Subgroups — #16017499

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718565	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718562	C₂	1	1	yes	maximal, characteristic, complemented
3	#13124557	C₂	1	2	yes	maximal, complemented
4	#13116967	C₂	1	4	yes	maximal, complemented
5	#13116962	C₂	1	2	yes	maximal, complemented
6	#13116957	C₂	1	2	yes	maximal, complemented
7	#13116583	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719453	—	2	2	no	—
10	C₂×#88158	C₂²	1	2	yes	—
11	C₂×#88155	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	2	yes	—
14	#11718793	—	2	2	no	—
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
17	#11718565	—	2	1	no	—
18	#11718565	—	2	1	no	—
19	#11718565	—	2	1	no	—
20	#11718565	—	2	1	no	—
21	C₂×#88040	C₂²	1	2	yes	complemented
22	C₂×D₂₀₈	C₂²	1	1	yes	characteristic, complemented
23	#11718562	—	2	1	no	—
24	#11718562	—	2	1	no	—
25	#11718562	C₂²	1	2	yes	—
26	#11718562	C₂²	1	2	yes	—
27	C₂×#88038	—	2	1	no	—
28	C₂×C₁₆×D₁₃	—	2	1	no	—
29	#11718559	—	2	4	no	—
30	#11718558	C₂²	1	4	yes	—
31	#11718557	C₂²	1	4	yes	—
32	C₁₃×#368	C₂²	1	2	yes	complemented
33	C₁₃×#368	C₂²	1	2	yes	complemented
34	#11718088	C₂²	1	4	yes	complemented
35	#11718087	C₂²	1	4	yes	complemented
36	C₂×#6127	—	2	2	no	—
37	C₂×#6127	D₄	1	1	yes	characteristic
38	C₂×C₁₃×M₁₆	—	2	2	no	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6099	D₄	1	1	yes	characteristic
41	#88163	—	2	4	no	—
42	#88163	—	4	2	no	—
43	#88162	—	2	4	no	—
44	#88162	—	4	2	no	—
45	C₂×Dic₅₂	C₂³	1	2	yes	—
46	#88158	—	2	2	no	—
47	#88158	—	2	2	no	—
48	#88158	—	2	2	no	—
49	#88158	—	2	2	no	—
50	#88158	—	4	4	no	—
51	C₂×D₁₀₄	C₂³	1	2	yes	—
52	C₂×#6095	—	2	2	no	—
53	#88155	—	2	1	no	—
54	#88155	—	2	1	no	—
55	#88155	D₄	1	2	yes	—
56	#88155	C₂³	1	2	yes	—
57	C₂×#6094	—	2	1	no	—
58	C₂×C₈×D₁₃	—	2	1	no	—
59	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
61	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₁₃×#108	—	2	2	no	—
66	C₁₃×#108	C₂³	1	4	yes	—
67	#88064	—	2	4	no	—
68	#88063	—	2	4	no	—
69	#88052	—	2	1	no	—
70	#88052	C₂³	1	2	yes	—
71	C₂×#6090	—	2	1	no	—
72	Dic₁₀₄	—	2	1	no	—
73	Dic₁₀₄	—	2	1	no	—
74	#88040	—	2	2	no	—
75	#88040	—	2	2	no	—
76	D₂₀₈	—	2	1	no	—
77	D₂₀₈	—	2	1	no	—
78	#88038	—	2	2	no	—
79	#88038	—	2	2	no	—
80	C₁₆×D₁₃	—	2	2	no	—
81	C₁₆×D₁₃	—	2	2	no	—
82	#33617	—	13	1	no	maximal, Sylow
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₃	—	2	1	no	—
85	C₂×C₂²⋊D₁₃	—	4	2	no	—
86	#6127	—	2	1	no	—
87	#6127	—	2	1	no	—
88	#6127	—	4	2	no	—
89	#6127	—	4	2	no	—
90	#6127	—	4	2	no	—
91	#6127	—	4	2	no	—
92	#6127	D₈	1	2	yes	—
93	#6127	C₂×D₄	1	2	yes	—
94	C₂×D₅₂	—	2	2	no	—
95	C₂×D₅₂	D₈	1	1	yes	characteristic
96	C₂×C₄×D₁₃	—	2	1	no	—
