Subgroups — #16020643

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 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	#13117063	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116988	C₂	1	2	yes	maximal, complemented
5	#13116973	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116888	C₂	1	2	yes	maximal
7	#13116878	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116745	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116698	C₂	1	2	yes	maximal, complemented
10	C₄×#88066	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#3393	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#556	—	2	1	no	—
13	Q₈×D₅₂	—	2	1	no	—
14	C₁₃×#495	C₂²	1	2	yes	—
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#464	—	2	2	no	—
17	C₁₃×#461	—	2	1	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
21	C₂×#88068	C₂²	1	2	yes	—
22	C₂×#88066	C₂²	1	1	yes	characteristic
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	2	yes	—
30	#11718156	C₂²	1	2	yes	—
31	#11718156	C₂²	1	2	yes	—
32	#11718154	C₂²	1	1	yes	characteristic
33	#11718154	C₂²	1	2	yes	—
34	#11718083	C₂²	1	1	yes	characteristic
35	#11718081	C₂²	1	1	yes	characteristic
36	#11718081	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	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×SD₁₆	—	2	2	no	—
45	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×#126	—	4	2	no	—
48	C₁₃×#124	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	4	1	no	—
50	C₄×Q₈×C₁₃	—	2	2	no	—
51	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×D₄×C₁₃	—	2	1	no	—
53	C₁₃×#117	—	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	1	no	—
58	C₄×D₅₂	—	2	2	no	—
59	#88123	—	2	1	no	—
60	#88123	—	2	2	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	2	no	—
65	C₁₃×#102	—	2	1	no	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	#88068	—	2	4	no	—
69	#88066	—	2	2	no	—
70	#88061	—	2	1	no	—
71	#88061	—	2	2	no	—
72	#88058	—	2	1	no	—
73	#88058	—	2	2	no	—
74	#88058	C₂³	1	1	yes	characteristic
75	#88058	C₂³	1	2	yes	—
76	C₂×#6090	C₂³	1	1	yes	characteristic
77	C₂×#6090	C₂³	1	2	yes	—
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	#33934	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
82	C₂×D₄×C₁₃	—	2	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	Q₈×D₁₃	—	8	4	no	—
85	C₂×D₅₂	—	2	1	no	—
86	C₂×C₄×D₁₃	—	4	2	no	—
87	C₂×C₄×D₁₃	—	4	1	no	—
88	C₂×Dic₂₆	—	4	2	no	—
89	C₂×Dic₂₆	—	4	1	no	—
90	C₁₃×Q₁₆	—	2	2	no	—
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	8	1	no	—
93	C₁₃×SD₁₆	—	8	2	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	—	2	1	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	2	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	1	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	2	yes	—
111	#6103	—	4	2	no	—
112	#6103	—	4	1	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	2	1	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	1	no	—
120	#6090	—	2	2	no	—
121	#6090	—	2	1	no	—
122	#3393	D₁₃	1	1	yes	characteristic, complemented
123	Q₈×D₈	—	13	1	no	—
124	#2243	—	13	2	no	—
125	#2241	—	13	1	no	—
126	#2227	—	13	2	no	—
127	#2224	—	13	1	no	—
128	#2210	—	13	2	no	—
129	#2208	—	13	1	no	—
130	#2183	—	13	1	no	—
131	#2175	—	13	2	no	—
132	C₄×D₁₆	—	13	1	no	—
133	C₂³×C₁₃	—	4	1	no	—
134	C₂²×D₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	2	2	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	8	2	no	—
143	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	D₅₂	—	2	2	no	—
151	D₅₂	—	4	1	no	—
152	C₄×D₁₃	—	8	4	no	—
153	C₄×D₁₃	—	8	2	no	—
154	Dic₂₆	—	8	4	no	—
155	Dic₂₆	—	8	2	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	#556	—	2	1	no	—
161	D₄×Q₈	—	26	1	no	—
162	C₂×SD₃₂	—	13	2	no	—
163	C₂×D₁₆	—	13	1	no	—
164	#500	—	13	1	no	—
165	#500	—	13	2	no	—
166	#495	C₂×D₁₃	1	2	yes	—
167	#494	C₂×D₁₃	1	1	yes	characteristic
168	#474	—	26	1	no	—
169	#474	—	26	2	no	—
170	#464	—	2	2	no	—
171	#461	—	2	1	no	—
172	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
173	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
174	C₄×D₈	—	13	1	no	—
175	C₄×D₈	—	13	2	no	—
176	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
177	#366	—	13	1	no	—
178	#366	—	13	2	no	—
179	C₄⋊C₁₆	—	13	1	no	—
180	C₄⋊C₁₆	—	13	2	no	—
181	#358	—	13	2	no	—
