Subgroups — #16807006

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	2	yes	maximal
2	#13131172	C₂	1	4	yes	maximal, complemented
3	#13131171	C₂	1	2	yes	maximal, complemented
4	#13131170	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131169	C₂	1	2	yes	maximal, complemented
6	C₂×#11718260	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃⋊(C₄⋊C₁₆)	C₂	1	2	yes	maximal, complemented
8	#13123258	C₂	1	1	yes	maximal, characteristic
9	C₂×#88161	C₂²	1	1	yes	characteristic
10	C₂×#88155	C₄	1	1	yes	characteristic
11	#11719227	C₄	1	1	yes	characteristic
12	C₂×#88106	C₂²	1	2	yes	—
13	C₂²×#6092	C₂²	1	2	yes	—
14	#11719096	—	2	8	no	—
15	C₂×#88098	C₂²	1	4	yes	—
16	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
17	C₁₃×#405	C₄	1	1	yes	characteristic
18	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	#11718658	C₂²	1	1	yes	characteristic
21	#11718369	C₂²	1	2	yes	—
22	#11718368	C₄	1	2	yes	—
23	#11718350	C₄	1	2	yes	—
24	C₄×C₈×D₁₃	C₂²	1	2	yes	—
25	#11718260	C₂²	1	1	yes	characteristic
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	1	yes	characteristic
28	#11718260	C₂²	1	2	yes	—
29	#11718260	C₂²	1	2	yes	—
30	C₄⋊#6092	C₂²	1	4	yes	—
31	C₄⋊#6092	C₂²	1	4	yes	—
32	C₂×#6127	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	#88161	D₄	1	4	yes	—
40	#88161	Q₈	1	4	yes	—
41	#88155	—	2	4	no	—
42	C₂×#6094	C₂×C₄	1	2	yes	—
43	C₂×#6094	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂³	1	2	yes	—
46	#88147	C₈	1	1	yes	characteristic
47	#88147	C₈	1	1	yes	characteristic
48	#88141	C₈	1	1	yes	characteristic
49	#88141	C₈	1	1	yes	characteristic
50	#88135	C₈	1	2	yes	—
51	#88135	C₈	1	2	yes	—
52	#88126	C₂×C₄	1	2	yes	—
53	C₄²×D₁₃	C₂×C₄	1	2	yes	—
54	#88106	—	2	4	no	—
55	C₂×#6092	—	2	4	no	—
56	C₂×#6092	C₂³	1	2	yes	—
57	C₂×#6092	C₂³	1	4	yes	—
58	C₂×#6092	C₂³	1	2	yes	—
59	#88098	—	2	8	no	—
60	D₁₃⋊C₁₆	—	2	8	no	—
61	C₁₃×#97	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
63	#88055	C₂×C₄	1	2	yes	—
64	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
69	#88043	C₂×C₄	1	2	yes	—
70	C₄×#1055	C₂³	1	2	yes	—
71	#33398	—	13	1	no	maximal, Sylow
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
74	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
76	#6127	C₄⋊C₄	1	4	yes	—
77	#6127	M₁₆	1	4	yes	—
78	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
83	C₁₃×M₁₆	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	#6103	C₂×C₈	1	2	yes	—
93	#6103	C₂×C₈	1	2	yes	—
94	#6101	C₂×C₈	1	2	yes	—
95	#6101	C₂×C₈	1	2	yes	—
96	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
97	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
98	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
102	#6099	—	2	2	no	—
103	C₂×#1055	C₂²×C₄	1	2	yes	—
104	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	2	yes	—
106	C₂×#1055	C₂×Q₈	1	2	yes	—
107	C₂×#1055	C₂⁴	1	1	yes	characteristic
108	#6094	—	2	4	no	—
109	#6094	C₄⋊C₄	1	8	yes	—
110	C₈×D₁₃	—	2	4	no	—
111	C₈×D₁₃	C₂×D₄	1	4	yes	—
112	C₈×D₁₃	C₂×Q₈	1	4	yes	—
113	#6092	—	2	2	no	—
114	#6092	—	2	4	no	—
115	#6092	—	2	2	no	—
116	C₂×#504	—	13	2	no	—
117	C₄×#131	—	13	1	no	—
118	#2151	—	13	2	no	—
119	#2151	—	13	4	no	—
120	#2150	—	13	2	no	—
121	#2150	—	13	1	no	—
122	C₂×C₄⋊C₁₆	—	13	2	no	—
123	C₂×C₄⋊C₁₆	—	13	1	no	—
124	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
125	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂²⋊D₁₃	C₄⋊C₈	1	8	yes	—
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
139	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
140	D₅₂	C₄⋊C₈	1	4	yes	—
141	C₄×D₁₃	—	2	4	no	—
142	C₄×D₁₃	—	4	2	no	—
143	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
144	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
145	Dic₂₆	C₄⋊C₈	1	4	yes	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
149	#1055	—	2	2	no	—
150	#1055	C₂×C₄⋊C₄	1	2	yes	—
151	#1055	C₂²×D₄	1	1	yes	characteristic
152	#1055	C₂²×Q₈	1	1	yes	characteristic
153	C₂×#131	—	13	1	no	—
154	C₂×#131	—	13	1	no	—
155	C₄×#41	—	13	1	no	—
156	#504	—	26	8	no	—
157	C₂×M₃₂	—	13	4	no	—
158	C₂×M₃₂	—	13	2	no	—
159	C₂²×C₁₆	—	13	4	no	—
160	C₂²×C₁₆	—	13	2	no	—
161	#405	—	13	1	no	—
162	#405	—	13	2	no	—
163	#405	—	13	2	no	—
164	#405	#261	1	1	yes	characteristic
165	C₄×M₁₆	—	13	1	no	—
166	C₄×M₁₆	—	13	2	no	—
167	C₂×C₄×C₈	—	13	1	no	—
168	C₂×C₄×C₈	—	13	2	no	—
169	C₄⋊C₁₆	—	13	4	no	—
170	C₄⋊C₁₆	—	13	4	no	—
171	C₄⋊C₁₆	—	13	1	no	—
172	C₄⋊C₁₆	—	13	2	no	—
173	C₄⋊C₁₆	—	13	1	no	—
174	C₄⋊C₁₆	—	13	2	no	—
