Subgroups — #16020518

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127184	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117116	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117027	C₂	1	2	yes	maximal, complemented
4	#13117019	C₂	1	2	yes	maximal, complemented
5	#13116980	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116973	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116937	C₂	1	2	yes	maximal, complemented
8	C₄×#88066	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116668	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116625	C₂	1	2	yes	maximal, complemented
11	C₁₃×#3341	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#555	—	2	1	no	—
13	#11719260	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#482	—	2	2	no	—
16	C₁₃×#476	—	2	1	no	—
17	C₁₃×#466	C₂²	1	2	yes	—
18	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
19	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#427	C₂²	1	2	yes	—
21	C₂×#88066	C₂²	1	1	yes	characteristic
22	#11718776	C₂²	1	2	yes	—
23	C₂×#88058	C₂²	1	2	yes	—
24	C₂×#88052	C₂²	1	2	yes	—
25	#11718507	—	2	1	no	—
26	#11718397	—	2	2	no	—
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	#11718343	C₂²	1	1	yes	characteristic
29	#11718154	C₂²	1	1	yes	characteristic
30	#11718154	C₂²	1	2	yes	—
31	C₂²⋊#6090	C₂²	1	2	yes	—
32	#11718083	C₂²	1	1	yes	characteristic
33	#11718083	C₂²	1	2	yes	—
34	#11718082	C₂²	1	2	yes	—
35	#11718082	C₂²	1	2	yes	—
36	#11718082	C₂²	1	2	yes	—
37	#11718081	C₂²	1	1	yes	characteristic
38	#11718081	C₂²	1	2	yes	—
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6129	—	4	1	no	—
42	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×#124	—	4	2	no	—
46	C₁₃×#121	—	2	2	no	—
47	C₁₃×#121	—	4	1	no	—
48	C₄×D₄×C₁₃	—	2	1	no	—
49	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
50	C₁₃×#117	—	2	2	no	—
51	C₁₃×#117	—	4	1	no	—
52	#88184	—	2	2	no	—
53	C₂×C₄⋊Dic₁₃	—	2	2	no	—
54	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
55	#88151	—	4	1	no	—
56	#88140	—	4	2	no	—
57	C₄×D₅₂	—	2	1	no	—
58	#88123	—	2	1	no	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#106	—	2	2	no	—
63	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#102	—	2	1	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	#88066	—	2	2	no	—
69	#88061	—	2	2	no	—
70	#88061	—	2	1	no	—
71	#88058	—	2	2	no	—
72	#88058	—	2	1	no	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88058	C₂³	1	2	yes	—
75	#88052	—	2	4	no	—
76	C₂×#6090	C₂³	1	2	yes	—
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	#33921	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	4	2	no	—
83	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₂²⋊D₁₃	—	4	2	no	—
87	C₂²×Dic₁₃	—	4	2	no	—
88	#6129	—	8	4	no	—
89	C₂×D₅₂	—	2	1	no	—
90	C₂×C₄×D₁₃	—	4	1	no	—
91	C₂×Dic₂₆	—	4	1	no	—
92	C₁₃×D₈	—	2	1	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	1	no	—
106	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
107	C₄²×C₁₃	—	4	2	no	—
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	#6103	—	4	1	no	—
110	#6103	—	4	2	no	—
111	C₄⋊Dic₁₃	—	2	2	no	—
112	C₄⋊Dic₁₃	—	2	1	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	C₄⋊Dic₁₃	—	4	2	no	—
115	D₁₀₄	—	2	1	no	—
116	D₁₀₄	—	2	1	no	—
117	#6090	—	2	2	no	—
118	#6090	—	2	1	no	—
119	#6090	#41	1	2	yes	complemented
120	#3341	D₁₃	1	1	yes	characteristic, complemented
121	#3334	—	13	1	no	—
122	#2254	—	13	1	no	—
123	#2234	—	13	2	no	—
124	#2233	—	13	2	no	—
125	#2226	—	13	1	no	—
126	#2224	—	13	1	no	—
127	#2218	—	13	2	no	—
128	C₄×D₁₆	—	13	1	no	—
129	#2169	—	13	1	no	—
130	#2159	—	13	2	no	—
131	C₂³×C₁₃	—	4	1	no	—
132	C₂³×C₁₃	C₂×D₈	1	2	yes	—
133	C₂²×D₁₃	—	4	1	no	—
134	Q₈×C₁₃	—	16	1	no	—
135	D₄×C₁₃	—	2	2	no	—
136	D₄×C₁₃	—	4	1	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	16	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₁₃	—	8	4	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	8	4	no	—
152	D₅₂	—	4	1	no	—
153	D₅₂	—	4	1	no	—
154	C₄×D₁₃	—	8	2	no	—
155	Dic₂₆	—	8	2	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	#555	—	2	1	no	—
161	#547	—	26	1	no	—
162	C₂×D₁₆	—	13	1	no	—
163	C₂×M₃₂	—	13	2	no	—
164	#500	—	13	1	no	—
165	#500	C₂×D₁₃	1	1	yes	characteristic
166	#482	—	2	2	no	—
167	#480	—	26	2	no	—
168	#476	—	2	1	no	—
169	#474	—	26	1	no	—
170	#466	—	13	2	no	—
171	#466	C₂×D₁₃	1	2	yes	—
172	C₄×D₈	—	13	1	no	—
173	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
174	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
175	#427	C₂×D₁₃	1	2	yes	—
176	C₂×#107	—	13	2	no	—
177	#367	—	13	2	no	—
178	#367	—	13	2	no	—
179	#367	—	13	2	no	—
180	#366	—	13	1	no	—
181	#366	—	13	2	no	—
182	C₄⋊C₁₆	—	13	1	no	—
