Subgroups — #15779528

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718097	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718101	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718101	C₂	1	1	yes	maximal, characteristic
5	#13115569	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112928	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₅₂	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88061	—	2	1	no	—
14	C₂×#88058	C₂²	1	1	yes	characteristic
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	#11718589	—	2	1	no	—
18	C₄×D₁₀₄	—	2	2	no	—
19	C₄×D₁₀₄	—	2	2	no	—
20	#11718143	—	2	1	no	—
21	#11718101	—	2	2	no	—
22	#11718101	C₄	1	4	yes	complemented
23	#11718097	—	2	2	no	—
24	C₂²×D₅₂	—	2	1	no	—
25	C₂²×C₄×D₁₃	—	4	1	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×#6103	—	4	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×#6101	—	4	1	no	—
34	C₂×D₁₀₄	—	2	1	no	—
35	C₂×D₁₀₄	—	2	1	no	—
36	C₂×D₁₀₄	—	2	1	no	—
37	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
38	C₂×D₁₀₄	D₄	1	2	yes	—
39	C₂×D₁₀₄	Q₈	1	2	yes	—
40	C₄×D₅₂	—	2	4	no	—
41	C₄×D₅₂	—	4	2	no	—
42	C₂×C₁₃×C₁₆	—	2	1	no	—
43	C₂×C₁₃×C₁₆	—	2	1	no	—
44	C₂×C₁₃×C₁₆	—	2	1	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	1	no	—
47	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
48	C₄×C₈×C₁₃	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	#88071	—	4	1	no	—
51	#88061	—	2	4	no	—
52	#88058	—	2	1	no	—
53	#88058	—	2	1	no	—
54	#88058	—	2	1	no	—
55	#88058	—	2	1	no	—
56	#88058	C₂×C₄	1	2	yes	complemented
57	#88058	D₄	1	2	yes	—
58	#12937	—	13	1	no	maximal, Sylow
59	C₂³×D₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	4	1	no	—
64	C₂²×Dic₁₃	—	4	1	no	—
65	C₂²×Dic₁₃	—	4	1	no	—
66	C₂×D₅₂	—	2	2	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	4	1	no	—
70	C₂×D₅₂	—	4	1	no	—
71	C₂×D₅₂	—	4	1	no	—
72	C₂×C₄×D₁₃	—	4	4	no	—
73	C₂×C₄×D₁₃	—	8	2	no	—
74	C₂×C₄×D₁₃	—	8	2	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	#6103	—	4	4	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	4	1	no	—
90	C₄⋊Dic₁₃	—	4	1	no	—
91	C₄⋊Dic₁₃	—	4	1	no	—
92	C₄⋊Dic₁₃	—	4	1	no	—
93	#6101	—	8	2	no	—
94	D₁₀₄	—	2	2	no	—
95	D₁₀₄	—	2	2	no	—
96	D₁₀₄	—	2	2	no	—
97	D₁₀₄	C₄⋊C₄	1	4	yes	complemented
98	C₁₃×C₁₆	—	2	2	no	—
99	C₁₃×C₁₆	—	4	1	no	—
100	C₂×C₄×D₈	—	13	1	no	—
101	C₂×#367	—	13	1	no	—
102	C₂×#357	—	13	1	no	—
103	C₂×#357	—	13	1	no	—
104	#1942	—	13	1	no	—
105	#1380	—	13	1	no	—
106	#1374	D₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂²×D₁₃	—	4	2	no	—
109	C₂²×D₁₃	—	4	2	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	—	8	1	no	—
112	C₂²×D₁₃	—	8	1	no	—
113	C₂²×D₁₃	—	8	1	no	—
114	C₂²×D₁₃	—	8	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
125	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
127	C₂×Dic₁₃	—	4	2	no	—
128	C₂×Dic₁₃	—	4	2	no	—
129	C₂×Dic₁₃	—	8	1	no	—
130	C₂×Dic₁₃	—	8	1	no	—
131	C₂×Dic₁₃	—	8	1	no	—
132	C₂×Dic₁₃	—	8	1	no	—
133	C₂×Dic₁₃	—	8	1	no	—
134	C₂×Dic₁₃	—	8	1	no	—
135	C₂×Dic₁₃	—	8	1	no	—
136	D₅₂	—	2	4	no	—
137	D₅₂	—	4	2	no	—
138	D₅₂	—	4	2	no	—
139	D₅₂	—	4	2	no	—
140	C₄×D₁₃	—	8	8	no	—
141	C₈×C₁₃	—	4	1	no	—
142	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₃	#123	1	1	yes	characteristic
146	C₂²×D₈	—	13	1	no	—
147	C₂×C₄×D₄	—	26	1	no	—
148	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
149	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
150	C₄×D₈	—	26	2	no	—
151	C₄×D₈	—	26	2	no	—
152	C₂×#107	—	13	1	no	—
153	C₂×#107	—	13	1	no	—
154	C₂×#107	—	13	1	no	—
155	C₂×#102	—	26	1	no	—
