Subgroups — #16020712

Order 3328.

476 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127842	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127704	C₂	1	1	yes	maximal, characteristic
3	#13117082	C₂	1	2	yes	maximal, complemented
4	#13117064	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117001	C₂	1	2	yes	maximal, complemented
6	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116882	C₂	1	1	yes	maximal, characteristic
8	#13116882	C₂	1	2	yes	maximal
9	#13116702	C₂	1	2	yes	maximal
10	C₄×#88040	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719284	—	2	1	no	—
13	#11719275	—	2	1	no	—
14	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88040	C₂²	1	1	yes	characteristic
16	C₂×#88040	C₂²	1	2	yes	—
17	#11718510	—	2	1	no	—
18	#11718510	—	2	2	no	—
19	#11718508	—	2	2	no	—
20	#11718503	—	2	1	no	—
21	#11718360	C₂²	1	1	yes	characteristic
22	#11718360	C₂²	1	2	yes	—
23	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₂	C₂²	1	2	yes	—
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	#11718344	C₂²	1	2	yes	—
28	#11718343	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#346	C₂²	1	2	yes	—
32	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
33	#11718101	C₂²	1	1	yes	characteristic
34	#11718097	C₂²	1	1	yes	characteristic
35	#11718096	C₂²	1	2	yes	—
36	#11718095	C₂²	1	1	yes	characteristic
37	#11718095	C₂²	1	2	yes	—
38	#11718095	C₂²	1	2	yes	—
39	#11718095	C₂²	1	2	yes	—
40	#11718095	C₂²	1	2	yes	—
41	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
42	#88200	—	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	1	yes	characteristic
47	C₂×Dic₅₂	C₂³	1	2	yes	—
48	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
49	#88150	—	4	2	no	—
50	C₄⋊C₄×D₁₃	—	4	1	no	—
51	#88144	—	4	1	no	—
52	#88144	—	4	2	no	—
53	C₄×D₅₂	—	2	1	no	—
54	C₄×D₅₂	—	2	2	no	—
55	#88124	—	2	2	no	—
56	#88123	—	2	1	no	—
57	#88123	—	2	1	no	—
58	#88123	—	2	2	no	—
59	C₄×Dic₂₆	—	2	1	no	—
60	C₄×Dic₂₆	—	2	2	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88061	—	2	2	no	—
69	#88061	—	2	1	no	—
70	#88058	—	2	1	no	—
71	#88058	C₂³	1	2	yes	—
72	#88058	C₂³	1	1	yes	characteristic
73	#88058	C₂³	1	1	yes	characteristic
74	#88057	—	2	2	no	—
75	#88056	—	2	2	no	—
76	#88056	—	2	1	no	—
77	#88040	—	2	2	no	—
78	#88040	—	2	4	no	—
79	#33939	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
81	C₂×D₅₂	—	2	1	no	—
82	C₂×C₄×D₁₃	—	4	1	no	—
83	C₂×C₄×D₁₃	—	4	2	no	—
84	C₂×Dic₂₆	—	2	1	no	—
85	C₂×Dic₂₆	—	2	2	no	—
86	C₂×Dic₂₆	—	4	1	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	2	yes	—
94	#6103	—	4	1	no	—
95	#6103	—	4	2	no	—
96	C₄⋊Dic₁₃	—	2	1	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	1	no	—
99	C₄⋊Dic₁₃	—	4	2	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	C₄⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	—	4	1	no	—
103	C₄⋊Dic₁₃	—	4	1	no	—
104	#6101	—	4	2	no	—
105	#6101	—	4	1	no	—
106	#6101	—	4	2	no	—
