Subgroups — #16789064

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal
2	#13129881	C₂	1	2	yes	maximal, complemented
3	#13129880	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	4	yes	—
9	D₁₃⋊M₃₂	—	2	4	no	—
10	D₁₃⋊M₃₂	C₄	1	8	yes	complemented
11	C₂×#88098	—	2	2	no	—
12	C₂×D₁₃⋊C₁₆	—	2	2	no	—
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₁₃	—	2	4	no	—
18	C₂²×C₄×D₁₃	—	2	1	no	—
19	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	—	2	1	no	—
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂³	1	1	yes	characteristic
27	C₂×C₈×D₁₃	—	2	2	no	—
28	C₂×C₈×D₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	C₈	1	2	yes	—
30	C₂×C₈×D₁₃	C₈	1	2	yes	—
31	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
33	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
34	C₂×C₈×D₁₃	D₄	1	4	yes	—
35	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
36	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
37	C₄²×D₁₃	—	2	2	no	—
38	C₄²×D₁₃	C₈	1	4	yes	—
39	#88106	—	2	4	no	—
40	#88106	C₂×C₄	1	8	yes	complemented
41	C₂×#6092	—	2	4	no	—
42	#88098	—	2	8	no	—
43	D₁₃⋊C₁₆	—	2	8	no	—
44	C₄×C₈×C₁₃	—	2	2	no	—
45	C₈×Dic₁₃	—	2	2	no	—
46	C₈×Dic₁₃	—	2	2	no	—
47	C₄×#1055	—	2	2	no	—
48	C₂×#1376	—	13	1	no	maximal, Sylow
49	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	2	1	no	—
53	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
54	C₂×C₄×D₁₃	—	2	8	no	—
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	C₄²	1	2	yes	—
57	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	C₄²	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂×C₈	1	2	yes	—
73	C₄×Dic₁₃	—	2	1	no	—
74	C₄×Dic₁₃	—	2	4	no	—
75	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
76	C₂×#1055	—	2	1	no	—
77	C₂×#1055	—	2	1	no	—
78	C₂×#1055	C₂×C₈	1	2	yes	—
79	C₂×#1055	C₂²×C₄	1	2	yes	—
80	C₂×#1055	C₂×D₄	1	1	yes	characteristic
81	C₂×#1055	C₂×D₄	1	2	yes	—
82	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
83	C₈×D₁₃	—	2	4	no	—
84	C₈×D₁₃	—	2	4	no	—
85	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
86	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
87	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
88	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
89	#6092	—	2	8	no	—
90	C₂²×M₃₂	—	13	2	no	—
91	C₂²×C₄×C₈	—	13	1	no	—
92	#1376	—	13	2	no	—
93	#1376	—	13	2	no	—
94	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂²×D₁₃	—	2	4	no	—
96	C₂²×D₁₃	#97	1	2	yes	—
97	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
98	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
99	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	C₂×M₁₆	1	1	yes	characteristic
108	C₂×Dic₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	#97	1	2	yes	—
111	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
112	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
113	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
114	C₄×D₁₃	—	2	16	no	—
115	C₄×D₁₃	#95	1	2	yes	—
116	C₄×D₁₃	#95	1	2	yes	—
117	C₄×D₁₃	C₄×C₈	1	2	yes	—
118	C₄×D₁₃	C₄×C₈	1	2	yes	—
119	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
120	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
121	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
122	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	#95	1	2	yes	—
125	C₈×C₁₃	#95	1	2	yes	—
126	#1055	—	2	2	no	—
127	#1055	C₂×C₂²⋊C₄	1	2	yes	—
128	#1055	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₈	—	13	1	no	—
130	C₂³×C₈	—	13	1	no	—
131	C₂²×C₄²	—	13	1	no	—
132	C₂×M₃₂	—	13	4	no	—
133	C₂×M₃₂	—	13	8	no	—
134	C₂×M₃₂	—	26	2	no	—
135	C₂×M₃₂	—	26	4	no	—
136	C₂²×C₁₆	—	26	2	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	—	13	1	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	C₂×C₄×C₈	—	26	4	no	—
141	C₂×C₄×C₈	#261	1	1	yes	characteristic
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂×D₁₃	—	2	2	no	—
147	C₂×D₁₃	—	2	4	no	—