182	#358	—	13	2	no	—
183	#358	—	13	2	no	—
184	#357	—	13	1	no	—
185	#357	—	13	2	no	—
186	#357	—	13	1	no	—
187	#346	—	13	2	no	—
188	C₄×C₁₆	—	13	1	no	—
189	C₂²×C₁₃	—	8	1	no	—
190	C₂²×C₁₃	—	8	1	no	—
191	C₂²×C₁₃	—	8	1	no	—
192	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
193	C₂×D₁₃	—	4	2	no	—
194	C₂×D₁₃	—	8	1	no	—
195	C₂×D₁₃	—	8	1	no	—
196	C₄×C₁₃	—	2	2	no	—
197	C₄×C₁₃	—	2	2	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	—	8	1	no	—
201	C₄×C₁₃	#509	1	2	yes	—
202	C₄×C₁₃	#548	1	1	yes	characteristic
203	C₄×C₁₃	#577	1	1	yes	characteristic
204	Dic₁₃	—	8	1	no	—
205	Dic₁₃	—	8	2	no	—
206	C₂²×Q₈	—	52	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	—	13	1	no	—
211	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	26	1	no	—
214	C₄⋊Q₈	—	26	2	no	—
215	#126	—	4	2	no	—
216	#124	—	2	2	no	—
217	C₂²⋊Q₈	—	4	1	no	—
218	C₂²⋊Q₈	—	52	1	no	—
219	C₂²⋊Q₈	—	52	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	—	26	1	no	—
224	C₄×D₄	—	26	2	no	—
225	#117	—	4	1	no	—
226	SD₃₂	—	26	4	no	—
227	D₁₆	—	26	2	no	—
228	C₂×C₁₆	—	13	1	no	—
229	C₂×C₁₆	—	13	1	no	—
230	C₂×C₁₆	—	13	2	no	—
231	#107	—	13	1	no	—
232	#107	—	13	2	no	—
233	#107	—	26	1	no	—
234	#107	—	26	2	no	—
235	#107	C₂²×D₁₃	1	1	yes	characteristic
236	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#102	—	2	1	no	—
240	#102	—	26	1	no	—
241	#102	—	26	2	no	—
242	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₃	1	2	yes	—
244	C₂×C₁₃	—	16	1	no	—
245	C₂×C₁₃	#3358	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
247	C₂×C₁₃	#3417	1	1	yes	characteristic
248	D₁₃	—	8	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	52	2	no	—
251	C₂×Q₈	—	52	1	no	—
252	C₂×Q₈	—	104	4	no	—
253	C₂×Q₈	#6104	1	1	yes	characteristic
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	26	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	52	2	no	—
258	C₂²×C₄	—	52	1	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	8	1	no	—
262	SD₁₆	—	8	2	no	—
263	D₈	—	2	1	no	—
264	D₈	—	2	1	no	—
265	D₈	—	13	2	no	—
266	D₈	—	26	1	no	—
267	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
269	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	26	1	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄⋊C₄	—	52	1	no	—
278	C₄⋊C₄	#6104	1	1	yes	characteristic
279	C₄⋊C₄	#6104	1	2	yes	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	52	2	no	—
284	C₂²⋊C₄	—	52	1	no	—
285	C₄²	—	4	2	no	—
286	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
288	C₁₆	—	26	1	no	—
289	C₁₆	—	26	2	no	—
290	C₁₆	—	26	1	no	—
291	C₁₃	#33934	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	4	1	no	—
293	C₂³	—	52	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	104	4	no	—
298	Q₈	—	104	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	26	2	no	—
302	D₄	—	52	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	1	no	—
308	C₂×C₄	—	104	4	no	—
309	C₂×C₄	—	104	2	no	—
310	C₂×C₄	C₂×#6104	1	2	yes	—
311	C₂×C₄	C₂×#6104	1	1	yes	characteristic
312	C₂×C₄	C₂×#6104	1	2	yes	—
313	C₂×C₄	C₂×#6104	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	#41×D₁₃	1	1	yes	characteristic
318	C₈	#88257	1	1	yes	characteristic
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	52	2	no	—
323	C₂²	—	104	1	no	—
324	C₂²	—	104	1	no	—
325	C₂²	C₂²×#6104	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	104	2	no	—
332	C₄	—	104	1	no	—
333	C₄	#11718854	1	2	yes	—
334	C₄	#11719536	1	1	yes	characteristic
335	C₄	#11719541	1	1	yes	characteristic
336	C₂	—	16	1	no	—
337	C₂	—	104	2	no	—
338	C₂	#13127467	1	1	yes	characteristic
339	C₂	C₂×#11718854	1	1	yes	characteristic
340	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3393	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88066	C₂²
hypercenter	#3393	D₁₃
hypocenter	C₁₃	#33934
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3393

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. #13117063	C₂	semidirect
4. #13116988	C₂	semidirect
5. #13116973	C₂	semidirect
7. #13116878	C₂	semidirect
8. #13116745	C₂	semidirect
9. #13116698	C₂	semidirect
11. C₁₃×#3393	C₂	semidirect
122. #3393	D₁₃	semidirect
291. C₁₃	#33934	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
#13117063	2
#13116988	2
#13116973	2
#13116888	2
#13116878	2
#13116745	2
#13116698	2
C₄×#88066	2
C₁₃×#3393	2
#33934	13