107	#6101	—	4	2	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	Dic₅₂	—	2	1	no	—
111	Dic₅₂	—	2	2	no	—
112	Dic₅₂	—	2	1	no	—
113	Dic₅₂	—	2	2	no	—
114	Dic₅₂	—	8	2	no	—
115	Dic₅₂	—	8	1	no	—
116	D₁₀₄	—	2	1	no	—
117	D₁₀₄	#41	1	2	yes	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	—	2	2	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	#3395	—	13	1	no	—
122	#3380	—	13	1	no	—
123	#2245	—	13	2	no	—
124	#2241	—	13	1	no	—
125	#2229	—	13	2	no	—
126	#2227	—	13	1	no	—
127	#2208	—	13	2	no	—
128	#2208	—	13	1	no	—
129	#2183	D₁₃	1	1	yes	characteristic, complemented
130	#2175	—	13	2	no	—
131	C₄×SD₃₂	—	13	1	no	—
132	C₂²×D₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	1	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	D₅₂	—	2	2	no	—
145	D₅₂	—	4	1	no	—
146	C₄×D₁₃	—	8	2	no	—
147	C₄×D₁₃	—	16	2	no	—
148	Dic₂₆	—	4	1	no	—
149	Dic₂₆	—	4	2	no	—
150	Dic₂₆	—	4	1	no	—
151	Dic₂₆	—	4	2	no	—
152	Dic₂₆	—	16	1	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
157	#557	—	26	1	no	—
158	#554	—	26	1	no	—
159	C₂×SD₃₂	—	13	1	no	—
160	C₂×SD₃₂	—	13	2	no	—
161	#500	—	13	1	no	—
162	#499	—	13	2	no	—
163	#494	—	13	1	no	—
164	#494	—	13	2	no	—
165	#478	—	26	2	no	—
166	#476	—	26	1	no	—
167	#462	—	26	1	no	—
168	#462	—	26	2	no	—
169	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
170	C₄×Q₁₆	—	13	1	no	—
171	C₄×Q₁₆	—	13	2	no	—
172	C₄×D₈	—	13	1	no	—
173	C₄×D₈	—	13	2	no	—
174	#367	—	13	1	no	—
175	#366	—	13	2	no	—
176	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
177	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
178	#358	—	13	1	no	—
179	#358	—	13	2	no	—
180	#358	—	13	2	no	—
181	#358	—	13	2	no	—
182	#358	—	13	2	no	—
183	#357	—	13	1	no	—
184	#346	C₂×D₁₃	1	2	yes	—
185	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	—	8	1	no	—
189	C₂×D₁₃	—	8	1	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	#510	1	2	yes	—
194	C₄×C₁₃	#548	1	1	yes	characteristic
195	C₄×C₁₃	#577	1	1	yes	characteristic
196	Dic₁₃	—	8	2	no	—
197	Dic₁₃	—	8	1	no	—
198	Dic₁₃	—	8	1	no	—
199	Dic₁₃	—	8	2	no	—
200	Dic₁₃	—	8	1	no	—
201	C₂×Q₁₆	—	13	1	no	—
202	C₂×Q₁₆	—	13	2	no	—
203	C₂×Q₁₆	—	26	2	no	—
204	C₂×Q₁₆	—	26	1	no	—
205	C₂×D₈	—	13	1	no	—
206	C₄⋊Q₈	—	26	1	no	—
207	C₄⋊Q₈	—	26	2	no	—
208	C₄⋊Q₈	—	26	1	no	—
209	#125	—	26	2	no	—
210	#125	—	52	1	no	—
211	#125	—	52	2	no	—
212	C₂²⋊Q₈	—	52	1	no	—
213	C₂²⋊Q₈	—	52	2	no	—
214	C₄×Q₈	—	26	2	no	—
215	C₄×Q₈	—	26	1	no	—
216	C₄×Q₈	—	52	1	no	—
217	C₄×Q₈	D₅₂	1	1	yes	characteristic
218	C₄×D₄	—	26	1	no	—
219	C₄×D₄	—	26	2	no	—
220	C₂×C₄⋊C₄	—	52	1	no	—
221	SD₃₂	—	26	2	no	—
222	SD₃₂	—	26	4	no	—
223	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
226	#107	—	13	1	no	—
227	#107	—	13	2	no	—
228	#107	—	13	1	no	—
229	#107	—	26	1	no	—
230	#106	—	26	2	no	—
231	C₄⋊C₈	D₅₂	1	1	yes	characteristic
232	C₄⋊C₈	D₅₂	1	2	yes	—
233	#103	—	26	2	no	—
234	#103	—	26	1	no	—
235	#102	—	26	2	no	—
236	#102	—	26	1	no	—
