Subgroups — #16018341

Order 3328.

826 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	C₂×#11718855	C₂	1	4	yes	maximal, complemented
2	#13125264	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117108	C₂	1	2	yes	maximal, complemented
4	#13117105	C₂	1	2	yes	maximal, complemented
5	#13117103	C₂	1	2	yes	maximal, complemented
6	#13117100	C₂	1	2	yes	maximal, complemented
7	C₂×#11718053	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	—	2	1	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₁₃×#532	—	2	1	no	—
12	C₂×#88158	—	2	1	no	—
13	C₂×#88158	C₂²	1	2	yes	—
14	#11719182	—	2	1	no	—
15	C₁₃×#495	—	2	1	no	—
16	C₁₃×#495	C₂²	1	2	yes	—
17	C₁₃×#494	C₂²	1	1	yes	characteristic
18	C₁₃×#493	C₂²	1	1	yes	characteristic
19	C₁₃×#492	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718855	—	2	16	no	—
22	C₂×#88069	C₂²	1	4	yes	—
23	C₂×#88068	C₂²	1	4	yes	—
24	C₂×#88067	C₂²	1	4	yes	—
25	C₂×#88066	C₂²	1	4	yes	—
26	C₂²×#6090	C₂²	1	2	yes	—
27	#11718360	C₂²	1	1	yes	characteristic
28	#11718357	—	2	1	no	—
29	#11718357	C₂²	1	2	yes	—
30	#11718356	C₂²	1	1	yes	characteristic
31	#11718354	—	2	1	no	—
32	#11718053	C₂²	1	2	yes	—
33	#11718053	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	—	2	2	no	—
35	C₂×#6127	—	2	2	no	—
36	C₁₃×#135	—	2	8	no	—
37	C₁₃×#135	—	4	4	no	—
38	C₂×C₁₃×Q₁₆	—	2	1	no	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×SD₁₆	—	2	2	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×#127	—	2	1	no	—
48	C₁₃×#124	—	2	2	no	—
49	C₁₃×#121	—	4	2	no	—
50	C₄×D₄×C₁₃	—	4	2	no	—
51	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
52	#88184	—	4	2	no	—
53	C₂×Dic₅₂	—	2	1	no	—
54	C₂×Dic₅₂	C₂³	1	2	yes	—
55	#88158	—	2	8	no	—
56	#88158	—	4	4	no	—
57	C₂×D₁₀₄	—	2	1	no	—
58	C₂×D₁₀₄	C₂³	1	2	yes	—
59	C₂×#6095	—	2	2	no	—
60	C₂×#6095	—	2	2	no	—
61	#88129	—	2	2	no	—
62	#88128	—	2	1	no	—
63	C₄×D₅₂	—	4	2	no	—
64	#88123	—	2	1	no	—
65	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88069	—	2	8	no	—
70	#88068	—	2	8	no	—
71	#88067	—	2	8	no	—
72	#88066	—	2	8	no	—
73	C₂×#6090	D₄	1	8	yes	complemented
74	C₂×#6090	C₂³	1	4	yes	—
75	#33716	—	13	1	no	maximal, Sylow
76	C₁₃×#41	—	4	4	no	—
77	C₁₃×#41	—	4	4	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	D₈	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₁₃	—	4	2	no	—
86	#6127	—	4	4	no	—
87	#6127	—	4	4	no	—
88	C₂×D₅₂	—	2	2	no	—
89	C₂×D₅₂	—	4	2	no	—
90	C₂×C₄×D₁₃	—	4	2	no	—
91	C₂×Dic₂₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	4	no	—
93	C₁₃×Q₁₆	—	4	2	no	—
94	C₁₃×SD₁₆	—	4	4	no	—
95	C₁₃×SD₁₆	—	4	4	no	—
96	C₁₃×D₈	—	2	4	no	—
97	C₁₃×D₈	—	4	2	no	—
98	C₂×C₈×C₁₃	D₈	1	4	yes	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	4	no	—
106	C₄²×C₁₃	D₈	1	2	yes	—
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	#6103	—	4	4	no	—
110	C₄⋊Dic₁₃	—	4	2	no	—
111	Dic₅₂	—	2	4	no	—
112	Dic₅₂	—	4	2	no	—
113	D₁₀₄	—	2	4	no	—
114	D₁₀₄	—	4	2	no	—
115	#6095	—	4	4	no	—
116	#6095	—	4	4	no	—
117	#6090	C₂×D₄	1	8	yes	—
118	C₂×#508	—	13	4	no	—
119	#3147	—	13	1	no	—
120	#3147	D₁₃	1	1	yes	characteristic, complemented
121	#2251	—	13	2	no	—
122	#2251	—	13	2	no	—
123	#2250	—	13	2	no	—
124	#2250	—	13	2	no	—
125	C₂×#346	—	13	1	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂²×D₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	Q₈×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	8	4	no	—
134	D₄×C₁₃	—	8	4	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	8	4	no	—
138	C₂×C₄×C₁₃	#127	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₁₃	—	8	4	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	D₅₂	—	4	2	no	—
147	D₅₂	—	4	2	no	—
148	D₅₂	—	8	4	no	—
149	C₄×D₁₃	—	8	4	no	—
150	Dic₂₆	—	4	2	no	—
151	Dic₂₆	—	4	2	no	—
152	C₈×C₁₃	#127	1	2	yes	—
153	C₈×C₁₃	C₂×D₈	1	2	yes	—
154	C₈×C₁₃	C₂×D₈	1	2	yes	—
155	C₈×C₁₃	C₂²×D₄	1	2	yes	—
156	C₂×#135	—	2	1	no	—
157	C₂×#135	—	13	2	no	—
158	C₂×#135	—	26	1	no	—
159	C₂×#135	C₂×D₁₃	1	2	yes	—
160	#532	—	2	1	no	—
161	#532	—	26	1	no	—
162	#508	—	26	16	no	—
163	C₂×Q₃₂	—	13	4	no	—
164	C₂×SD₃₂	—	13	4	no	—
165	C₂×SD₃₂	—	13	4	no	—
166	C₂×D₁₆	—	13	4	no	—
167	C₂²×C₁₆	—	13	2	no	—
168	#495	—	2	1	no	—
169	#495	—	13	2	no	—
170	#495	—	26	1	no	—
171	#495	C₂×D₁₃	1	2	yes	—
172	#494	—	13	1	no	—
