Subgroups — #16807426

Order 3328.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	2	yes	maximal, complemented
2	#13131221	C₂	1	2	yes	maximal, complemented
3	#13131218	C₂	1	2	yes	maximal, complemented
4	#13131216	C₂	1	2	yes	maximal, complemented
5	#13131202	C₂	1	2	yes	maximal, complemented
6	#13131199	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131173	C₂	1	2	yes	maximal, complemented
8	#13131169	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123814	C₂	1	1	yes	maximal, characteristic
10	#11719262	C₄	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	2	yes	—
12	C₂²×#6092	C₂²	1	2	yes	—
13	C₂×#88098	C₂²	1	1	yes	characteristic
14	C₂×#88098	C₂²	1	1	yes	characteristic
15	C₁₃×#436	C₄	1	1	yes	characteristic
16	#11718788	C₄	1	2	yes	—
17	#11718783	C₂²	1	2	yes	—
18	D₄×#1055	C₂²	1	1	yes	characteristic
19	#11718496	C₄	1	1	yes	characteristic
20	#11718491	C₂²	1	1	yes	characteristic
21	#11718386	C₂²	1	2	yes	—
22	#11718381	C₄	1	2	yes	—
23	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
24	#11718340	C₄	1	1	yes	characteristic
25	C₂²⋊#6092	C₂²	1	2	yes	—
26	C₂²⋊#6092	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718260	C₂²	1	1	yes	characteristic
29	#11718260	C₂²	1	2	yes	—
30	#11718260	C₂²	1	2	yes	—
31	#11718260	C₂²	1	1	yes	characteristic
32	#11718259	C₂²	1	2	yes	—
33	#11718258	C₂²	1	2	yes	—
34	#11718257	C₂²	1	2	yes	—
35	#11718231	C₂²	1	2	yes	—
36	C₄⋊#6092	C₂²	1	2	yes	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
38	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	#88192	C₈	1	1	yes	characteristic
40	#88190	C₈	1	1	yes	characteristic
41	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
42	#88180	C₂×C₄	1	2	yes	—
43	C₂²×#1055	C₂³	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
46	#88151	C₈	1	1	yes	characteristic
47	#88144	C₈	1	1	yes	characteristic
48	#88137	C₈	1	2	yes	—
49	#88133	C₈	1	2	yes	—
50	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
52	#88106	—	4	2	no	—
53	C₂×#6092	—	4	2	no	—
54	C₂×#6092	C₂³	1	2	yes	—
55	C₂×#6092	C₂³	1	2	yes	—
56	C₂×#6092	C₂³	1	2	yes	—
57	C₂×#6092	C₂³	1	2	yes	—
58	#88098	—	4	1	no	—
59	#88098	—	4	1	no	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊#1055	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88055	C₂×C₄	1	2	yes	—
66	#88055	C₂×C₄	1	2	yes	—
67	#88054	C₂×C₄	1	2	yes	—
68	#88054	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	2	yes	—
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
73	C₄×#1055	C₂³	1	1	yes	characteristic
74	#33509	—	13	1	no	maximal, Sylow
75	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
82	C₁₃×M₁₆	—	4	2	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
90	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
91	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
92	#6103	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
96	#6101	C₂×C₈	1	2	yes	—
97	#6101	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	2	yes	—
101	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
105	C₂×#1055	C₂²×C₄	1	2	yes	—
106	C₂×#1055	C₂⁴	1	1	yes	characteristic
107	C₈×D₁₃	—	2	1	no	—
108	C₈×D₁₃	—	2	1	no	—
109	C₈×D₁₃	—	4	1	no	—
110	#6092	—	2	2	no	—
111	#6092	—	2	2	no	—
112	#6092	—	2	2	no	—
113	#6092	—	2	2	no	—
114	#2966	—	13	1	no	—
115	#2183	—	13	2	no	—
116	#2169	—	13	2	no	—
117	#2168	—	13	2	no	—
118	#2168	—	13	2	no	—
119	#2164	—	13	2	no	—
120	#2163	—	13	1	no	—
121	#2151	—	13	2	no	—
122	#2150	—	13	1	no	—
123	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
124	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
125	D₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂²⋊D₁₃	—	4	2	no	—
133	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
137	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
138	C₄×D₁₃	—	2	1	no	—
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	4	1	no	—
141	C₄×D₁₃	—	4	1	no	—
142	Dic₂₆	—	4	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	2	no	—
146	#1055	—	2	2	no	—
147	#1055	—	2	1	no	—
148	#1055	H(5,𝔽₂)	1	1	yes	characteristic
149	#1055	2₋¹⁺⁴	1	1	yes	characteristic
150	#529	—	13	1	no	—
151	C₂×M₃₂	—	13	1	no	—
152	C₂×M₃₂	—	13	1	no	—
153	C₂×M₃₂	—	13	2	no	—
154	C₂²×C₁₆	—	13	2	no	—
155	#446	—	13	1	no	—
156	C₈×Q₈	—	13	1	no	—
157	#436	#261	1	1	yes	characteristic
158	#435	—	13	2	no	—
159	C₈×D₄	—	13	1	no	—
160	C₈×D₄	—	13	2	no	—
161	#433	—	13	2	no	—