97	C₂×C₄×D₁₃	—	4	2	no	—
98	C₂×Dic₂₆	—	2	2	no	—
99	C₂×Dic₂₆	D₈	1	1	yes	characteristic
100	C₁₃×M₁₆	—	2	4	no	—
101	C₁₃×M₁₆	—	4	2	no	—
102	C₂×C₈×C₁₃	—	4	2	no	—
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
107	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	#6099	—	2	1	no	—
110	#6099	C₂×D₄	1	2	yes	—
111	C₂×#1055	—	2	1	no	—
112	Dic₅₂	—	2	2	no	—
113	Dic₅₂	—	2	2	no	—
114	Dic₅₂	—	4	4	no	—
115	D₁₀₄	—	2	2	no	—
116	D₁₀₄	—	2	2	no	—
117	D₁₀₄	—	4	4	no	—
118	#6095	—	4	4	no	—
119	#6095	—	4	4	no	—
120	#6095	—	4	2	no	—
121	#6095	—	4	2	no	—
122	#6094	—	2	2	no	—
123	#6094	—	2	2	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
127	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
128	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
129	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
130	#6090	—	2	2	no	—
131	C₂×#508	—	13	1	no	—
132	C₂×#504	—	13	1	no	—
133	#3067	—	13	2	no	—
134	#2223	—	13	4	no	—
135	#2222	—	13	2	no	—
136	#2221	—	13	2	no	—
137	C₂×#368	D₁₃	1	1	yes	characteristic, complemented
138	#2148	—	13	2	no	—
139	C₂³×C₁₃	#135	1	1	yes	characteristic
140	C₂²×D₁₃	—	2	1	no	—
141	C₂²×D₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₁₃	—	2	2	no	—
149	C₂²⋊D₁₃	—	2	2	no	—
150	C₂²⋊D₁₃	—	8	2	no	—
151	C₂²⋊D₁₃	—	8	2	no	—
152	C₂×Dic₁₃	—	2	1	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	D₅₂	—	2	1	no	—
155	D₅₂	—	4	2	no	—
156	D₅₂	—	4	2	no	—
157	D₅₂	C₂×D₈	1	2	yes	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	8	2	no	—
161	C₄×D₁₃	—	8	2	no	—
162	Dic₂₆	—	2	1	no	—
163	Dic₂₆	—	4	2	no	—
164	Dic₂₆	—	4	2	no	—
165	Dic₂₆	C₂×D₈	1	2	yes	—
166	C₈×C₁₃	—	4	4	no	—
167	C₈×C₁₃	#121	1	1	yes	characteristic
168	C₈×C₁₃	#121	1	1	yes	characteristic
169	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
171	#1055	—	2	2	no	—
172	#575	—	26	2	no	—
173	C₂×#135	—	13	2	no	—
174	C₂×#131	—	13	1	no	—
175	#508	—	26	1	no	—
176	#508	—	26	1	no	—
177	#508	—	26	1	no	—
178	#508	—	26	1	no	—
179	C₂×Q₃₂	—	13	1	no	—
180	C₂×SD₃₂	—	13	2	no	—
181	C₂×D₁₆	—	13	1	no	—
182	#504	—	13	2	no	—
183	#504	—	13	2	no	—
184	#504	—	26	1	no	—
185	#504	—	26	1	no	—
186	C₂×M₃₂	—	13	1	no	—
187	C₂×M₃₂	—	26	1	no	—
188	C₂²×C₁₆	—	26	1	no	—
189	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
190	#473	—	13	4	no	—
191	#472	—	13	4	no	—
192	#471	—	26	4	no	—
193	C₂×#108	C₂×D₁₃	1	2	yes	—
194	#413	—	26	2	no	—
195	#368	C₂×D₁₃	1	2	yes	complemented
196	#368	C₂×D₁₃	1	2	yes	complemented
197	#362	—	13	4	no	—
198	#361	—	13	4	no	—
199	C₂²×C₁₃	—	4	1	no	—
200	C₂²×C₁₃	#466	1	1	yes	characteristic
201	C₂²×C₁₃	#466	1	1	yes	characteristic
202	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
203	C₂×D₁₃	—	2	2	no	—
204	C₂×D₁₃	—	4	2	no	—
