Subgroups — #16016048

Order 3328.

520 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123262	C₂	1	1	yes	maximal, characteristic
2	#13116649	C₂	1	4	yes	maximal, complemented
3	#13116645	C₂	1	4	yes	maximal, complemented
4	#13116601	C₂	1	2	yes	maximal, complemented
5	#13116457	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718104	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2115	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	#41×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×#88202	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	#11718783	C₄	1	1	yes	characteristic
13	#11718783	C₂²	1	2	yes	—
14	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
15	M₁₆×Dic₁₃	C₂²	1	2	yes	—
16	#11718762	C₄	1	2	yes	—
17	#11718762	C₂²	1	1	yes	characteristic
18	C₂×#88055	C₄	1	2	yes	—
19	C₂×#88055	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	2	yes	—
21	C₁₃×#349	C₂²	1	4	yes	—
22	#11718145	C₂²	1	4	yes	—
23	#11718104	C₂²	1	4	yes	—
24	#11718104	C₂²	1	4	yes	—
25	#11718103	C₂²	1	8	yes	—
26	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
27	C₁₃×#131	—	2	2	no	—
28	C₁₃×#131	—	2	2	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	#88202	—	2	4	no	—
34	Q₈×Dic₁₃	—	2	1	no	—
35	D₄×Dic₁₃	—	2	2	no	—
36	D₄×Dic₁₃	—	2	1	no	—
37	#88180	C₂×C₄	1	2	yes	—
38	#88180	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#6099	C₂×C₄	1	2	yes	—
42	C₂×#6099	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	2	yes	—
44	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	#88055	C₂×C₄	1	1	yes	characteristic
49	#88055	C₂×C₄	1	2	yes	—
50	#88055	C₂×C₄	1	2	yes	—
51	#88055	C₂×C₄	1	2	yes	—
52	#88055	C₂×C₄	1	1	yes	characteristic
53	#88055	C₂³	1	1	yes	characteristic
54	#88055	C₂³	1	1	yes	characteristic
55	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	#88052	—	2	2	no	—
59	#88052	C₂³	1	4	yes	—
60	C₂×#6090	—	2	2	no	—
61	#33426	—	13	1	no	maximal, Sylow
62	C₁₃×#41	—	2	2	no	—
63	C₁₃×#41	—	2	2	no	—
64	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
65	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
66	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	—	2	1	no	—
71	C₁₃×M₁₆	—	2	2	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₁₃	—	2	2	no	—
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄×Dic₁₃	—	2	2	no	—
85	C₄×Dic₁₃	—	2	1	no	—
86	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
89	#6099	—	2	4	no	—
90	#6099	M₁₆	1	4	yes	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
94	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	C₁₃×C₁₆	—	2	4	no	—
97	#6090	—	2	4	no	—
98	#2875	—	13	1	no	—
99	#2166	—	13	4	no	—
100	#2165	—	13	4	no	—
101	#2153	—	13	2	no	—
102	#2115	—	13	1	no	—
103	#2115	D₁₃	1	1	yes	characteristic
104	C₂×#347	—	13	2	no	—
105	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
106	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	2	no	—
108	D₄×C₁₃	—	2	4	no	—
109	D₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	#117	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	2	no	—
120	C₂×Dic₁₃	—	8	1	no	—
121	C₂×Dic₁₃	—	8	2	no	—
122	C₈×C₁₃	—	2	2	no	—
123	C₈×C₁₃	#117	1	1	yes	characteristic
124	C₈×C₁₃	#117	1	1	yes	characteristic
125	C₈×C₁₃	C₂×#41	1	2	yes	—
126	#1055	—	2	1	no	—
127	#1055	—	2	1	no	—
128	#1055	C₂×M₁₆	1	4	yes	—
129	C₂×#131	—	13	1	no	—
130	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
131	C₄×#41	—	13	1	no	—
132	C₂×M₃₂	—	13	2	no	—
133	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
134	#405	—	13	1	no	—
135	#405	—	13	2	no	—
136	#405	—	13	2	no	—
137	#405	—	13	1	no	—
138	C₄×M₁₆	—	13	1	no	—
139	C₄×M₁₆	—	13	2	no	—
140	C₂×#97	—	13	2	no	—
141	C₂×#97	—	13	1	no	—
142	#364	—	13	8	no	—