162	#432	—	13	1	no	—
163	#405	—	13	2	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₄⋊C₁₆	—	13	2	no	—
166	C₄⋊C₁₆	—	13	2	no	—
167	C₄⋊C₁₆	—	13	1	no	—
168	C₄⋊C₁₆	—	13	2	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	C₄⋊C₁₆	—	13	1	no	—
171	C₂²⋊C₁₆	—	13	2	no	—
172	C₂²⋊C₁₆	—	13	2	no	—
173	C₂²⋊C₁₆	—	13	2	no	—
174	#346	—	13	2	no	—
175	#346	—	13	2	no	—
176	C₄×C₁₆	—	13	2	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
181	C₂×D₁₃	—	2	1	no	—
182	C₂×D₁₃	—	2	1	no	—
183	C₂×D₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	#520	1	1	yes	characteristic
188	C₄×C₁₃	#568	1	1	yes	characteristic
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	—	2	1	no	—
192	Dic₁₃	—	2	2	no	—
193	Dic₁₃	—	2	1	no	—
194	C₂×M₁₆	C₂×#261	1	2	yes	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	13	1	no	—
198	#126	—	13	2	no	—
199	#125	—	13	1	no	—
200	#124	—	13	1	no	—
201	#123	—	13	2	no	—
202	C₂²⋊Q₈	—	13	1	no	—
203	#121	—	13	1	no	—
204	C₄×Q₈	—	13	1	no	—
205	C₄×D₄	—	13	2	no	—
206	C₄×D₄	C₂×#261	1	1	yes	characteristic
207	#117	—	13	2	no	—
208	C₂×C₄²	—	13	1	no	—
209	M₃₂	—	52	1	no	—
210	M₃₂	—	52	1	no	—
211	M₃₂	—	52	2	no	—
212	C₂×C₁₆	—	13	2	no	—
213	C₂×C₁₆	—	13	2	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	13	2	no	—
216	C₂×C₁₆	—	52	2	no	—
217	C₄⋊C₈	—	13	2	no	—
218	C₄⋊C₈	—	13	2	no	—
219	C₄⋊C₈	—	13	1	no	—
220	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
221	C₂²⋊C₈	—	13	2	no	—
222	C₂²⋊C₈	—	13	2	no	—
223	C₂²⋊C₈	C₂×#261	1	2	yes	—
224	#97	—	13	2	no	—
225	#97	—	13	2	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	C₂×#261	1	1	yes	characteristic
231	C₂×C₁₃	—	4	2	no	—
232	C₂×C₁₃	#2988	1	1	yes	characteristic
233	C₂×C₁₃	#3407	1	1	yes	characteristic
234	C₂×C₁₃	#3407	1	1	yes	characteristic
235	D₁₃	—	4	1	no	—
236	C₂×Q₈	—	13	1	no	—
237	C₂×D₄	—	13	2	no	—
238	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	C₂²×#261	1	2	yes	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	26	1	no	—
248	C₂×C₈	—	26	1	no	—
249	C₂×C₈	—	26	2	no	—
250	C₂×C₈	—	26	2	no	—
251	C₂×C₈	—	52	1	no	—
252	C₂×C₈	C₂²×#261	1	1	yes	characteristic
253	C₂×C₈	C₂²×#261	1	1	yes	characteristic
254	C₂×C₈	C₂²×#261	1	2	yes	—
255	C₄⋊C₄	—	13	2	no	—
256	C₄⋊C₄	—	13	2	no	—
257	C₄⋊C₄	—	13	2	no	—
258	C₄⋊C₄	—	13	1	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
261	C₂²⋊C₄	—	13	2	no	—
262	C₂²⋊C₄	—	13	2	no	—
263	C₂²⋊C₄	—	13	2	no	—
264	C₂²⋊C₄	—	13	2	no	—
265	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
266	C₄²	—	13	1	no	—
267	C₄²	—	13	1	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	2	no	—
270	C₄²	C₂²×#261	1	1	yes	characteristic
271	C₁₆	—	26	2	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₆	—	26	2	no	—
275	C₁₃	#33509	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	13	1	no	—
277	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
278	Q₈	—	52	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	52	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	13	1	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	26	1	no	—
289	C₂×C₄	—	26	1	no	—
290	C₂×C₄	—	52	1	no	—
291	C₂×C₄	—	52	1	no	—
292	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
295	C₂×C₄	C₂³×#261	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	1	no	—
299	C₈	—	13	1	no	—
300	C₈	—	13	1	no	—
301	C₈	—	26	1	no	—
302	C₈	—	26	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	26	1	no	—
307	C₂²	—	26	1	no	—
308	C₂²	—	26	1	no	—
309	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	26	1	no	—
314	C₄	—	26	2	no	—
315	C₄	—	26	1	no	—
316	C₄	—	26	2	no	—
317	C₄	—	26	1	no	—
318	C₄	#11719621	1	1	yes	characteristic
319	C₄	#11719631	1	1	yes	characteristic
320	C₂	—	4	2	no	—
321	C₂	—	52	1	no	—
322	C₂	#13131752	1	1	yes	characteristic
323	C₂	#13131804	1	1	yes	characteristic
324	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#436	C₄
prime radical	#88192	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#436	#261
hypocenter	C₁₃	#33509
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131221	C₂	semidirect
3. #13131218	C₂	semidirect
4. #13131216	C₂	semidirect
5. #13131202	C₂	semidirect
6. #13131199	C₂	semidirect
7. #13131173	C₂	semidirect
8. #13131169	C₂	semidirect
275. C₁₃	#33509	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131221	2
#13131218	2
#13131216	2
#13131202	2
#13131199	2
#13131173	2
#13131169	2
#13123814	2
#33509	13