175	C₄⋊C₁₆	—	13	2	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
179	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
186	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
188	Dic₁₃	—	2	1	no	—
189	Dic₁₃	—	2	1	no	—
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
192	Dic₁₃	#504	1	2	yes	complemented
193	C₂×#41	—	13	1	no	—
194	#131	—	13	4	no	—
195	#131	—	13	4	no	—
196	#131	—	26	4	no	—
197	C₂×M₁₆	—	13	1	no	—
198	C₂×M₁₆	—	13	2	no	—
199	C₂×M₁₆	—	13	2	no	—
200	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	C₂×#261	1	1	yes	characteristic
205	C₄×Q₈	—	13	1	no	—
206	C₄×Q₈	—	13	1	no	—
207	C₄×D₄	—	13	1	no	—
208	C₄×D₄	—	13	1	no	—
209	C₄×D₄	—	13	2	no	—
210	C₄×D₄	—	13	2	no	—
211	#117	—	13	2	no	—
212	#117	C₂×#261	1	1	yes	characteristic
213	C₂×C₄²	—	13	1	no	—
214	C₂×C₄²	—	13	2	no	—
215	M₃₂	—	26	8	no	—
216	M₃₂	—	26	4	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	13	4	no	—
219	C₂×C₁₆	—	13	2	no	—
220	C₂×C₁₆	—	26	8	no	—
221	C₂×C₁₆	—	26	4	no	—
222	#97	—	13	2	no	—
223	#97	—	13	2	no	—
224	#97	C₂×#261	1	2	yes	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	13	1	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	2	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	2	no	—
231	C₄×C₈	C₂×#261	1	2	yes	—
232	C₂×C₁₃	—	2	2	no	—
233	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
234	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
235	C₂×C₁₃	#3407	1	1	yes	characteristic
236	D₁₃	—	2	4	no	—
237	#41	—	13	4	no	—
238	#41	—	13	4	no	—
239	C₂×Q₈	—	13	1	no	—
240	C₂×D₄	—	13	2	no	—
241	C₂×D₄	—	13	1	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
248	M₁₆	—	13	8	no	—
249	M₁₆	—	26	4	no	—
250	M₁₆	—	26	2	no	—
251	M₁₆	C₄⋊#261	1	4	yes	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	1	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	4	no	—
258	C₂×C₈	—	13	4	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	C₂²×#261	1	1	yes	characteristic
262	C₂×C₈	C₂²×#261	1	1	yes	characteristic
263	C₂×C₈	C₂²×#261	1	2	yes	—
264	C₄⋊C₄	—	13	2	no	—
265	C₄⋊C₄	—	13	2	no	—
266	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
267	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
268	C₂²⋊C₄	—	13	2	no	—
269	C₂²⋊C₄	—	13	2	no	—
270	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
271	C₄²	—	13	2	no	—
272	C₄²	—	13	1	no	—
273	C₄²	—	13	2	no	—
274	C₄²	—	13	1	no	—
275	C₄²	—	13	1	no	—
276	C₄²	—	13	1	no	—
277	C₄²	C₂²×#261	1	2	yes	—
278	C₁₆	—	26	2	no	—
279	C₁₆	—	26	4	no	—
280	C₁₆	—	26	2	no	—
281	C₁₃	#33398	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	13	2	no	—
283	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
284	Q₈	—	13	4	no	—
285	D₄	—	13	8	no	—
286	D₄	—	13	4	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	4	no	—
290	C₂×C₄	—	13	4	no	—
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	26	4	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	52	2	no	—
297	C₂×C₄	—	52	1	no	—
298	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
299	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
300	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
301	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
303	C₂×C₄	C₂³×#261	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	13	1	no	—
307	C₈	—	13	2	no	—
308	C₈	—	13	1	no	—
309	C₈	—	26	2	no	—
310	C₈	C₂×C₄⋊#261	1	4	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	13	4	no	—
314	C₂²	—	26	2	no	—
315	C₂²	—	26	2	no	—
316	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
317	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	13	2	no	—
321	C₄	—	13	4	no	—
322	C₄	—	26	1	no	—
323	C₄	—	26	1	no	—
324	C₄	—	26	1	no	—
325	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
326	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
327	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	26	4	no	—
330	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
331	C₂	D₁₃⋊#504	1	1	yes	characteristic
332	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#405	C₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#33398
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13131172	C₂	semidirect
3. #13131171	C₂	semidirect
4. #13131170	C₂	semidirect
5. #13131169	C₂	semidirect
6. C₂×#11718260	C₂	semidirect
7. D₁₃⋊(C₄⋊C₁₆)	C₂	semidirect
192. Dic₁₃	#504	semidirect
281. C₁₃	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
#13131172	2
#13131171	2
#13131170	2
#13131169	2
C₂×#11718260	2
D₁₃⋊(C₄⋊C₁₆)	2
#13123258	2
#33398	13