205	C₂×D₁₃	—	8	2	no	—
206	C₂×D₁₃	—	16	2	no	—
207	C₄×C₁₃	#466	1	1	yes	characteristic
208	C₄×C₁₃	#466	1	1	yes	characteristic
209	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
210	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
211	Dic₁₃	—	2	2	no	—
212	Dic₁₃	—	8	2	no	—
213	C₂×#41	—	13	1	no	—
214	C₂×#41	—	26	2	no	—
215	#137	—	26	4	no	—
216	#137	—	52	2	no	—
217	AGL(1,ℤ/8)	—	26	4	no	—
218	AGL(1,ℤ/8)	—	52	2	no	—
219	#135	—	26	2	no	—
220	#135	—	26	2	no	—
221	#135	—	26	2	no	—
222	#135	—	26	2	no	—
223	#135	—	52	4	no	—
224	C₂×Q₁₆	—	13	2	no	—
225	C₂×SD₁₆	—	26	2	no	—
226	C₂×D₈	—	13	2	no	—
227	#131	—	13	2	no	—
228	#131	—	13	2	no	—
229	#131	—	26	1	no	—
230	#131	—	26	1	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	13	1	no	—
233	C₂×M₁₆	—	26	1	no	—
234	C₂²×C₈	—	26	1	no	—
235	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	Q₃₂	—	26	1	no	—
237	Q₃₂	—	26	1	no	—
238	SD₃₂	—	26	2	no	—
239	SD₃₂	—	26	2	no	—
240	D₁₆	—	26	1	no	—
241	D₁₆	—	26	1	no	—
242	M₃₂	—	13	2	no	—
243	M₃₂	—	26	1	no	—
244	M₃₂	—	26	2	no	—
245	M₃₂	—	26	2	no	—
246	C₂×C₁₆	—	26	1	no	—
247	C₂×C₁₆	—	26	2	no	—
248	C₂×C₁₆	—	26	2	no	—
249	C₂×C₁₆	D₅₂	1	1	yes	characteristic
250	C₂×C₁₆	D₅₂	1	1	yes	characteristic
251	C₂×C₁₆	D₅₂	1	1	yes	characteristic
252	C₂×C₁₆	D₅₂	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
254	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
255	#108	—	2	2	no	—
256	#108	C₂²×D₁₃	1	4	yes	—
257	#101	—	26	4	no	—
258	#100	—	26	4	no	—
259	C₂×C₁₃	—	2	1	no	—
260	C₂×C₁₃	—	2	1	no	—
261	C₂×C₁₃	—	2	1	no	—
262	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
263	D₁₃	—	4	2	no	—
264	D₁₃	—	16	2	no	—
265	#41	—	13	2	no	—
266	#41	—	13	2	no	—
267	#41	—	26	1	no	—
268	#41	—	26	1	no	—
269	#41	—	52	2	no	—
270	#41	—	52	2	no	—
271	#41	—	52	2	no	—
272	#41	—	52	2	no	—
273	C₂×Q₈	—	13	1	no	—
274	C₂×Q₈	—	26	2	no	—
275	C₂×D₄	—	13	1	no	—
276	C₂×D₄	—	26	1	no	—
277	C₂×D₄	—	26	2	no	—
278	C₂×D₄	—	52	2	no	—
279	C₂²×C₄	—	26	1	no	—
280	C₂²×C₄	—	52	2	no	—
281	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
282	Q₁₆	—	26	2	no	—
283	Q₁₆	—	26	2	no	—
284	Q₁₆	—	52	4	no	—
285	SD₁₆	—	52	4	no	—
286	SD₁₆	—	52	4	no	—
287	SD₁₆	—	52	2	no	—
288	SD₁₆	—	52	2	no	—
289	D₈	—	26	2	no	—
290	D₈	—	26	2	no	—
291	D₈	—	52	4	no	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	4	2	no	—
294	M₁₆	—	13	2	no	—
295	M₁₆	—	26	1	no	—
296	M₁₆	—	26	2	no	—
297	M₁₆	—	26	2	no	—
298	C₂×C₈	—	4	2	no	—
299	C₂×C₈	—	26	1	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	26	2	no	—
302	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
303	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
304	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
