Subgroups — #16016426

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	#13123770	C₂	1	1	yes	maximal, characteristic
2	#13116744	C₂	1	2	yes	maximal, complemented
3	#13116742	C₂	1	4	yes	maximal, complemented
4	C₁₆×Dic₂₆	C₂	1	2	yes	maximal, complemented
5	#13116596	C₂	1	2	yes	maximal
6	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
7	#13116435	C₂	1	2	yes	maximal
8	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
9	#11719175	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718763	C₄	1	2	yes	—
14	#11718762	C₂²	1	2	yes	—
15	C₂²×#6090	C₂²	1	2	yes	—
16	#11718585	C₄	1	1	yes	characteristic
17	#11718340	C₄	1	2	yes	—
18	#11718340	C₄	1	2	yes	—
19	C₈×Dic₂₆	C₂²	1	2	yes	—
20	C₈×Dic₂₆	C₂²	1	2	yes	—
21	C₁₃×#346	C₂²	1	2	yes	—
22	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
23	#11718094	C₂²	1	4	yes	—
24	#11718093	C₂²	1	4	yes	—
25	#11718093	C₂²	1	4	yes	—
26	C₁₆×Dic₁₃	C₂²	1	4	yes	—
27	C₄⋊#6090	C₂²	1	1	yes	characteristic
28	C₄⋊#6090	C₂²	1	2	yes	—
29	C₄⋊#6090	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	#88180	C₂×C₄	1	2	yes	—
34	#88178	C₈	1	2	yes	—
35	#88178	C₈	1	2	yes	—
36	C₂×#6099	C₂×C₄	1	2	yes	—
37	#88124	C₈	1	2	yes	—
38	#88123	C₈	1	1	yes	characteristic
39	#88123	C₈	1	1	yes	characteristic
40	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
42	C₁₃×M₃₂	—	2	2	no	—
43	C₁₃×M₃₂	Q₈	1	4	yes	complemented
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	#88055	C₂×C₄	1	4	yes	—
51	#88054	C₂×C₄	1	4	yes	—
52	#88054	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₃	C₂³	1	4	yes	—
54	C₂×#6090	—	2	4	no	—
55	C₂×#6090	C₂³	1	2	yes	—
56	C₂×#6090	C₂³	1	2	yes	—
57	C₄⋊#1055	C₂×C₄	1	2	yes	—
58	C₄⋊#1055	C₂×C₄	1	2	yes	—
59	#33476	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
64	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
77	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
78	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
79	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
80	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
81	#6101	C₂×C₈	1	4	yes	—
82	#6101	C₂×C₈	1	4	yes	—
83	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
84	#6099	—	2	4	no	—
85	C₂×#1055	C₂²×C₄	1	4	yes	—
86	C₁₃×C₁₆	—	2	2	no	—
87	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
88	#6090	—	2	2	no	—
89	#6090	—	2	2	no	—
90	#2960	—	13	1	no	—
91	#2183	—	13	4	no	—
92	#2183	—	13	2	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	D₁₃	1	1	yes	characteristic
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	4	1	no	—
101	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
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	#131	1	2	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
110	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
111	Dic₂₆	—	2	4	no	—
112	Dic₂₆	—	4	2	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	C₄×Q₈	1	2	yes	—
116	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
118	#1055	—	2	4	no	—
119	#533	—	13	1	no	—
120	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
121	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	—	13	2	no	—
123	#446	—	13	2	no	—
124	#446	—	13	2	no	—
125	C₈×Q₈	—	13	2	no	—
126	C₈×Q₈	—	13	2	no	—
127	#424	—	13	2	no	—
128	#423	—	13	1	no	—
129	#405	—	13	2	no	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	C₄⋊C₁₆	—	13	4	no	—
132	C₄⋊C₁₆	—	13	4	no	—
133	C₄⋊C₁₆	—	13	1	no	—
134	C₄⋊C₁₆	—	13	1	no	—
135	C₄⋊C₁₆	—	13	2	no	—