143	#349	—	13	4	no	—
144	#349	C₂×D₁₃	1	4	yes	—
145	#347	—	13	4	no	—
146	#347	—	13	4	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	#432	1	2	yes	—
151	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	#432	1	2	yes	—
154	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
156	Dic₁₃	—	4	1	no	—
157	Dic₁₃	—	4	2	no	—
158	Dic₁₃	—	4	1	no	—
159	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	—	26	4	no	—
163	C₂×M₁₆	—	13	1	no	—
164	C₂×M₁₆	—	13	2	no	—
165	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
166	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
167	C₂²×C₈	—	13	2	no	—
168	C₂²×C₈	—	13	1	no	—
169	C₂²×C₈	C₄×D₁₃	1	2	yes	—
170	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₄×Q₈	—	26	1	no	—
172	C₄×D₄	—	26	2	no	—
173	C₄×D₄	—	26	1	no	—
174	#117	—	13	2	no	—
175	#117	—	13	1	no	—
176	C₂×C₄²	—	13	2	no	—
177	C₂×C₄²	—	13	1	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	—	13	4	no	—
180	M₃₂	—	26	2	no	—
181	M₃₂	C₂²×D₁₃	1	4	yes	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	26	2	no	—
184	#97	—	13	1	no	—
185	#97	—	13	2	no	—
186	#97	—	13	2	no	—
187	#97	—	13	1	no	—
188	#97	—	13	2	no	—
189	#97	—	13	1	no	—
190	#97	—	13	1	no	—
191	C₄×C₈	—	13	2	no	—
192	C₄×C₈	—	13	2	no	—
193	C₄×C₈	—	13	2	no	—
194	C₂×C₁₃	—	2	1	no	—
195	C₂×C₁₃	—	2	2	no	—
196	C₂×C₁₃	—	4	2	no	—
197	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	C₂×Q₈	#6094	1	1	yes	characteristic
201	C₂×D₄	#6094	1	2	yes	—
202	C₂×D₄	#6094	1	1	yes	characteristic
203	C₂²×C₄	—	26	2	no	—
204	C₂²×C₄	—	26	1	no	—
205	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	13	4	no	—
210	M₁₆	—	26	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	26	4	no	—
217	C₂×C₈	—	26	2	no	—
218	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₈	#6129	1	2	yes	—
221	C₂×C₈	#6129	1	2	yes	—
222	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
223	C₄⋊C₄	—	26	2	no	—
224	C₄⋊C₄	—	26	1	no	—
225	C₂²⋊C₄	—	26	2	no	—
226	C₂²⋊C₄	—	26	1	no	—
227	C₄²	—	13	1	no	—
228	C₄²	—	13	2	no	—
229	C₄²	—	13	1	no	—
230	C₄²	—	26	2	no	—
231	C₄²	—	26	1	no	—
232	C₁₆	—	2	4	no	—
233	C₁₆	—	26	4	no	—
234	C₁₃	#33426	1	1	yes	characteristic, complemented, Sylow
235	C₂³	C₂×#6094	1	2	yes	—
236	C₂³	C₂×#6094	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	26	1	no	—
243	C₂×C₄	—	26	1	no	—
244	C₂×C₄	—	26	2	no	—
245	C₂×C₄	—	104	1	no	—
246	C₂×C₄	—	104	2	no	—
247	C₂×C₄	#88131	1	2	yes	—
248	C₂×C₄	C₂×#6094	1	2	yes	—
249	C₂×C₄	C₂×#6094	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	13	4	no	—
254	C₈	—	26	1	no	—
255	C₈	—	26	1	no	—
256	C₈	#88131	1	1	yes	characteristic
257	C₈	#88131	1	1	yes	characteristic
258	C₈	C₂×#6129	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	#11718380	1	2	yes	—
263	C₂²	C₂²×#6094	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	52	1	no	—
266	C₄	—	52	2	no	—
267	C₄	—	52	1	no	—
268	C₄	#11718380	1	2	yes	—
269	C₄	C₂×#88131	1	1	yes	characteristic
270	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#11718380	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88131
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2115	C₂
socle	C₂×C₁₃	C₂×#432
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2115	D₁₃
hypocenter	C₁₃	#33426
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2115

Complements

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

N	H ≅ G/N	type
2. #13116649	C₂	semidirect
3. #13116645	C₂	semidirect
4. #13116601	C₂	semidirect
6. C₂×#11718104	C₂	semidirect
234. C₁₃	#33426	semidirect

Maximal subgroups

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

subgroup	index
#13123262	2
#13116649	2
#13116645	2
#13116601	2
#13116457	2
C₂×#11718104	2
C₁₃×#2115	2
#33426	13