Subgroups — #16020570

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127841	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127688	C₂	1	1	yes	maximal, characteristic
3	#13117065	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117062	C₂	1	2	yes	maximal, complemented
5	#13116977	C₂	1	2	yes	maximal, complemented
6	#13116882	C₂	1	1	yes	maximal, characteristic
7	#13116880	C₂	1	2	yes	maximal
8	#13116738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#88068	C₂	1	1	yes	maximal, characteristic
10	#13116687	C₂	1	2	yes	maximal
11	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719432	—	2	1	no	—
13	#11719272	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	#11718842	C₂²	1	2	yes	—
17	C₂×#88068	C₂²	1	1	yes	characteristic
18	#11718748	—	2	1	no	—
19	#11718692	—	2	1	no	—
20	C₂×#88040	C₂²	1	2	yes	—
21	#11718551	—	2	2	no	—
22	#11718535	C₂²	1	2	yes	—
23	#11718534	C₂²	1	2	yes	—
24	#11718533	C₂²	1	2	yes	—
25	#11718454	—	2	2	no	—
26	#11718360	C₂²	1	1	yes	characteristic
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₁₃×#358	C₂²	1	2	yes	—
32	#11718156	C₂²	1	1	yes	characteristic
33	#11718095	C₂²	1	2	yes	—
34	#11718093	C₂²	1	2	yes	—
35	C₁₆×Dic₁₃	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718084	C₂²	1	2	yes	—
38	#11718083	C₂²	1	1	yes	characteristic
39	#11718082	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₁₃	—	4	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₄×Q₈×C₁₃	—	2	1	no	—
45	#88200	—	4	1	no	—
46	#88198	—	2	2	no	—
47	C₂×#6107	—	2	2	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	C₂³	1	2	yes	—
50	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
51	#88150	—	4	2	no	—
52	#88147	—	2	2	no	—
53	#88143	—	4	2	no	—
54	#88142	—	2	2	no	—
55	#88141	—	2	2	no	—
56	C₄×D₅₂	—	2	1	no	—
57	#88124	—	4	1	no	—
58	#88123	—	2	1	no	—
59	C₄×Dic₂₆	—	4	1	no	—
60	C₄×Dic₂₆	D₄	1	1	yes	characteristic
61	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
62	C₁₃×#107	—	2	1	no	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×#103	—	2	1	no	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	#88068	—	2	2	no	—
68	#88061	—	2	1	no	—
69	#88058	C₂³	1	1	yes	characteristic
70	#88054	D₄	1	2	yes	—
71	C₈×Dic₁₃	C₂³	1	2	yes	—
72	C₂×#6090	C₂³	1	1	yes	characteristic
73	C₂×#6090	C₂³	1	1	yes	characteristic
74	#88050	—	2	2	no	—
75	#88049	—	2	2	no	—
76	#88047	—	2	2	no	—
77	C₄⋊#1055	D₄	1	1	yes	characteristic
78	#88040	—	2	4	no	—
79	#33926	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	Q₈×D₁₃	—	8	4	no	—
83	C₂×D₅₂	—	2	1	no	—
84	C₂×C₄×D₁₃	—	4	1	no	—
85	C₂×C₄×D₁₃	—	4	2	no	—
86	C₂×Dic₂₆	—	2	2	no	—
87	C₂×Dic₂₆	—	4	2	no	—
88	C₂×Dic₂₆	D₈	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₄²×C₁₃	—	4	1	no	—
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	#6107	—	8	2	no	—
100	#6103	—	4	1	no	—
101	#6103	—	4	2	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	4	1	no	—
104	C₄⋊Dic₁₃	—	4	1	no	—
105	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
