Subgroups — #16020984

Order 3328.

510 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	#13127849	C₂	1	1	yes	maximal, characteristic
2	#13127842	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117107	C₂	1	2	yes	maximal, complemented
4	#13117092	C₂	1	1	yes	maximal, characteristic
5	#13116899	C₂	1	2	yes	maximal, complemented
6	#13116887	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116882	C₂	1	2	yes	maximal, complemented
8	#13116877	C₂	1	1	yes	maximal, characteristic
9	#13116702	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719290	—	2	1	no	—
13	#11719275	—	2	1	no	—
14	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₀₄	C₂²	1	2	yes	—
17	C₂×#88040	C₂²	1	2	yes	—
18	C₂×#88040	C₂²	1	2	yes	—
19	C₂×#88040	C₂²	1	2	yes	—
20	#11718510	—	2	2	no	—
21	#11718510	—	2	1	no	—
22	#11718509	—	2	1	no	—
23	#11718505	—	2	2	no	—
24	#11718360	C₂²	1	1	yes	characteristic
25	#11718360	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	C₄×Dic₅₂	C₂²	1	2	yes	—
29	#11718357	C₂²	1	2	yes	—
30	#11718356	C₂²	1	1	yes	characteristic
31	C₄×D₁₀₄	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×#346	C₂²	1	2	yes	—
35	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
36	#11718101	C₂²	1	2	yes	—
37	#11718096	C₂²	1	1	yes	characteristic
38	#11718095	C₂²	1	1	yes	characteristic
39	#11718095	C₂²	1	1	yes	characteristic
40	#11718095	C₂²	1	2	yes	—
41	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
42	#88201	—	4	1	no	—
43	Q₈×Dic₁₃	—	4	1	no	—
44	C₂×Dic₅₂	—	2	2	no	—
45	C₂×Dic₅₂	—	2	1	no	—
46	C₂×Dic₅₂	C₂³	1	2	yes	—
47	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
49	C₂×D₁₀₄	—	2	1	no	—
50	C₂×D₁₀₄	C₂³	1	2	yes	—
51	C₂×#6095	—	2	2	no	—
52	#88148	—	4	2	no	—
53	#88147	—	4	1	no	—
54	#88144	—	4	1	no	—
55	#88144	—	4	2	no	—
56	#88129	—	2	2	no	—
57	#88128	—	2	1	no	—
58	C₄×D₅₂	—	2	2	no	—
59	#88123	—	2	1	no	—
60	#88123	—	2	2	no	—
61	C₄×Dic₂₆	—	2	1	no	—
62	C₄×Dic₂₆	—	2	2	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
68	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88061	—	2	2	no	—
72	#88058	C₂³	1	1	yes	characteristic
73	#88058	C₂³	1	1	yes	characteristic
74	#88056	—	2	1	no	—
75	#88056	—	2	2	no	—
76	#88056	—	2	1	no	—
77	Dic₁₀₄	—	2	2	no	—
78	Dic₁₀₄	—	4	2	no	—
79	#88040	—	2	4	no	—
80	#88040	—	4	2	no	—
81	#88040	—	4	2	no	—
82	#33967	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
84	C₂×D₅₂	—	2	2	no	—
85	C₂×D₅₂	—	4	1	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×C₄×D₁₃	—	4	2	no	—
88	C₂×Dic₂₆	—	2	2	no	—
89	C₂×Dic₂₆	—	2	1	no	—
90	C₂×Dic₂₆	—	2	1	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	2	yes	—
98	#6103	—	4	2	no	—
99	#6103	—	4	2	no	—
100	#6103	—	4	2	no	—
101	C₄⋊Dic₁₃	—	2	1	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	4	1	no	—
104	C₄⋊Dic₁₃	—	4	2	no	—
105	C₄⋊Dic₁₃	—	4	2	no	—
106	C₄⋊Dic₁₃	—	4	1	no	—
107	#6101	—	4	1	no	—
108	#6101	—	4	2	no	—
109	#6101	—	4	1	no	—
110	C₄×Dic₁₃	—	4	2	no	—
111	C₄×Dic₁₃	—	4	1	no	—
112	C₄×Dic₁₃	—	4	1	no	—
113	Dic₅₂	—	2	2	no	—
114	Dic₅₂	—	2	1	no	—
115	Dic₅₂	—	2	1	no	—
116	Dic₅₂	—	2	1	no	—
117	Dic₅₂	—	2	2	no	—
118	Dic₅₂	—	2	1	no	—
119	Dic₅₂	—	8	2	no	—
