Subgroups — #16020616

Order 3328.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127825	C₂	1	1	yes	maximal, characteristic
2	#13127728	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117068	C₂	1	2	yes	maximal
4	#13117064	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117001	C₂	1	2	yes	maximal, complemented
6	#13116974	C₂	1	1	yes	maximal, characteristic
7	#13116899	C₂	1	2	yes	maximal
8	#13116887	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×Dic₁₀₄	C₂	1	2	yes	maximal, complemented
10	C₄×#88040	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719289	—	2	1	no	—
13	#11719272	—	2	1	no	—
14	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₀₄	C₂²	1	2	yes	—
16	C₂×#88040	C₂²	1	1	yes	characteristic
17	#11718512	—	2	1	no	—
18	#11718506	—	2	2	no	—
19	#11718502	—	2	1	no	—
20	#11718490	—	2	2	no	—
21	#11718360	C₂²	1	1	yes	characteristic
22	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
23	C₄×Dic₅₂	C₂²	1	2	yes	—
24	C₄×Dic₅₂	C₂²	1	2	yes	—
25	#11718357	C₂²	1	2	yes	—
26	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
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	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
32	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
33	#11718101	C₂²	1	1	yes	characteristic
34	#11718101	C₂²	1	2	yes	—
35	#11718097	C₂²	1	1	yes	characteristic
36	#11718096	C₂²	1	2	yes	—
37	#11718095	C₂²	1	1	yes	characteristic
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	#88198	—	4	1	no	—
44	C₂×Dic₅₂	—	2	1	no	—
45	C₂×Dic₅₂	C₂³	1	2	yes	—
46	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
47	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
48	C₂×#6095	—	2	2	no	—
49	#88152	—	4	2	no	—
50	#88146	—	4	1	no	—
51	#88143	—	4	2	no	—
52	#88141	—	4	1	no	—
53	#88129	—	2	2	no	—
54	C₄×D₅₂	—	2	1	no	—
55	#88124	—	2	2	no	—
56	#88123	—	2	1	no	—
57	#88123	—	2	1	no	—
58	C₄×Dic₂₆	—	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	2	yes	—
65	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88061	—	2	1	no	—
69	#88058	—	2	1	no	—
70	#88058	C₂³	1	2	yes	—
71	#88058	C₂³	1	1	yes	characteristic
72	#88058	C₂³	1	1	yes	characteristic
73	#88057	—	2	2	no	—
74	#88056	—	2	1	no	—
75	#88056	—	2	2	no	—
76	#88056	—	2	2	no	—
77	Dic₁₀₄	—	2	4	no	—
78	#88040	—	2	2	no	—
79	#33931	—	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×Dic₂₆	—	2	1	no	—
84	C₂×Dic₂₆	—	2	2	no	—
85	C₂×Dic₂₆	—	4	1	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	#6103	—	4	1	no	—
94	#6103	—	4	2	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	1	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	C₄⋊Dic₁₃	—	4	1	no	—
102	#6101	—	4	1	no	—
103	#6101	—	4	2	no	—
104	#6101	—	4	2	no	—
105	#6101	—	4	2	no	—
106	#6101	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	—	4	1	no	—
109	C₄×Dic₁₃	—	4	2	no	—
110	Dic₅₂	—	2	2	no	—
111	Dic₅₂	—	2	1	no	—
112	Dic₅₂	—	2	2	no	—
113	Dic₅₂	—	8	1	no	—
114	Dic₅₂	#41	1	2	yes	—