136	#346	—	13	4	no	—
137	#346	C₂×D₁₃	1	2	yes	—
138	C₄×C₁₆	—	13	4	no	—
139	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
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	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	C₈×Q₈	1	2	yes	—
148	C₄×C₁₃	#504	1	2	yes	—
149	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	—	2	4	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	13	1	no	—
157	C₄⋊Q₈	—	13	1	no	—
158	#125	—	13	2	no	—
159	C₂²⋊Q₈	—	13	2	no	—
160	C₂²⋊Q₈	—	13	2	no	—
161	C₄×Q₈	—	13	2	no	—
162	C₄×Q₈	—	13	2	no	—
163	#117	—	13	2	no	—
164	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
165	M₃₂	—	2	2	no	—
166	M₃₂	Dic₂₆	1	4	yes	complemented
167	C₂×C₁₆	—	13	2	no	—
168	C₂×C₁₆	—	13	2	no	—
169	C₂×C₁₆	—	26	4	no	—
170	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
171	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
172	C₄⋊C₈	—	13	4	no	—
173	C₄⋊C₈	—	13	4	no	—
174	C₄⋊C₈	—	13	2	no	—
175	C₄⋊C₈	—	13	2	no	—
176	#97	—	13	4	no	—
177	C₄×C₈	—	13	4	no	—
178	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
179	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
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	—	13	2	no	—
187	C₂×Q₈	—	13	2	no	—
188	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
191	M₁₆	—	26	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	13	4	no	—
194	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
195	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
198	C₂×C₈	#6127	1	2	yes	—
199	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
200	C₄⋊C₄	—	13	2	no	—
201	C₄⋊C₄	—	13	2	no	—
202	C₄⋊C₄	—	13	4	no	—
203	C₄⋊C₄	—	13	4	no	—
204	C₄⋊C₄	—	13	2	no	—
205	C₄⋊C₄	—	13	2	no	—
206	C₂²⋊C₄	—	13	4	no	—
207	C₄²	—	13	4	no	—
208	C₄²	C₈×D₁₃	1	1	yes	characteristic
209	C₄²	C₈×D₁₃	1	1	yes	characteristic
210	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₁₆	—	2	2	no	—
213	C₁₆	—	26	2	no	—
214	C₁₆	—	26	2	no	—
215	C₁₆	C₂×Dic₂₆	1	4	yes	—
216	C₁₃	#33476	1	1	yes	characteristic, complemented, Sylow
217	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
218	Q₈	—	26	4	no	—
219	Q₈	—	52	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	13	4	no	—
223	C₂×C₄	—	13	4	no	—
224	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
225	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
230	C₂×C₄	#88155	1	2	yes	—
231	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	26	4	no	—
235	C₈	C₄×Dic₂₆	1	2	yes	—
236	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
237	C₈	C₂×#6127	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₈×Dic₂₆	1	2	yes	—
241	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	26	4	no	—
246	C₄	—	26	4	no	—
247	C₄	C₈×Dic₂₆	1	2	yes	—
248	C₄	#11718568	1	2	yes	—
249	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
250	C₄	C₂×#88155	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×C₈×Dic₂₆	1	1	yes	characteristic
253	C₂	C₂×#11718568	1	1	yes	characteristic
254	C₂	#13127938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×M₃₂	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33476
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13116744	C₂	semidirect
3. #13116742	C₂	semidirect
4. C₁₆×Dic₂₆	C₂	semidirect
43. C₁₃×M₃₂	Q₈	semidirect
166. M₃₂	Dic₂₆	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
#13123770	2
#13116744	2
#13116742	2
C₁₆×Dic₂₆	2
#13116596	2
C₂×C₄⋊#6090	2
#13116435	2
C₄×C₁₃×M₃₂	2
#33476	13