106	#6101	—	4	2	no	—
107	#6101	—	4	2	no	—
108	#6101	—	4	2	no	—
109	#6101	—	4	2	no	—
110	#6101	D₈	1	2	yes	—
111	C₄×Dic₁₃	—	4	2	no	—
112	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
113	C₂×#1055	C₂×D₄	1	2	yes	—
114	Dic₅₂	—	2	2	no	—
115	Dic₅₂	—	2	2	no	—
116	Dic₅₂	—	8	1	no	—
117	D₁₀₄	—	2	1	no	—
118	D₁₀₄	#41	1	2	yes	—
119	C₁₃×C₁₆	—	2	2	no	—
120	#6090	—	2	1	no	—
121	#6090	—	2	1	no	—
122	#3395	—	13	1	no	—
123	Q₈×D₈	—	13	1	no	—
124	#2241	—	13	1	no	—
125	#2241	—	13	2	no	—
126	#2225	—	13	2	no	—
127	#2225	D₁₃	1	1	yes	characteristic, complemented
128	#2208	—	13	1	no	—
129	#2208	—	13	2	no	—
130	Q₈×C₁₆	—	13	1	no	—
131	C₄×SD₃₂	—	13	1	no	—
132	C₄×SD₃₂	—	13	2	no	—
133	C₂²×D₁₃	—	4	1	no	—
134	Q₈×C₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	1	no	—
145	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
146	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
147	D₅₂	—	2	2	no	—
148	D₅₂	—	4	1	no	—
149	C₄×D₁₃	—	8	2	no	—
150	C₄×D₁₃	—	8	4	no	—
151	Dic₂₆	—	4	2	no	—
152	Dic₂₆	—	4	2	no	—
153	Dic₂₆	—	8	4	no	—
154	Dic₂₆	SD₃₂	1	4	yes	complemented
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
158	#1055	—	2	2	no	—
159	#557	—	26	1	no	—
160	D₄×Q₈	—	26	1	no	—
161	C₂×SD₃₂	—	13	1	no	—
162	C₂×SD₃₂	—	13	2	no	—
163	#500	—	13	2	no	—
164	#500	C₂×D₁₃	1	1	yes	characteristic
165	#494	—	13	2	no	—
166	#494	—	13	1	no	—
167	#474	—	26	2	no	—
168	#474	—	26	1	no	—
169	#462	—	26	2	no	—
170	#462	—	26	1	no	—
171	C₈×Q₈	—	13	1	no	—
172	C₄×Q₁₆	—	13	2	no	—
173	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
174	C₄×D₈	—	13	2	no	—
175	C₄×D₈	—	13	1	no	—
176	#367	—	13	1	no	—
177	#367	C₂×D₁₃	1	2	yes	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
180	#358	—	13	1	no	—
181	#358	—	13	2	no	—
182	#358	—	13	2	no	—
183	#358	—	13	2	no	—
184	#358	C₂×D₁₃	1	2	yes	—
185	#357	—	13	1	no	—
186	C₄×C₁₆	—	13	2	no	—
187	C₄×C₁₆	—	13	1	no	—
188	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
189	C₂×D₁₃	—	4	2	no	—
190	C₂×D₁₃	—	8	1	no	—
191	C₂×D₁₃	—	8	1	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	8	1	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	—	8	1	no	—
196	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
197	C₄×C₁₃	#548	1	1	yes	characteristic
198	C₄×C₁₃	#577	1	1	yes	characteristic
199	Dic₁₃	—	2	2	no	—
200	Dic₁₃	—	8	2	no	—
201	Dic₁₃	—	8	1	no	—
202	Dic₁₃	C₂×SD₃₂	1	4	yes	—
203	C₂²×Q₈	—	52	1	no	—
204	C₂×Q₁₆	—	13	2	no	—
205	C₂×Q₁₆	—	26	2	no	—
206	C₂×Q₁₆	—	26	1	no	—
207	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×D₈	—	13	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	26	2	no	—
211	C₄⋊Q₈	—	26	2	no	—
212	C₄⋊Q₈	—	26	1	no	—
213	#125	—	52	1	no	—
214	#125	—	52	2	no	—
215	C₂²⋊Q₈	—	52	2	no	—
216	C₂²⋊Q₈	—	52	1	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×Q₈	—	13	1	no	—
219	C₄×Q₈	—	26	2	no	—
220	C₄×Q₈	—	52	1	no	—
221	C₄×D₄	—	26	2	no	—
222	C₄×D₄	—	26	1	no	—
223	SD₃₂	—	26	4	no	—
224	SD₃₂	—	26	2	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	1	no	—