173	#494	C₂×D₁₃	1	1	yes	characteristic
174	#493	—	13	1	no	—
175	#493	C₂×D₁₃	1	1	yes	characteristic
176	#492	—	2	1	no	—
177	#492	—	26	1	no	—
178	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
179	#346	—	13	2	no	—
180	#346	—	13	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	4	no	—
184	C₂²×C₁₃	#493	1	2	yes	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₂×D₁₃	—	8	2	no	—
187	C₂×D₁₃	—	8	4	no	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	#493	1	2	yes	—
192	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
193	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
194	Dic₁₃	—	8	2	no	—
195	C₂×#41	—	2	2	no	—
196	C₂×#41	—	26	2	no	—
197	#135	—	2	8	no	—
198	#135	—	4	4	no	—
199	#135	—	26	8	no	—
200	#135	—	52	4	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	13	2	no	—
203	C₂×Q₁₆	—	26	1	no	—
204	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	26	2	no	—
208	C₂×SD₁₆	—	26	2	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	13	2	no	—
211	C₂×D₈	—	26	1	no	—
212	C₂×D₈	C₂²×D₁₃	1	2	yes	—
213	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	26	1	no	—
217	#127	—	2	1	no	—
218	#127	—	26	1	no	—
219	#124	—	2	2	no	—
220	#124	—	26	2	no	—
221	#121	—	4	2	no	—
222	#121	—	52	2	no	—
223	C₄×D₄	—	4	2	no	—
224	C₄×D₄	—	52	2	no	—
225	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
226	Q₃₂	—	26	8	no	—
227	SD₃₂	—	26	8	no	—
228	SD₃₂	—	26	8	no	—
229	D₁₆	—	26	8	no	—
230	C₂×C₁₆	—	13	8	no	—
231	C₂×C₁₆	—	13	4	no	—
232	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	—	16	2	no	—
238	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
239	C₂×C₁₃	#3414	1	2	yes	—
240	D₁₃	—	16	2	no	—
241	#41	—	4	4	no	—
242	#41	—	4	4	no	—
243	#41	—	52	4	no	—
244	#41	—	52	4	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	26	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	26	2	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂×D₄	—	52	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	52	2	no	—
255	C₂²×C₄	#6104	1	2	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	4	2	no	—
259	Q₁₆	—	26	4	no	—
260	Q₁₆	—	52	2	no	—
261	SD₁₆	—	4	4	no	—
262	SD₁₆	—	4	4	no	—
263	SD₁₆	—	52	4	no	—
264	SD₁₆	—	52	4	no	—
265	D₈	—	2	4	no	—
266	D₈	—	4	2	no	—
267	D₈	—	26	4	no	—
268	D₈	—	52	2	no	—
269	C₂×C₈	#6104	1	4	yes	—
270	C₂×C₈	D₄×D₁₃	1	4	yes	—
271	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	52	4	no	—
279	C₄²	#6104	1	2	yes	—
280	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
282	C₁₆	—	13	8	no	—
283	C₁₃	#33716	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	4	2	no	—
285	C₂³	—	52	2	no	—
286	C₂³	C₂×#6104	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	52	2	no	—
290	Q₈	—	52	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	52	2	no	—
296	D₄	—	52	2	no	—
297	D₄	—	104	4	no	—
298	D₄	—	104	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	104	4	no	—
304	C₂×C₄	C₂×#6104	1	2	yes	—
305	C₂×C₄	C₂×#6104	1	2	yes	—
306	C₂×C₄	C₂×#6104	1	2	yes	—
307	C₂×C₄	C₂×#6104	1	1	yes	characteristic
308	C₂×C₄	#88194	1	2	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₈	C₂×#6104	1	2	yes	—
311	C₈	C₂×#6104	1	2	yes	—
312	C₈	#88194	1	2	yes	—
313	C₈	C₂×D₄×D₁₃	1	2	yes	—
314	C₂²	—	2	2	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	104	4	no	—
318	C₂²	—	104	2	no	—
319	C₂²	#11718734	1	2	yes	—
320	C₂²	C₂²×#6104	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	104	2	no	—
325	C₄	#11718734	1	2	yes	—
326	C₄	C₂²×#6104	1	1	yes	characteristic
327	C₄	C₂×#88194	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	16	2	no	—
330	C₂	—	208	2	no	—
331	C₂	C₂×#11718734	1	1	yes	characteristic
332	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#33716
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718855	C₂	semidirect
2. #13125264	C₂	semidirect
3. #13117108	C₂	semidirect
4. #13117105	C₂	semidirect
5. #13117103	C₂	semidirect
6. #13117100	C₂	semidirect
7. C₂×#11718053	C₂	semidirect
8. C₁₃×#3147	C₂	semidirect
73. C₂×#6090	D₄	semidirect
120. #3147	D₁₃	semidirect
283. C₁₃	#33716	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718855	2
#13125264	2
#13117108	2
#13117105	2
#13117103	2
#13117100	2
C₂×#11718053	2
C₁₃×#3147	2
#33716	13