183	#357	—	13	1	no	—
184	#357	—	13	2	no	—
185	#357	—	13	1	no	—
186	#357	—	13	2	no	—
187	C₂²⋊C₁₆	—	13	2	no	—
188	C₄×C₁₆	—	13	1	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	2	2	no	—
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	—	8	1	no	—
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₁₃	—	8	1	no	—
197	C₂×D₁₃	—	8	1	no	—
198	C₂×D₁₃	—	8	1	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	#509	1	2	yes	—
204	C₄×C₁₃	#547	1	1	yes	characteristic
205	C₄×C₁₃	#578	1	1	yes	characteristic
206	Dic₁₃	—	8	2	no	—
207	Dic₁₃	—	8	1	no	—
208	C₂×#41	—	52	1	no	—
209	C₂×D₈	—	13	1	no	—
210	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	26	1	no	—
214	#124	—	4	2	no	—
215	#123	—	52	2	no	—
216	C₂²⋊Q₈	—	52	1	no	—
217	#121	—	2	2	no	—
218	#121	—	4	1	no	—
219	#121	—	26	2	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	—	26	1	no	—
222	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
223	#117	—	2	2	no	—
224	#117	—	4	1	no	—
225	C₂×C₄⋊C₄	—	26	2	no	—
226	D₁₆	—	26	2	no	—
227	M₃₂	—	26	4	no	—
228	C₂×C₁₆	—	13	1	no	—
229	C₂×C₁₆	—	13	1	no	—
230	C₂×C₁₆	—	13	2	no	—
231	#107	—	2	1	no	—
232	#107	—	13	1	no	—
233	#107	—	13	2	no	—
234	#107	—	26	2	no	—
235	#107	—	26	1	no	—
236	#107	C₂²×D₁₃	1	1	yes	characteristic
237	#107	C₂²×D₁₃	1	2	yes	—
238	#106	—	2	2	no	—
239	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
240	#102	—	2	1	no	—
241	#102	—	2	2	no	—
242	#102	—	26	1	no	—
243	#102	—	26	2	no	—
244	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
245	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
246	C₂×C₁₃	—	4	2	no	—
247	C₂×C₁₃	—	16	1	no	—
248	C₂×C₁₃	#3334	1	1	yes	characteristic
249	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
250	C₂×C₁₃	#3416	1	1	yes	characteristic
251	D₁₃	—	16	1	no	—
252	#41	—	104	4	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	52	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	26	1	no	—
258	C₂×D₄	—	52	2	no	—
259	C₂×D₄	#6104	1	1	yes	characteristic
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	52	2	no	—
262	C₂²×C₄	—	52	1	no	—
263	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	26	1	no	—
267	D₈	—	26	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
272	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	26	1	no	—
277	C₄⋊C₄	—	26	2	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	1	no	—
280	C₄⋊C₄	#6104	1	1	yes	characteristic
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	52	1	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	#6104	1	2	yes	—
287	C₄²	—	4	2	no	—
288	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₁₆	—	13	2	no	—
290	C₁₆	—	26	2	no	—
291	C₁₆	—	26	1	no	—
292	C₁₃	#33921	1	1	yes	characteristic, complemented, Sylow
293	C₂³	—	4	1	no	—
294	C₂³	—	52	1	no	—
295	C₂³	C₂×#6104	1	2	yes	—
296	Q₈	—	16	1	no	—
297	Q₈	—	104	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	16	2	no	—
302	D₄	—	52	1	no	—
303	D₄	—	52	1	no	—
304	D₄	—	104	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	52	1	no	—
311	C₂×C₄	—	52	2	no	—
312	C₂×C₄	—	104	4	no	—
313	C₂×C₄	—	104	2	no	—
314	C₂×C₄	C₂×#6104	1	2	yes	—
315	C₂×C₄	C₂×#6104	1	1	yes	characteristic
316	C₂×C₄	C₂×#6104	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	#41×D₁₃	1	1	yes	characteristic
321	C₈	#88257	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	104	1	no	—
329	C₂²	—	104	1	no	—
330	C₂²	—	104	1	no	—
331	C₂²	C₂²×#6104	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	104	1	no	—
337	C₄	—	104	2	no	—
338	C₄	#11718854	1	2	yes	—
339	C₄	#11719529	1	1	yes	characteristic
340	C₄	#11719543	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	16	1	no	—
343	C₂	—	208	1	no	—
344	C₂	#13127179	1	1	yes	characteristic
345	C₂	C₂×#11718854	1	1	yes	characteristic
346	C₂	#13127983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3341	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3341	D₁₃
hypocenter	C₁₃	#33921
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3341

Complements

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

N	H ≅ G/N	type
1. #13127184	C₂	semidirect
2. #13117116	C₂	semidirect
3. #13117027	C₂	semidirect
4. #13117019	C₂	semidirect
5. #13116980	C₂	semidirect
6. #13116973	C₂	semidirect
7. #13116937	C₂	semidirect
8. C₄×#88066	C₂	semidirect
9. #13116668	C₂	semidirect
10. #13116625	C₂	semidirect
11. C₁₃×#3341	C₂	semidirect
119. #6090	#41	semidirect
120. #3341	D₁₃	semidirect
292. C₁₃	#33921	semidirect

Maximal subgroups

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

subgroup	index
#13127184	2
#13117116	2
#13117027	2
#13117019	2
#13116980	2
#13116973	2
#13116937	2
C₄×#88066	2
#13116668	2
#13116625	2
C₁₃×#3341	2
#33921	13