120	Dic₅₂	—	8	1	no	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	—	2	2	no	—
123	D₁₀₄	—	8	1	no	—
124	#6095	—	8	2	no	—
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	—	2	2	no	—
127	C₁₃×C₁₆	#41	1	2	yes	—
128	#3396	—	13	1	no	—
129	#3395	—	13	1	no	—
130	#2251	—	13	2	no	—
131	#2248	—	13	1	no	—
132	#2211	—	13	2	no	—
133	#2209	—	13	1	no	—
134	#2208	—	13	2	no	—
135	#2207	—	13	1	no	—
136	#2183	D₁₃	1	1	yes	characteristic, complemented
137	#2175	—	13	2	no	—
138	C₄×Q₃₂	—	13	1	no	—
139	C₂²×D₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₁₃	—	4	1	no	—
147	C₂×Dic₁₃	—	4	1	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	—	4	1	no	—
151	D₅₂	—	4	2	no	—
152	D₅₂	—	4	2	no	—
153	D₅₂	—	8	2	no	—
154	D₅₂	—	16	1	no	—
155	C₄×D₁₃	—	16	2	no	—
156	Dic₂₆	—	4	2	no	—
157	Dic₂₆	—	4	1	no	—
158	Dic₂₆	—	4	1	no	—
159	Dic₂₆	—	4	1	no	—
160	Dic₂₆	—	4	1	no	—
161	Dic₂₆	—	4	2	no	—
162	C₈×C₁₃	—	2	2	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
166	#559	—	26	1	no	—
167	#557	—	26	1	no	—
168	C₂×Q₃₂	—	13	1	no	—
169	C₂×Q₃₂	—	13	2	no	—
170	C₂×SD₃₂	—	13	2	no	—
171	C₂×SD₃₂	—	13	2	no	—
172	C₂×SD₃₂	—	13	2	no	—
173	#495	—	13	2	no	—
174	#494	—	13	2	no	—
175	#494	—	13	1	no	—
176	#493	—	13	1	no	—
177	#463	—	26	2	no	—
178	#462	—	26	1	no	—
179	#462	—	26	2	no	—
180	#460	—	26	1	no	—
181	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
182	C₄×Q₁₆	—	13	1	no	—
183	C₄×Q₁₆	—	13	2	no	—
184	C₄×Q₁₆	—	13	1	no	—
185	C₄×D₈	—	13	2	no	—
186	#366	—	13	1	no	—
187	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
188	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
189	#358	—	13	1	no	—
190	#358	—	13	1	no	—
191	#358	—	13	2	no	—
192	#357	—	13	2	no	—
193	#346	C₂×D₁₃	1	2	yes	—
194	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₁₃	—	8	2	no	—
197	C₂×D₁₃	—	8	2	no	—
198	C₂×D₁₃	—	8	2	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	#510	1	2	yes	—
203	C₄×C₁₃	#550	1	1	yes	characteristic
204	C₄×C₁₃	#576	1	1	yes	characteristic
205	Dic₁₃	—	8	1	no	—
206	Dic₁₃	—	8	2	no	—
207	Dic₁₃	—	8	1	no	—
208	Dic₁₃	—	8	1	no	—
209	Dic₁₃	—	8	1	no	—
210	C₂×Q₁₆	—	13	2	no	—
211	C₂×Q₁₆	—	13	1	no	—
212	C₂×Q₁₆	—	13	1	no	—
213	C₂×Q₁₆	—	26	2	no	—
214	C₂×Q₁₆	—	26	1	no	—
215	C₂×SD₁₆	—	26	2	no	—
216	C₂×D₈	—	13	2	no	—
217	C₂×D₈	—	26	1	no	—
218	C₄⋊Q₈	—	26	2	no	—
219	C₄⋊Q₈	—	26	1	no	—
220	#127	—	26	1	no	—
221	#125	—	52	1	no	—
222	#125	—	52	2	no	—
223	#124	—	26	2	no	—
224	#124	—	52	1	no	—
225	#123	—	52	2	no	—
226	C₄×Q₈	—	26	2	no	—
227	C₄×Q₈	—	26	1	no	—
228	C₄×Q₈	—	26	1	no	—
229	C₄×Q₈	—	52	1	no	—
230	C₄×Q₈	D₅₂	1	1	yes	characteristic
231	C₄×D₄	—	26	2	no	—
232	C₄×D₄	—	52	1	no	—
233	Q₃₂	—	26	2	no	—
234	Q₃₂	—	52	2	no	—
235	SD₃₂	—	26	4	no	—
236	SD₃₂	—	52	2	no	—
237	SD₃₂	—	52	2	no	—
238	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
240	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
241	#107	—	13	1	no	—
242	#107	—	13	1	no	—
243	C₄⋊C₈	D₅₂	1	1	yes	characteristic
244	C₄⋊C₈	D₅₂	1	2	yes	—
245	#103	—	26	2	no	—
246	#103	—	26	1	no	—
247	#103	—	26	1	no	—
248	#102	—	26	2	no	—
249	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₁₃	1	2	yes	—
251	C₂×C₁₃	#3391	1	1	yes	characteristic
252	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
