no	—
98	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
99	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	#1055	C₂×Q₁₆	1	4	yes	—
102	#544	—	26	2	no	—
103	C₂×M₃₂	—	13	1	no	—
104	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
105	#501	—	26	4	no	—
106	#500	—	13	4	no	—
107	#499	—	13	4	no	—
108	#428	—	26	2	no	—
109	#427	—	13	2	no	—
110	C₂×#107	C₂×D₁₃	1	2	yes	—
111	#405	—	13	1	no	—
112	#367	—	13	2	no	—
113	#367	—	13	4	no	—
114	#366	—	13	1	no	—
115	#366	—	13	1	no	—
116	#366	—	13	4	no	—
117	#366	C₂×D₁₃	1	2	yes	—
118	#366	C₂×D₁₃	1	2	yes	—
119	#346	—	13	2	no	—
120	C₄×C₁₆	—	13	2	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	#500	1	2	yes	—
124	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₃	—	8	4	no	—
126	C₄×C₁₃	#500	1	2	yes	—
127	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	—	2	2	no	—
131	Dic₁₃	—	8	2	no	—
132	Dic₁₃	—	8	2	no	—
133	C₂×M₁₆	—	13	1	no	—
134	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	26	4	no	—
136	C₄⋊Q₈	—	52	2	no	—
137	#125	—	26	4	no	—
138	#125	—	52	2	no	—
139	C₂²⋊Q₈	—	52	4	no	—
140	#117	—	13	1	no	—
141	#117	—	26	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	M₃₂	—	13	4	no	—
144	C₂×C₁₆	—	13	2	no	—
145	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
146	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
148	#107	—	2	2	no	—
149	#107	—	13	2	no	—
150	#107	—	13	2	no	—
151	#107	—	26	4	no	—
152	#107	C₂²×D₁₃	1	4	yes	—
153	#106	—	26	4	no	—
154	#106	—	26	2	no	—
155	#97	—	13	2	no	—
156	C₄×C₈	—	13	2	no	—
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
159	C₂×C₁₃	#3415	1	1	yes	characteristic
160	C₂×C₁₃	#3417	1	1	yes	characteristic
161	C₂×Q₈	—	52	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
164	M₁₆	—	13	4	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
167	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₁₃	1	2	yes	—
169	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	13	1	no	—
173	C₄⋊C₄	—	13	1	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₄⋊C₄	—	52	4	no	—
177	C₄⋊C₄	—	52	4	no	—
178	C₄⋊C₄	—	52	4	no	—
179	C₂²⋊C₄	—	13	2	no	—
180	C₂²⋊C₄	—	52	2	no	—
181	C₄²	—	13	2	no	—
182	C₄²	—	52	2	no	—
183	C₁₆	—	13	4	no	—
184	C₁₆	C₂×Dic₂₆	1	2	yes	—
185	C₁₆	C₂×Dic₂₆	1	2	yes	—
186	C₁₃	#33729	1	1	yes	characteristic, complemented, Sylow
187	C₂³	D₈×D₁₃	1	1	yes	characteristic
188	Q₈	—	208	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	13	2	no	—
191	C₂×C₄	—	13	2	no	—
192	C₂×C₄	—	16	2	no	—
193	C₂×C₄	—	52	2	no	—
194	C₂×C₄	—	52	2	no	—
195	C₂×C₄	#88142	1	2	yes	—
196	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
199	C₈	—	13	4	no	—
200	C₈	#88142	1	2	yes	—
201	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
202	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#11718535	1	2	yes	—
206	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
207	C₄	—	8	4	no	—
208	C₄	—	26	2	no	—
209	C₄	—	26	2	no	—
210	C₄	—	104	2	no	—
211	C₄	—	104	2	no	—
212	C₄	#11718535	1	2	yes	—
213	C₄	C₂×#88142	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#11718535	1	1	yes	characteristic
217	C₂	#13127973	1	1	yes	characteristic
218	C₂	#13127985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#366	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#366	D₁₃
hypocenter	C₁₃	#33729
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #13117120	C₂	semidirect
3. #13117113	C₂	semidirect
4. #13117110	C₂	semidirect
38. C₂×C₁₃×C₁₆	Q₈	semidirect
48. #88052	Q₈	semidirect
145. C₂×C₁₆	Dic₂₆	semidirect
186. C₁₃	#33729	semidirect

Maximal subgroups

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

subgroup	index
#13125372	2
#13117120	2
#13117113	2
#13117110	2
#13116622	2
#13116620	2
#13116432	2
C₂×C₁₃×#366	2
#33729	13