Subgroups — #16020717

Order 3328.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117072	C₂	1	2	yes	maximal, complemented
3	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116983	C₂	1	2	yes	maximal, complemented
5	#13116978	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116888	C₂	1	2	yes	maximal, complemented
7	#13116868	C₂	1	1	yes	maximal, characteristic
8	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116710	C₂	1	2	yes	maximal, complemented
10	C₄×#88068	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3396	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#559	—	2	1	no	—
13	Q₈×D₅₂	—	2	1	no	—
14	C₁₃×#495	C₂²	1	2	yes	—
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₁₃×#463	—	2	2	no	—
17	C₁₃×#460	—	2	1	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×D₈	C₂²	1	2	yes	—
21	C₂×#88068	C₂²	1	1	yes	characteristic
22	C₂×#88066	C₂²	1	2	yes	—
23	#11718507	—	2	1	no	—
24	#11718507	—	2	2	no	—
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	#11718343	C₂²	1	1	yes	characteristic
28	#11718343	C₂²	1	2	yes	—
29	#11718156	C₂²	1	1	yes	characteristic
30	#11718156	C₂²	1	2	yes	—
31	#11718154	C₂²	1	2	yes	—
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718083	C₂²	1	1	yes	characteristic
35	#11718082	C₂²	1	1	yes	characteristic
36	#11718082	C₂²	1	2	yes	—
37	C₄⋊#6090	C₂²	1	1	yes	characteristic
38	C₄⋊#6090	C₂²	1	2	yes	—
39	#11718053	C₂²	1	2	yes	—
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	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#124	—	2	2	no	—
48	C₁₃×#124	—	4	1	no	—
49	C₁₃×#123	—	4	2	no	—
50	C₄×Q₈×C₁₃	—	2	1	no	—
51	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
55	#88151	—	4	1	no	—
56	#88151	—	4	2	no	—
57	C₄×D₅₂	—	2	2	no	—
58	C₄×D₅₂	—	2	1	no	—
59	#88123	—	2	2	no	—
60	#88123	—	2	1	no	—
61	C₁₃×#107	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₁₃×#103	—	2	1	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₄×C₈×C₁₃	C₂³	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	#88068	—	2	2	no	—
69	#88066	—	2	4	no	—
70	#88061	—	2	2	no	—
71	#88061	—	2	1	no	—
72	#88058	—	2	2	no	—
73	#88058	—	2	1	no	—
74	#88058	C₂³	1	2	yes	—
75	#88058	C₂³	1	1	yes	characteristic
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	2	yes	—
79	#33940	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	Q₈×D₁₃	—	8	4	no	—
87	C₂×D₅₂	—	2	1	no	—
88	C₂×C₄×D₁₃	—	4	2	no	—
89	C₂×C₄×D₁₃	—	4	1	no	—
90	C₂×Dic₂₆	—	4	2	no	—
91	C₂×Dic₂₆	—	4	1	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×SD₁₆	—	8	2	no	—
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	2	2	no	—
97	C₁₃×D₈	—	8	1	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₁₃	—	4	1	no	—
109	C₄²×C₁₃	C₂×D₄	1	2	yes	—
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	#6103	—	4	2	no	—
112	#6103	—	4	1	no	—
113	C₄⋊Dic₁₃	—	2	1	no	—
114	C₄⋊Dic₁₃	—	2	2	no	—
115	C₄⋊Dic₁₃	—	4	2	no	—
116	C₄⋊Dic₁₃	—	4	1	no	—
117	D₁₀₄	—	2	1	no	—
118	D₁₀₄	#41	1	2	yes	—
119	#6090	—	2	2	no	—
120	#6090	—	2	1	no	—
121	#6090	—	2	1	no	—
122	#3396	D₁₃	1	1	yes	characteristic, complemented
123	Q₈×D₈	—	13	1	no	—
124	#2243	—	13	2	no	—
125	#2240	—	13	1	no	—
126	#2226	—	13	2	no	—
127	#2225	—	13	1	no	—
128	#2210	—	13	2	no	—
129	#2206	—	13	1	no	—
130	#2183	—	13	1	no	—
131	#2177	—	13	2	no	—
132	C₄×SD₃₂	—	13	1	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂²×D₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	2	2	no	—
136	Q₈×C₁₃	—	4	1	no	—
137	Q₈×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	8	2	no	—
141	D₄×C₁₃	—	16	1	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	16	2	no	—
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	2	no	—
152	D₅₂	—	2	2	no	—
153	D₅₂	—	4	1	no	—
154	C₄×D₁₃	—	8	4	no	—
155	C₄×D₁₃	—	8	2	no	—
156	Dic₂₆	—	8	4	no	—
157	Dic₂₆	—	8	2	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	#559	—	2	1	no	—
163	D₄×Q₈	—	26	1	no	—
164	C₂×SD₃₂	—	13	1	no	—
165	C₂×D₁₆	—	13	2	no	—
166	#500	—	13	1	no	—
167	#500	—	13	2	no	—
168	#495	C₂×D₁₃	1	2	yes	—
169	#493	C₂×D₁₃	1	1	yes	characteristic
170	#474	—	26	1	no	—
171	#474	—	26	2	no	—
172	#463	—	2	2	no	—
173	#460	—	2	1	no	—
174	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
175	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
176	C₄×D₈	—	13	1	no	—
177	C₄×D₈	—	13	2	no	—
178	C₄×D₈	C₂×D₁₃	1	2	yes	—
179	#367	—	13	1	no	—
180	#367	—	13	2	no	—
181	C₄⋊C₁₆	—	13	1	no	—
182	C₄⋊C₁₆	—	13	2	no	—
183	#358	—	13	1	no	—