115	D₁₀₄	—	2	1	no	—
116	D₁₀₄	—	2	1	no	—
117	#6095	—	8	2	no	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	—	2	2	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	#3393	—	13	1	no	—
122	#3383	—	13	1	no	—
123	#2242	—	13	2	no	—
124	#2241	—	13	1	no	—
125	#2229	—	13	2	no	—
126	#2224	—	13	1	no	—
127	#2211	—	13	2	no	—
128	#2209	—	13	1	no	—
129	Q₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
130	C₄×Q₃₂	—	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	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	D₅₂	—	4	1	no	—
145	D₅₂	—	4	1	no	—
146	C₄×D₁₃	—	8	2	no	—
147	Dic₂₆	—	2	2	no	—
148	Dic₂₆	—	4	1	no	—
149	Dic₂₆	—	4	2	no	—
150	Dic₂₆	—	4	2	no	—
151	Dic₂₆	—	8	2	no	—
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
156	#557	—	26	1	no	—
157	#556	—	26	1	no	—
158	C₂×Q₃₂	—	13	2	no	—
159	C₂×SD₃₂	—	13	1	no	—
160	#500	—	13	1	no	—
161	#499	—	13	2	no	—
162	#495	—	13	2	no	—
163	#494	—	13	1	no	—
164	#479	—	26	2	no	—
165	#475	—	26	1	no	—
166	#464	—	26	2	no	—
167	#461	—	26	1	no	—
168	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
169	C₄×Q₁₆	—	13	1	no	—
170	C₄×Q₁₆	—	13	2	no	—
171	C₄×Q₁₆	—	13	2	no	—
172	C₄×D₈	—	13	1	no	—
173	#367	—	13	1	no	—
174	#366	—	13	2	no	—
175	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
176	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
177	#358	—	13	1	no	—
178	#358	—	13	2	no	—
179	#358	—	13	2	no	—
180	#358	—	13	2	no	—
181	#357	—	13	1	no	—
182	#357	—	13	2	no	—
183	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
184	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₂×D₁₃	—	8	1	no	—
187	C₂×D₁₃	—	8	1	no	—
188	C₂×D₁₃	—	8	1	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	#508	1	2	yes	—
193	C₄×C₁₃	#548	1	1	yes	characteristic
194	C₄×C₁₃	#577	1	1	yes	characteristic
195	Dic₁₃	—	4	2	no	—
196	Dic₁₃	—	8	2	no	—
197	Dic₁₃	—	8	2	no	—
198	Dic₁₃	—	8	1	no	—
199	Dic₁₃	—	8	1	no	—
200	C₂×Q₁₆	—	13	1	no	—
201	C₂×Q₁₆	—	13	2	no	—
202	C₂×Q₁₆	—	26	1	no	—
203	C₂×SD₁₆	—	26	2	no	—
204	C₂×D₈	—	13	1	no	—
205	C₄⋊Q₈	—	26	1	no	—
206	C₄⋊Q₈	—	26	1	no	—
207	C₄⋊Q₈	—	52	1	no	—
208	#126	—	52	2	no	—
209	#125	—	26	2	no	—
210	#125	—	52	2	no	—
211	#124	—	26	2	no	—
212	C₂²⋊Q₈	—	52	1	no	—
213	C₄×Q₈	—	26	2	no	—
214	C₄×Q₈	—	26	1	no	—
215	C₄×Q₈	—	26	2	no	—
216	C₄×Q₈	—	52	1	no	—
217	C₄×Q₈	D₅₂	1	1	yes	characteristic
218	C₄×D₄	—	26	1	no	—
219	#117	—	52	1	no	—
220	Q₃₂	—	26	4	no	—
221	SD₃₂	—	26	2	no	—
222	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
224	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
225	#107	—	13	2	no	—
226	#107	—	13	1	no	—
227	#107	—	13	1	no	—
228	#107	—	26	1	no	—
229	#106	—	26	2	no	—
230	C₄⋊C₈	D₅₂	1	1	yes	characteristic
231	C₄⋊C₈	D₅₂	1	2	yes	—
232	#103	—	26	2	no	—
233	#103	—	26	2	no	—
234	#103	—	26	1	no	—
235	#102	—	26	1	no	—
236	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₃	1	2	yes	—
238	C₂×C₁₃	#3358	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