237	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₃	#3358	1	1	yes	characteristic
240	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
241	C₂×C₁₃	#3416	1	1	yes	characteristic
242	D₁₃	—	8	2	no	—
243	C₂×Q₈	—	26	2	no	—
244	C₂×Q₈	—	26	1	no	—
245	C₂×Q₈	—	52	1	no	—
246	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
247	C₂×D₄	—	26	1	no	—
248	C₂²×C₄	—	52	1	no	—
249	C₂²×C₄	—	52	2	no	—
250	Q₁₆	—	26	1	no	—
251	Q₁₆	—	26	2	no	—
252	Q₁₆	—	26	1	no	—
253	Q₁₆	—	26	2	no	—
254	Q₁₆	—	104	2	no	—
255	Q₁₆	—	104	1	no	—
256	D₈	—	13	2	no	—
257	D₈	—	26	1	no	—
258	C₂×C₈	C₂×D₅₂	1	2	yes	—
259	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	26	1	no	—
262	C₄⋊C₄	—	26	2	no	—
263	C₄⋊C₄	—	26	1	no	—
264	C₄⋊C₄	—	52	2	no	—
265	C₄⋊C₄	—	52	1	no	—
266	C₄⋊C₄	—	52	1	no	—
267	C₄⋊C₄	—	52	2	no	—
268	C₄⋊C₄	—	52	2	no	—
269	C₄⋊C₄	—	52	1	no	—
270	C₄⋊C₄	—	52	2	no	—
271	C₄⋊C₄	—	52	1	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
274	C₄⋊C₄	D₁₀₄	1	2	yes	—
275	C₂²⋊C₄	—	52	1	no	—
276	C₂²⋊C₄	—	52	2	no	—
277	C₄²	—	52	2	no	—
278	C₄²	—	52	1	no	—
279	C₄²	C₂×D₅₂	1	2	yes	—
280	C₄²	C₂×D₅₂	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₃	#33939	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	52	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	52	1	no	—
288	Q₈	—	52	2	no	—
289	Q₈	—	52	1	no	—
290	Q₈	—	52	2	no	—
291	Q₈	—	208	1	no	—
292	D₄	—	26	2	no	—
293	D₄	—	52	1	no	—
294	C₂×C₄	—	52	1	no	—
295	C₂×C₄	—	52	2	no	—
296	C₂×C₄	—	52	1	no	—
297	C₂×C₄	—	52	1	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	—	104	2	no	—
300	C₂×C₄	—	208	2	no	—
301	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
302	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
303	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
304	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	#88254	1	1	yes	characteristic
309	C₈	#41×D₁₃	1	1	yes	characteristic
310	C₂²	—	52	2	no	—
311	C₂²	—	104	1	no	—
312	C₂²	—	104	1	no	—
313	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	104	1	no	—
318	C₄	—	104	2	no	—
319	C₄	—	104	1	no	—
320	C₄	—	104	1	no	—
321	C₄	—	104	2	no	—
322	C₄	#11718570	1	2	yes	—
323	C₄	#11719281	1	1	yes	characteristic
324	C₄	#11719456	1	1	yes	characteristic
325	C₂	—	104	2	no	—
326	C₂	#13127472	1	1	yes	characteristic
327	C₂	C₂×#11718570	1	1	yes	characteristic
328	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	C₂×D₁₀₄	C₂³
hypercenter	#2183	D₁₃
hypocenter	C₁₃	#33939
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
1. #13127842	C₂	semidirect
3. #13117082	C₂	semidirect
4. #13117064	C₂	semidirect
5. #13117001	C₂	semidirect
6. #13116989	C₂	semidirect
11. C₁₃×#2183	C₂	semidirect
129. #2183	D₁₃	semidirect
284. C₁₃	#33939	semidirect

Maximal subgroups

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

subgroup	index
#13127842	2
#13127704	2
#13117082	2
#13117064	2
#13117001	2
#13116989	2
#13116882	2
#13116702	2
C₄×#88040	2
C₁₃×#2183	2
#33939	13