305	C₂×C₈	#6131	1	1	yes	characteristic
306	C₂×C₈	#6131	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₁₆	—	26	2	no	—
309	C₁₆	C₂×D₅₂	1	1	yes	characteristic
310	C₁₆	C₂×D₅₂	1	1	yes	characteristic
311	C₁₆	C₂×D₅₂	1	1	yes	characteristic
312	C₁₆	C₂×D₅₂	1	1	yes	characteristic
313	C₁₃	#33617	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	26	1	no	—
315	C₂³	—	52	2	no	—
316	C₂³	#88173	1	1	yes	characteristic
317	Q₈	—	13	2	no	—
318	Q₈	—	26	1	no	—
319	Q₈	—	52	2	no	—
320	Q₈	—	52	2	no	—
321	D₄	—	13	2	no	—
322	D₄	—	26	2	no	—
323	D₄	—	26	2	no	—
324	D₄	—	26	1	no	—
325	D₄	—	52	2	no	—
326	D₄	—	52	2	no	—
327	D₄	—	104	2	no	—
328	D₄	—	104	2	no	—
329	C₂×C₄	—	26	1	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	2	no	—
332	C₂×C₄	—	52	2	no	—
333	C₂×C₄	—	104	2	no	—
334	C₂×C₄	—	104	2	no	—
335	C₂×C₄	#88149	1	1	yes	characteristic
336	C₂×C₄	#88149	1	1	yes	characteristic
337	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
338	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
339	C₂×C₄	#88173	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	26	2	no	—
343	C₈	#88149	1	1	yes	characteristic
344	C₈	#88149	1	1	yes	characteristic
345	C₈	C₂²×D₅₂	1	1	yes	characteristic
346	C₈	C₂×#6131	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	26	2	no	—
349	C₂²	—	52	2	no	—
350	C₂²	—	104	2	no	—
351	C₂²	—	208	2	no	—
352	C₂²	#11718544	1	1	yes	characteristic
353	C₂²	#11718544	1	1	yes	characteristic
354	C₂²	C₂×#88173	1	1	yes	characteristic
355	C₄	—	26	2	no	—
356	C₄	—	104	2	no	—
357	C₄	#11718544	1	1	yes	characteristic
358	C₄	#11718544	1	1	yes	characteristic
359	C₄	C₂×#88149	1	1	yes	characteristic
360	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	52	2	no	—
365	C₂	—	208	2	no	—
366	C₂	C₂×#11718544	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718544
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#368	C₂
socle	C₂×C₁₃	C₂×#466
hypercenter	C₂×#368	D₁₃
hypocenter	C₁₃	#33617
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#11718565	C₂	semidirect
2. C₂×#11718562	C₂	semidirect
3. #13124557	C₂	semidirect
4. #13116967	C₂	semidirect
5. #13116962	C₂	semidirect
6. #13116957	C₂	semidirect
7. #13116583	C₂	semidirect
8. C₂×C₁₃×#368	C₂	semidirect
16. C₂×Dic₁₀₄	C₂²	semidirect
21. C₂×#88040	C₂²	semidirect
22. C₂×D₂₀₈	C₂²	semidirect
32. C₁₃×#368	C₂²	semidirect
33. C₁₃×#368	C₂²	semidirect
34. #11718088	C₂²	semidirect
35. #11718087	C₂²	semidirect
137. C₂×#368	D₁₃	semidirect
195. #368	C₂×D₁₃	semidirect
196. #368	C₂×D₁₃	semidirect
313. C₁₃	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718565	2
C₂×#11718562	2
#13124557	2
#13116967	2
#13116962	2
#13116957	2
#13116583	2
C₂×C₁₃×#368	2
#33617	13