Subgroups — #16016428

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116745	C₂	1	2	yes	maximal, complemented
2	#13116745	C₂	1	4	yes	maximal, complemented
3	Q₈×#6090	C₂	1	2	yes	maximal, complemented
4	#13116595	C₂	1	2	yes	maximal
5	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
6	#13116433	C₂	1	2	yes	maximal
7	C₄×#88052	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₄	1	1	yes	characteristic
10	C₁₃×#446	C₄	1	2	yes	—
11	C₁₃×#446	C₄	1	2	yes	—
12	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#424	C₄	1	2	yes	—
15	C₁₃×#423	C₄	1	1	yes	characteristic
16	C₁₃×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	1	yes	characteristic
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	2	yes	—
21	C₄⋊#6090	C₂²	1	1	yes	characteristic
22	C₄⋊#6090	C₂²	1	4	yes	—
23	C₄⋊#6090	C₂²	1	1	yes	characteristic
24	C₄⋊#6090	C₂²	1	2	yes	—
25	C₄⋊#6090	C₂²	1	4	yes	—
26	#11718053	C₂²	1	2	yes	—
27	#11718053	C₂²	1	4	yes	—
28	C₄×#6090	C₂²	1	2	yes	—
29	C₄×#6090	C₂²	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
35	C₁₃×#125	C₈	1	2	yes	—
36	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
37	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
38	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₁₃×#117	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₃×#97	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	4	yes	—
52	#88052	—	2	2	no	—
53	#88052	Q₈	1	4	yes	complemented
54	C₂×#6090	—	2	4	no	—
55	C₂×#6090	C₂³	1	2	yes	—
56	C₂×#6090	C₂³	1	2	yes	—
57	C₂×#6090	C₂³	1	2	yes	—
58	C₂×#6090	C₂³	1	2	yes	—
59	#33476	—	13	1	no	maximal, Sylow
60	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
61	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
77	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
78	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
81	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
86	#6090	—	2	2	no	—
87	#6090	—	2	2	no	—
88	#6090	—	2	2	no	—
89	#6090	C₂×Q₈	1	4	yes	—
90	#2960	D₁₃	1	1	yes	characteristic
91	#2183	—	13	2	no	—
92	#2183	—	13	4	no	—
93	Q₈×C₁₆	—	13	2	no	—
94	#2151	—	13	2	no	—
95	C₂×C₄⋊C₁₆	—	13	1	no	—
96	#2108	—	13	2	no	—
97	C₄×M₃₂	—	13	1	no	—
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	4	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#131	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	C₄×Q₈	1	2	yes	—
117	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
119	#533	Dic₁₃	1	1	yes	characteristic
120	C₂×M₃₂	—	13	1	no	—
121	C₂×M₃₂	—	13	1	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	#446	Dic₁₃	1	2	yes	—
124	#446	Dic₁₃	1	2	yes	—
125	C₈×Q₈	C₂×D₁₃	1	2	yes	—
126	C₈×Q₈	C₂×D₁₃	1	2	yes	—
127	#424	Dic₁₃	1	2	yes	—
128	#423	Dic₁₃	1	1	yes	characteristic
129	#405	C₂×D₁₃	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	13	1	no	—
132	C₄⋊C₁₆	—	13	1	no	—
133	C₄⋊C₁₆	—	13	2	no	—
134	C₄⋊C₁₆	—	13	4	no	—
135	C₄⋊C₁₆	—	13	4	no	—
136	#346	—	13	2	no	—
137	#346	—	13	4	no	—
138	C₄×C₁₆	—	13	2	no	—
139	C₄×C₁₆	—	13	4	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	C₈×Q₈	1	2	yes	—
150	C₄×C₁₃	#504	1	2	yes	—
151	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
153	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
154	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₄⋊Q₈	#1055	1	1	yes	characteristic
157	C₄⋊Q₈	#1055	1	1	yes	characteristic
158	#125	#1055	1	2	yes	—
159	C₂²⋊Q₈	#1055	1	2	yes	—
160	C₂²⋊Q₈	#1055	1	2	yes	—
161	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
162	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
163	#117	C₂×Dic₁₃	1	2	yes	—
164	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
165	M₃₂	—	13	4	no	—
166	M₃₂	—	26	2	no	—
167	C₂×C₁₆	—	13	2	no	—
168	C₂×C₁₆	—	13	2	no	—
169	C₂×C₁₆	—	13	2	no	—
170	C₂×C₁₆	—	13	2	no	—
171	C₂×C₁₆	—	26	4	no	—
172	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
173	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
174	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
175	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
176	#97	C₂×Dic₁₃	1	4	yes	—
177	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₁₃	1	4	yes	—
181	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
185	C₂×C₁₃	#3407	1	1	yes	characteristic
186	C₂×Q₈	C₂×#1055	1	2	yes	—
187	C₂×Q₈	C₂×#1055	1	2	yes	—
188	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
189	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	Q₈×D₁₃	1	2	yes	—
194	C₂×C₈	#6131	1	2	yes	—
195	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
198	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂×#1055	1	2	yes	—
201	C₄⋊C₄	C₂×#1055	1	2	yes	—
202	C₄⋊C₄	C₂×#1055	1	4	yes	—
203	C₄⋊C₄	C₂×#1055	1	4	yes	—
204	C₄⋊C₄	C₂×#1055	1	2	yes	—
205	C₄⋊C₄	C₂×#1055	1	2	yes	—
206	C₂²⋊C₄	C₂×#1055	1	4	yes	—
207	C₄²	C₂×#1055	1	1	yes	characteristic
208	C₄²	C₂×#1055	1	1	yes	characteristic
209	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₄²	C₂²×Dic₁₃	1	4	yes	—
212	C₁₆	—	13	4	no	—
213	C₁₆	—	26	2	no	—
214	C₁₆	—	26	2	no	—
215	C₁₆	—	26	2	no	—
216	C₁₃	#33476	1	1	yes	characteristic, complemented, Sylow
217	C₂³	C₂²×#1055	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
223	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
224	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
225	C₂×C₄	C₂²×#1055	1	4	yes	—
226	C₂×C₄	C₂²×#1055	1	4	yes	—
227	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
228	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
229	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
230	C₂×C₄	#88202	1	2	yes	—
231	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	Q₈×Dic₁₃	1	2	yes	—
236	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
237	C₈	C₂×#6131	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×#1055	1	2	yes	—
241	C₂²	C₂³×#1055	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×#1055	1	2	yes	—
248	C₄	#11718801	1	2	yes	—
249	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
250	C₄	C₂×#88202	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×#1055	1	1	yes	characteristic
253	C₂	C₂×#11718801	1	1	yes	characteristic
254	C₂	#13127939	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2960	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#33476
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #13116745	C₂	semidirect
2. #13116745	C₂	semidirect
3. Q₈×#6090	C₂	semidirect
53. #88052	Q₈	semidirect
216. C₁₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#13116745	2
Q₈×#6090	2
#13116595	2
C₂×C₄⋊#6090	2
#13116433	2
C₄×#88052	2
C₁₃×#2960	2
#33476	13