Subgroups — #16017662

Order 3328.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117012	C₂	1	4	yes	maximal, complemented
3	#13117007	C₂	1	4	yes	maximal, complemented
4	#13116610	C₂	1	2	yes	maximal, complemented
5	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116572	C₂	1	2	yes	maximal
7	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#131	—	2	1	no	—
10	#11719175	—	2	1	no	—
11	C₁₃×#443	—	2	1	no	—
12	C₁₃×#443	—	2	1	no	—
13	C₁₃×#443	C₂²	1	2	yes	—
14	C₁₃×#443	C₂²	1	2	yes	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#104	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#11718767	—	2	1	no	—
19	#11718767	C₂²	1	2	yes	—
20	C₂×#88052	C₂²	1	2	yes	—
21	#11718344	—	2	1	no	—
22	#11718344	C₂²	1	2	yes	complemented
23	#11718343	—	2	1	no	—
24	#11718341	C₂²	1	2	yes	complemented
25	#11718158	C₂²	1	8	yes	—
26	#11718086	C₂²	1	8	yes	complemented
27	#11718055	C₂²	1	2	yes	complemented
28	#11718055	C₂²	1	2	yes	complemented
29	C₂×C₁₃×#41	—	2	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	—	2	1	no	—
32	C₁₃×#135	Q₈	1	2	yes	complemented
33	C₁₃×#135	C₂³	1	2	yes	—
34	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
35	C₂×C₁₃×SD₁₆	—	2	1	no	—
36	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
37	C₁₃×#131	—	2	2	no	—
38	C₁₃×#131	—	2	2	no	—
39	C₁₃×#131	—	4	1	no	—
40	C₁₃×#131	—	4	1	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂×C₁₃×M₁₆	—	4	1	no	—
43	C₂²×C₈×C₁₃	—	4	1	no	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
47	#88180	—	2	2	no	—
48	#88178	—	4	2	no	—
49	#88124	—	2	2	no	—
50	#88123	—	2	2	no	—
51	C₄×Dic₂₆	—	4	2	no	—
52	C₁₃×#108	—	2	1	no	—
53	C₁₃×#108	C₂³	1	2	yes	—
54	C₁₃×#104	—	2	2	no	—
55	C₁₃×#104	—	2	2	no	—
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	#88058	—	2	2	no	—
61	#88058	—	2	2	no	—
62	#88057	—	2	2	no	—
63	#88057	C₂³	1	4	yes	—
64	#88052	—	2	2	no	—
65	#88052	C₂³	1	4	yes	—
66	C₂×#6090	—	2	2	no	—
67	#33637	—	13	1	no	maximal, Sylow
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₂₆	—	4	2	no	—
79	C₁₃×Q₁₆	—	2	1	no	—
80	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
81	C₁₃×SD₁₆	—	2	2	no	—
82	C₁₃×SD₁₆	—	2	2	no	—
83	C₁₃×D₈	—	2	1	no	—
84	C₁₃×D₈	C₂×Q₈	1	2	yes	—
85	C₁₃×M₁₆	—	2	2	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₁₃×M₁₆	—	4	2	no	—
88	C₁₃×M₁₆	—	4	2	no	—
89	C₂×C₈×C₁₃	—	4	1	no	—
90	C₂×C₈×C₁₃	—	4	2	no	—
91	C₂×C₈×C₁₃	—	4	2	no	—
92	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
95	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂²⋊Dic₁₃	—	4	2	no	—
108	C₄⋊Dic₁₃	—	2	4	no	—
109	C₄⋊Dic₁₃	—	4	2	no	—
110	#6101	—	4	4	no	—
111	C₄×Dic₁₃	—	4	2	no	—
112	#6090	—	2	4	no	—
113	#3160	—	13	1	no	—
114	C₂×#443	D₁₃	1	1	yes	characteristic
115	#2231	—	13	4	no	—
116	#2230	—	13	4	no	—
117	#2155	—	13	2	no	—
118	C₂×#364	—	13	1	no	—
119	#2145	—	13	2	no	—
120	C₂³×C₁₃	—	4	1	no	—
121	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	2	no	—
123	Q₈×C₁₃	—	4	1	no	—
124	D₄×C₁₃	—	2	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	4	4	no	—
141	Dic₂₆	—	8	4	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
146	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
149	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
152	C₂×#131	—	2	1	no	—
153	#533	—	26	1	no	—
154	C₂×M₃₂	—	13	2	no	—
155	#500	—	26	1	no	—
156	#499	—	13	2	no	—
157	#499	—	26	1	no	—
158	#498	—	13	2	no	—
159	#443	—	2	1	no	—
160	#443	—	2	1	no	—
161	#443	C₂×D₁₃	1	2	yes	—
162	#443	C₂×D₁₃	1	2	yes	—
163	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
164	#427	—	13	2	no	—
165	#427	—	26	1	no	—
166	C₂×#104	—	2	1	no	—
167	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
168	#365	—	13	8	no	—
169	#364	—	13	2	no	—
170	#364	—	13	2	no	—
171	#360	—	13	8	no	—
172	C₂²×C₁₃	—	4	1	no	—
173	C₂²×C₁₃	—	4	2	no	—
174	C₂²×C₁₃	—	8	2	no	—
175	C₂²×C₁₃	#476	1	1	yes	characteristic
176	C₂²×C₁₃	#476	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	4	1	no	—
180	C₄×C₁₃	—	4	2	no	—
181	C₄×C₁₃	#476	1	1	yes	characteristic
182	C₄×C₁₃	#476	1	1	yes	characteristic