184	#358	—	13	2	no	—
185	#357	—	13	2	no	—
186	#357	—	13	2	no	—
187	#357	—	13	2	no	—
188	#357	—	13	1	no	—
189	#346	—	13	2	no	—
190	C₄×C₁₆	—	13	1	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
195	C₂×D₁₃	—	4	2	no	—
196	C₂×D₁₃	—	8	1	no	—
197	C₂×D₁₃	—	8	1	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	#509	1	2	yes	—
204	C₄×C₁₃	#548	1	1	yes	characteristic
205	C₄×C₁₃	#577	1	1	yes	characteristic
206	Dic₁₃	—	8	1	no	—
207	Dic₁₃	—	8	2	no	—
208	C₂²×Q₈	—	52	1	no	—
209	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	13	1	no	—
213	C₂×D₈	C₂²×D₁₃	1	2	yes	—
214	C₄⋊Q₈	—	26	2	no	—
215	C₄⋊Q₈	—	26	1	no	—
216	#127	—	2	1	no	—
217	#124	—	2	2	no	—
218	#124	—	4	1	no	—
219	#123	—	4	2	no	—
220	C₂²⋊Q₈	—	52	1	no	—
221	C₂²⋊Q₈	—	52	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	4	1	no	—
226	C₄×D₄	—	26	2	no	—
227	C₄×D₄	—	26	1	no	—
228	SD₃₂	—	26	2	no	—
229	D₁₆	—	26	4	no	—
230	C₂×C₁₆	—	13	1	no	—
231	C₂×C₁₆	—	13	1	no	—
232	C₂×C₁₆	—	13	2	no	—
233	#107	—	13	2	no	—
234	#107	—	13	1	no	—
235	#107	—	26	2	no	—
236	#107	—	26	1	no	—
237	#107	C₂²×D₁₃	1	1	yes	characteristic
238	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
239	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
240	#103	—	2	1	no	—
241	#102	—	2	2	no	—
242	#102	—	26	2	no	—
243	#102	—	26	1	no	—
244	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₁₃	1	2	yes	—
246	C₂×C₁₃	—	16	2	no	—
247	C₂×C₁₃	#3358	1	1	yes	characteristic
248	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
249	C₂×C₁₃	#3416	1	1	yes	characteristic
250	D₁₃	—	8	2	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	52	2	no	—
253	C₂×Q₈	—	52	1	no	—
254	C₂×Q₈	—	104	4	no	—
255	C₂×Q₈	#6104	1	1	yes	characteristic
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	26	1	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	52	2	no	—
262	C₂²×C₄	—	52	1	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	1	no	—
265	SD₁₆	—	8	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	8	1	no	—
269	D₈	—	13	2	no	—
270	D₈	—	26	1	no	—
271	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
272	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄⋊C₄	—	26	1	no	—
278	C₄⋊C₄	—	52	2	no	—
279	C₄⋊C₄	—	52	1	no	—
280	C₄⋊C₄	#6104	1	1	yes	characteristic
281	C₄⋊C₄	#6104	1	2	yes	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	—	52	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
289	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₁₆	—	26	2	no	—
291	C₁₆	—	26	1	no	—
292	C₁₆	—	26	1	no	—
293	C₁₃	#33940	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	4	2	no	—
295	C₂³	—	52	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	104	4	no	—
300	Q₈	—	104	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	16	1	no	—
305	D₄	—	26	2	no	—
306	D₄	—	52	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	52	2	no	—
311	C₂×C₄	—	52	1	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	2	yes	—
316	C₂×C₄	C₂×#6104	1	1	yes	characteristic
317	C₂×C₄	C₂×#6104	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	#41×D₁₃	1	1	yes	characteristic
322	C₈	#88257	1	1	yes	characteristic
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	52	2	no	—
327	C₂²	—	104	1	no	—
328	C₂²	—	104	1	no	—
329	C₂²	C₂²×#6104	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	104	2	no	—
336	C₄	—	104	1	no	—
337	C₄	#11718854	1	2	yes	—
338	C₄	#11719536	1	1	yes	characteristic
339	C₄	#11719541	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	104	2	no	—
342	C₂	#13127467	1	1	yes	characteristic
343	C₂	C₂×#11718854	1	1	yes	characteristic
344	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×#3396	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13116868	C₂
hypercenter	#3396	D₁₃
hypocenter	C₁₃	#33940
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3396

Complements

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

N	H ≅ G/N	type
1. Q₈×D₁₀₄	C₂	semidirect
2. #13117072	C₂	semidirect
3. #13117059	C₂	semidirect
4. #13116983	C₂	semidirect
5. #13116978	C₂	semidirect
6. #13116888	C₂	semidirect
8. #13116745	C₂	semidirect
9. #13116710	C₂	semidirect
10. C₄×#88068	C₂	semidirect
11. C₁₃×#3396	C₂	semidirect
122. #3396	D₁₃	semidirect
293. C₁₃	#33940	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₁₀₄	2
#13117072	2
#13117059	2
#13116983	2
#13116978	2
#13116888	2
#13116868	2
#13116745	2
#13116710	2
C₄×#88068	2
C₁₃×#3396	2
#33940	13