148	C₂×D₁₃	—	2	4	no	—
149	C₂×D₁₃	#336	1	2	yes	—
150	C₂×D₁₃	#336	1	2	yes	—
151	C₂×D₁₃	#364	1	4	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
154	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
155	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
157	Dic₁₃	—	2	4	no	—
158	Dic₁₃	#336	1	2	yes	—
159	Dic₁₃	#336	1	2	yes	—
160	C₂³×C₄	—	13	1	no	—
161	C₂³×C₄	—	26	1	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	4	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	13	1	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	13	2	no	—
171	C₂²×C₈	—	13	2	no	—
172	C₂²×C₈	—	26	2	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	C₂×#261	1	1	yes	characteristic
175	C₂²×C₈	C₂×#261	1	1	yes	characteristic
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	13	4	no	—
178	C₂×C₄²	—	26	2	no	—
179	C₂×C₄²	—	26	1	no	—
180	C₂×C₄²	C₂×#261	1	1	yes	characteristic
181	M₃₂	—	13	8	no	—
182	M₃₂	—	26	8	no	—
183	M₃₂	—	26	4	no	—
184	C₂×C₁₆	—	26	8	no	—
185	C₂×C₁₆	—	26	4	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	26	2	no	—
188	C₄×C₈	—	26	2	no	—
189	C₄×C₈	—	26	2	no	—
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	1	no	—
192	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#364	1	2	yes	—
194	D₁₃	—	2	2	no	—
195	D₁₃	#1376	1	4	yes	complemented
196	C₂⁴	—	13	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	2	no	—
203	C₂²×C₄	—	13	2	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	1	no	—
206	C₂²×C₄	—	13	2	no	—
207	C₂²×C₄	—	26	4	no	—
208	C₂²×C₄	—	26	8	no	—
209	C₂²×C₄	—	26	1	no	—
210	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	13	4	no	—
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	13	4	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	26	4	no	—
223	C₂×C₈	—	26	4	no	—
224	C₂×C₈	—	26	1	no	—
225	C₂×C₈	—	26	1	no	—
226	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
227	C₂×C₈	C₄×#261	1	2	yes	—
228	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
229	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊#261	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₄²	—	13	2	no	—
233	C₄²	—	26	4	no	—
234	C₄²	—	26	1	no	—
235	C₄²	D₁₃⋊C₈	1	2	yes	—
236	C₁₆	—	26	8	no	—
237	C₁₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	13	1	no	—
239	C₂³	—	13	2	no	—
240	C₂³	—	13	1	no	—
241	C₂³	—	13	2	no	—
242	C₂³	—	26	4	no	—
243	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	13	2	no	—
247	C₂×C₄	—	13	1	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	13	2	no	—
254	C₂×C₄	—	13	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	26	16	no	—
259	C₂×C₄	—	26	4	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
263	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	13	2	no	—
269	C₈	—	13	2	no	—
270	C₈	—	26	2	no	—
271	C₈	(C₂×#261)⋊C₄	1	2	yes	—
272	C₈	(C₂×#261)⋊C₄	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	13	2	no	—
275	C₂²	—	13	4	no	—
276	C₂²	—	13	2	no	—
277	C₂²	—	26	4	no	—
278	C₂²	—	26	4	no	—
279	C₂²	—	26	2	no	—
280	C₂²	#11719047	1	1	yes	characteristic
281	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
282	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	13	2	no	—
285	C₄	—	13	2	no	—
286	C₄	—	26	4	no	—
287	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
288	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
289	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
290	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	13	4	no	—
294	C₂	—	26	2	no	—
295	C₂	#13130497	1	1	yes	characteristic
296	C₂	D₁₃⋊#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1376
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129881	C₂	semidirect
3. #13129880	C₂	semidirect
10. D₁₃⋊M₃₂	C₄	semidirect
40. #88106	C₂×C₄	semidirect
195. D₁₃	#1376	semidirect
237. C₁₃	C₂×#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊M₃₂	2
#13129881	2
#13129880	2
C₂×C₄×C₈×D₁₃	2
C₂×#1376	13