183	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
185	Dic₁₃	—	8	4	no	—
186	C₂×#41	—	2	1	no	—
187	#135	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	Dic₂₆	1	2	yes	complemented
190	#135	C₂²×D₁₃	1	2	yes	—
191	C₂×Q₁₆	Dic₂₆	1	1	yes	characteristic, complemented
192	C₂×SD₁₆	—	2	1	no	—
193	C₂×D₈	Dic₂₆	1	1	yes	characteristic, complemented
194	#131	—	2	2	no	—
195	#131	—	2	2	no	—
196	#131	—	4	1	no	—
197	#131	—	4	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	4	1	no	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	26	2	no	—
204	#125	—	26	2	no	—
205	C₂²⋊Q₈	—	52	2	no	—
206	C₄×Q₈	—	52	2	no	—
207	#117	—	26	2	no	—
208	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
209	M₃₂	—	13	4	no	—
210	M₃₂	—	26	2	no	—
211	C₂×C₁₆	—	26	2	no	—
212	#108	—	2	1	no	—
213	#108	C₂²×D₁₃	1	2	yes	—
214	#107	—	26	2	no	—
215	#107	—	26	2	no	—
216	#106	—	13	4	no	—
217	#106	—	26	2	no	—
218	#104	—	2	2	no	—
219	#104	—	2	2	no	—
220	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	8	2	no	—
228	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	52	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
240	Q₁₆	—	2	1	no	—
241	Q₁₆	C₂×Dic₂₆	1	2	yes	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	2	2	no	—
244	D₈	—	2	1	no	—
245	D₈	C₂×Dic₂₆	1	2	yes	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	#6105	1	1	yes	characteristic
254	C₂×C₈	#6105	1	1	yes	characteristic
255	C₂×C₈	#6105	1	1	yes	characteristic
256	C₂×C₈	#6105	1	1	yes	characteristic
257	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
258	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
259	C₂×C₈	#6127	1	1	yes	characteristic
260	C₂×C₈	#6127	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	26	4	no	—
266	C₄⋊C₄	—	52	4	no	—
267	C₄⋊C₄	—	52	2	no	—
268	C₂²⋊C₄	—	52	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	52	2	no	—
271	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₁₆	—	26	4	no	—
274	C₁₃	#33637	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	4	1	no	—
276	C₂³	#88203	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	104	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	52	4	no	—
291	C₂×C₄	#88178	1	1	yes	characteristic
292	C₂×C₄	#88178	1	1	yes	characteristic
293	C₂×C₄	C₂×#6105	1	1	yes	characteristic
294	C₂×C₄	C₂×#6105	1	1	yes	characteristic
295	C₂×C₄	#88203	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	#88178	1	1	yes	characteristic
299	C₈	#88178	1	1	yes	characteristic
300	C₈	C₂×#6105	1	1	yes	characteristic
301	C₈	C₂×#6105	1	1	yes	characteristic
302	C₈	C₂×#6105	1	1	yes	characteristic
303	C₈	C₂×#6105	1	1	yes	characteristic
304	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
305	C₈	C₂×#6127	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	#11718669	1	1	yes	characteristic
310	C₂²	#11718669	1	1	yes	characteristic
311	C₂²	C₂×#88203	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	104	4	no	—
316	C₄	#11718669	1	1	yes	characteristic
317	C₄	#11718669	1	1	yes	characteristic
318	C₄	C₂×#88178	1	1	yes	characteristic
319	C₄	C₂²×#6105	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#11718669	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718669
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#443	C₂
socle	C₂×C₁₃	C₂×#476
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	C₂×#443	D₁₃
hypocenter	C₁₃	#33637
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #13125350	C₂	semidirect
2. #13117012	C₂	semidirect
3. #13117007	C₂	semidirect
4. #13116610	C₂	semidirect
5. C₂×#11718055	C₂	semidirect
22. #11718344	C₂²	semidirect
24. #11718341	C₂²	semidirect
26. #11718086	C₂²	semidirect
27. #11718055	C₂²	semidirect
28. #11718055	C₂²	semidirect
32. C₁₃×#135	Q₈	semidirect
34. C₂×C₁₃×Q₁₆	Q₈	semidirect
36. C₂×C₁₃×D₈	Q₈	semidirect
189. #135	Dic₂₆	semidirect
191. C₂×Q₁₆	Dic₂₆	semidirect
193. C₂×D₈	Dic₂₆	semidirect
274. C₁₃	#33637	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13117012	2
#13117007	2
#13116610	2
C₂×#11718055	2
#13116572	2
C₂×C₁₃×#443	2
#33637	13