253	C₂×C₁₃	#3416	1	1	yes	characteristic
254	D₁₃	—	16	2	no	—
255	C₂×Q₈	—	26	1	no	—
256	C₂×Q₈	—	26	2	no	—
257	C₂×Q₈	—	26	1	no	—
258	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
259	C₂×D₄	—	26	2	no	—
260	C₂×D₄	—	52	1	no	—
261	C₂×D₄	—	52	1	no	—
262	C₂²×C₄	—	52	2	no	—
263	Q₁₆	—	26	2	no	—
264	Q₁₆	—	26	1	no	—
265	Q₁₆	—	26	1	no	—
266	Q₁₆	—	26	1	no	—
267	Q₁₆	—	26	2	no	—
268	Q₁₆	—	26	1	no	—
269	Q₁₆	—	104	1	no	—
270	Q₁₆	—	104	2	no	—
271	SD₁₆	—	104	2	no	—
272	D₈	—	26	2	no	—
273	D₈	—	26	2	no	—
274	D₈	—	104	1	no	—
275	C₂×C₈	C₂×D₅₂	1	2	yes	—
276	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	26	1	no	—
279	C₄⋊C₄	—	26	1	no	—
280	C₄⋊C₄	—	52	1	no	—
281	C₄⋊C₄	—	52	2	no	—
282	C₄⋊C₄	—	52	1	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	—	52	1	no	—
285	C₄⋊C₄	—	52	2	no	—
286	C₄⋊C₄	—	52	1	no	—
287	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
288	C₄⋊C₄	D₁₀₄	1	2	yes	—
289	C₂²⋊C₄	—	52	2	no	—
290	C₂²⋊C₄	—	52	2	no	—
291	C₂²⋊C₄	—	52	2	no	—
292	C₄²	—	52	1	no	—
293	C₄²	—	52	2	no	—
294	C₄²	—	52	1	no	—
295	C₄²	C₂×D₅₂	1	2	yes	—
296	C₄²	C₂×D₅₂	1	1	yes	characteristic
297	C₁₆	—	2	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	#6131	1	2	yes	—
300	C₁₃	#33967	1	1	yes	characteristic, complemented, Sylow
301	C₂³	—	52	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	52	1	no	—
304	Q₈	—	52	2	no	—
305	Q₈	—	52	1	no	—
306	Q₈	—	52	2	no	—
307	Q₈	—	52	1	no	—
308	Q₈	—	52	1	no	—
309	D₄	—	52	2	no	—
310	D₄	—	52	2	no	—
311	D₄	—	104	2	no	—
312	D₄	—	208	1	no	—
313	C₂×C₄	—	52	1	no	—
314	C₂×C₄	—	52	1	no	—
315	C₂×C₄	—	52	1	no	—
316	C₂×C₄	—	52	2	no	—
317	C₂×C₄	—	52	1	no	—
318	C₂×C₄	—	208	2	no	—
319	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
320	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
321	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
322	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	C₂×#6131	1	1	yes	characteristic
327	C₈	#88256	1	1	yes	characteristic
328	C₂²	—	104	2	no	—
329	C₂²	—	104	2	no	—
330	C₂²	—	104	2	no	—
331	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	104	1	no	—
336	C₄	—	104	1	no	—
337	C₄	—	104	1	no	—
338	C₄	—	104	2	no	—
339	C₄	—	104	1	no	—
340	C₄	#11718570	1	2	yes	—
341	C₄	#11719282	1	1	yes	characteristic
342	C₄	#11719457	1	1	yes	characteristic
343	C₂	—	208	2	no	—
344	C₂	#13127811	1	1	yes	characteristic
345	C₂	C₂×#11718570	1	1	yes	characteristic
346	C₂	#13127984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2183	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#11718356	C₂²
hypercenter	#2183	D₁₃
hypocenter	C₁₃	#33967
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #13127842	C₂	semidirect
3. #13117107	C₂	semidirect
5. #13116899	C₂	semidirect
6. #13116887	C₂	semidirect
7. #13116882	C₂	semidirect
9. #13116702	C₂	semidirect
10. C₄×Dic₁₀₄	C₂	semidirect
11. C₁₃×#2183	C₂	semidirect
136. #2183	D₁₃	semidirect
300. C₁₃	#33967	semidirect

Maximal subgroups

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

subgroup	index
#13127849	2
#13127842	2
#13117107	2
#13117092	2
#13116899	2
#13116887	2
#13116882	2
#13116877	2
#13116702	2
C₄×Dic₁₀₄	2
C₁₃×#2183	2
#33967	13