227	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	13	1	no	—
230	#107	—	26	2	no	—
231	#107	C₂²×D₁₃	1	1	yes	characteristic
232	#107	C₂²×D₁₃	1	2	yes	—
233	C₄⋊C₈	—	13	2	no	—
234	C₄⋊C₈	—	13	1	no	—
235	#103	—	2	1	no	—
236	#103	—	26	2	no	—
237	#102	—	26	2	no	—
238	#102	—	26	1	no	—
239	C₄×C₈	—	13	2	no	—
240	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×C₁₃	#3358	1	1	yes	characteristic
242	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
243	C₂×C₁₃	#3417	1	1	yes	characteristic
244	D₁₃	—	8	2	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	13	1	no	—
248	C₂×Q₈	—	26	2	no	—
249	C₂×Q₈	—	52	2	no	—
250	C₂×Q₈	—	104	4	no	—
251	C₂×D₄	—	26	1	no	—
252	C₂²×C₄	—	52	2	no	—
253	C₂²×C₄	—	52	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	26	2	no	—
257	Q₁₆	—	26	2	no	—
258	Q₁₆	—	104	1	no	—
259	Q₁₆	—	104	2	no	—
260	D₈	—	13	2	no	—
261	D₈	—	26	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	13	1	no	—
271	C₄⋊C₄	—	26	1	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₄⋊C₄	—	52	2	no	—
275	C₄⋊C₄	—	52	2	no	—
276	C₄⋊C₄	—	52	1	no	—
277	C₄⋊C₄	—	52	1	no	—
278	C₂²⋊C₄	—	52	2	no	—
279	C₂²⋊C₄	—	52	1	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	52	2	no	—
283	C₄²	D₄×D₁₃	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	26	1	no	—
286	C₁₆	—	26	1	no	—
287	C₁₃	#33926	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	52	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	13	4	no	—
293	Q₈	—	52	2	no	—
294	Q₈	—	52	2	no	—
295	Q₈	—	104	4	no	—
296	D₄	—	26	2	no	—
297	D₄	—	52	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	—	52	1	no	—
305	C₂×C₄	—	104	4	no	—
306	C₂×C₄	—	104	2	no	—
307	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
308	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	26	2	no	—
312	C₈	#88254	1	1	yes	characteristic
313	C₈	#41×D₁₃	1	1	yes	characteristic
314	C₂²	—	52	2	no	—
315	C₂²	—	104	1	no	—
316	C₂²	—	104	1	no	—
317	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	13	4	no	—
323	C₄	—	26	2	no	—
324	C₄	—	104	1	no	—
325	C₄	—	104	2	no	—
326	C₄	D₁₃×SD₃₂	1	2	yes	—
327	C₄	#11719338	1	1	yes	characteristic
328	C₄	#11719483	1	1	yes	characteristic
329	C₂	—	104	2	no	—
330	C₂	#13127470	1	1	yes	characteristic
331	C₂	C₂×D₁₃×SD₃₂	1	1	yes	characteristic
332	C₂	#13127988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2225	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₁₀₄	C₂³
hypercenter	#2225	D₁₃
hypocenter	C₁₃	#33926
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
1. #13127841	C₂	semidirect
3. #13117065	C₂	semidirect
4. #13117062	C₂	semidirect
5. #13116977	C₂	semidirect
8. #13116738	C₂	semidirect
11. C₁₃×#2225	C₂	semidirect
127. #2225	D₁₃	semidirect
154. Dic₂₆	SD₃₂	semidirect
287. C₁₃	#33926	semidirect

Maximal subgroups

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

subgroup	index
#13127841	2
#13127688	2
#13117065	2
#13117062	2
#13116977	2
#13116882	2
#13116880	2
#13116738	2
C₄×#88068	2
#13116687	2
C₁₃×#2225	2
#33926	13