240	C₂×C₁₃	#3417	1	1	yes	characteristic
241	D₁₃	—	16	1	no	—
242	C₂×Q₈	—	26	1	no	—
243	C₂×Q₈	—	26	2	no	—
244	C₂×Q₈	—	52	1	no	—
245	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
246	C₂×D₄	—	26	1	no	—
247	C₂²×C₄	—	52	1	no	—
248	Q₁₆	—	13	2	no	—
249	Q₁₆	—	26	1	no	—
250	Q₁₆	—	26	2	no	—
251	Q₁₆	—	26	2	no	—
252	Q₁₆	—	104	1	no	—
253	SD₁₆	—	104	2	no	—
254	D₈	—	26	1	no	—
255	D₈	—	26	1	no	—
256	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
257	C₂×C₈	C₂×D₅₂	1	2	yes	—
258	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
259	C₄⋊C₄	—	26	1	no	—
260	C₄⋊C₄	—	26	1	no	—
261	C₄⋊C₄	—	26	2	no	—
262	C₄⋊C₄	—	52	1	no	—
263	C₄⋊C₄	—	52	2	no	—
264	C₄⋊C₄	—	52	1	no	—
265	C₄⋊C₄	—	52	2	no	—
266	C₄⋊C₄	—	52	2	no	—
267	C₄⋊C₄	—	52	2	no	—
268	C₄⋊C₄	—	52	1	no	—
269	C₄⋊C₄	—	52	1	no	—
270	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
271	C₄⋊C₄	D₁₀₄	1	2	yes	—
272	C₂²⋊C₄	—	52	2	no	—
273	C₂²⋊C₄	—	52	1	no	—
274	C₂²⋊C₄	—	52	2	no	—
275	C₄²	—	52	2	no	—
276	C₄²	—	52	2	no	—
277	C₄²	—	52	1	no	—
278	C₄²	C₂×D₅₂	1	1	yes	characteristic
279	C₄²	C₂×D₅₂	1	2	yes	—
280	C₁₆	—	2	1	no	—
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	2	no	—
283	C₁₃	#33931	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	52	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	26	2	no	—
287	Q₈	—	52	1	no	—
288	Q₈	—	52	2	no	—
289	Q₈	—	52	2	no	—
290	Q₈	—	104	2	no	—
291	D₄	—	52	1	no	—
292	D₄	—	52	1	no	—
293	C₂×C₄	—	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	2	no	—
298	C₂×C₄	—	104	2	no	—
299	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
300	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
301	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
302	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	#88254	1	1	yes	characteristic
307	C₈	#41×D₁₃	1	1	yes	characteristic
308	C₂²	—	104	1	no	—
309	C₂²	—	104	1	no	—
310	C₂²	—	104	1	no	—
311	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	52	2	no	—
316	C₄	—	104	1	no	—
317	C₄	—	104	2	no	—
318	C₄	—	104	2	no	—
319	C₄	—	104	1	no	—
320	C₄	#11718565	1	2	yes	—
321	C₄	#11719281	1	1	yes	characteristic
322	C₄	#11719456	1	1	yes	characteristic
323	C₂	—	208	1	no	—
324	C₂	#13127472	1	1	yes	characteristic
325	C₂	C₂×#11718565	1	1	yes	characteristic
326	C₂	#13127989	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	Q₈×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₁₀₄	C₂³
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	#33931
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13127728	C₂	semidirect
4. #13117064	C₂	semidirect
5. #13117001	C₂	semidirect
8. #13116887	C₂	semidirect
9. C₄×Dic₁₀₄	C₂	semidirect
11. Q₈×C₁₃×C₁₆	C₂	semidirect
129. Q₈×C₁₆	D₁₃	semidirect
283. C₁₃	#33931	semidirect

Maximal subgroups

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

subgroup	index
#13127825	2
#13127728	2
#13117068	2
#13117064	2
#13117001	2
#13116974	2
#13116899	2
#13116887	2
C₄×Dic₁₀₄	2
C₄×#88040	2
Q₈×C₁₃×C₁₆	2
#33931	13