156	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	#389	—	26	1	no	—
158	#367	—	26	2	no	—
159	#357	—	13	4	no	—
160	#357	—	26	2	no	—
161	#340	—	26	1	no	—
162	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
163	C₂²×C₁₃	#447	1	1	yes	characteristic
164	C₂²×C₁₃	#474	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
166	C₂²×C₁₃	#508	1	1	yes	characteristic
167	C₂²×C₁₃	#509	1	1	yes	characteristic
168	C₂²×C₁₃	#509	1	1	yes	characteristic
169	C₂×D₁₃	—	4	4	no	—
170	C₂×D₁₃	—	8	2	no	—
171	C₂×D₁₃	—	8	2	no	—
172	C₂×D₁₃	—	8	2	no	—
173	C₂×D₁₃	—	8	2	no	—
174	C₂×D₁₃	—	8	2	no	—
175	C₂×D₁₃	—	8	2	no	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	4	1	no	—
178	C₄×C₁₃	#385	1	1	yes	characteristic
179	C₄×C₁₃	#440	1	1	yes	characteristic
180	C₄×C₁₃	#452	1	1	yes	characteristic
181	C₄×C₁₃	#478	1	1	yes	characteristic
182	Dic₁₃	—	8	2	no	—
183	Dic₁₃	—	8	2	no	—
184	Dic₁₃	—	16	1	no	—
185	C₂²×D₄	—	26	1	no	—
186	C₂³×C₄	—	52	1	no	—
187	C₂×D₈	—	13	2	no	—
188	C₂×D₈	—	13	2	no	—
189	C₂×D₈	—	13	2	no	—
190	C₂×D₈	—	26	1	no	—
191	C₂×D₈	—	26	1	no	—
192	C₂×D₈	—	26	1	no	—
193	C₂²×C₈	D₅₂	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×D₄	—	26	4	no	—
196	C₄×D₄	—	52	2	no	—
197	C₂×C₄⋊C₄	—	26	1	no	—
198	C₂×C₄⋊C₄	—	26	1	no	—
199	C₂×C₄⋊C₄	—	26	1	no	—
200	C₂×C₄⋊C₄	—	52	1	no	—
201	C₂×C₂²⋊C₄	—	52	1	no	—
202	C₂×C₄²	D₅₂	1	1	yes	characteristic
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	C₄×D₁₃	1	2	yes	complemented
209	#107	—	13	2	no	—
210	#107	—	13	2	no	—
211	#107	—	26	1	no	—
212	#107	—	26	1	no	—
213	#107	—	26	1	no	—
214	#107	—	26	1	no	—
215	#102	—	26	4	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	#95	—	52	1	no	—
219	C₂×C₁₃	#1862	1	1	yes	characteristic
220	C₂×C₁₃	C₄×SD₃₂	1	1	yes	characteristic
221	C₂×C₁₃	#2177	1	1	yes	characteristic
222	C₂×C₁₃	#2186	1	1	yes	characteristic
223	C₂×C₁₃	#2187	1	1	yes	characteristic
224	C₂×C₁₃	#2226	1	1	yes	characteristic
225	C₂×C₁₃	#2228	1	1	yes	characteristic
226	D₁₃	—	8	4	no	—
227	C₂⁴	—	52	1	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂×D₄	—	26	2	no	—
230	C₂×D₄	—	26	2	no	—
231	C₂×D₄	—	52	1	no	—
232	C₂×D₄	—	52	1	no	—
233	C₂×D₄	—	52	1	no	—
234	C₂²×C₄	—	52	4	no	—
235	C₂²×C₄	—	52	1	no	—
236	C₂²×C₄	—	52	1	no	—
237	C₂²×C₄	—	52	1	no	—
238	C₂²×C₄	—	104	2	no	—
239	C₂²×C₄	—	104	2	no	—
240	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
241	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
243	D₈	—	13	4	no	—
244	D₈	—	26	2	no	—
245	D₈	—	26	2	no	—
246	D₈	—	26	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂×C₈	#6127	1	1	yes	characteristic
252	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
253	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
254	C₂×C₈	#6129	1	1	yes	characteristic
255	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
256	C₄⋊C₄	—	26	2	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	52	1	no	—
259	C₄⋊C₄	—	52	1	no	—
260	C₄⋊C₄	—	52	1	no	—
261	C₄⋊C₄	—	52	1	no	—
262	C₄⋊C₄	—	104	2	no	—
263	C₂²⋊C₄	—	52	4	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	4	1	no	—
268	C₁₃	#12937	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	52	2	no	—
270	C₂³	—	52	2	no	—
271	C₂³	—	52	2	no	—
272	C₂³	—	104	1	no	—
273	C₂³	—	104	1	no	—
274	C₂³	—	104	1	no	—
275	C₂³	—	104	1	no	—
276	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
277	D₄	—	26	4	no	—
278	D₄	—	52	2	no	—
279	D₄	—	52	2	no	—
280	D₄	—	52	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	52	2	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	—	104	8	no	—
289	C₂×C₄	—	104	1	no	—
290	C₂×C₄	—	104	1	no	—
291	C₂×C₄	—	104	1	no	—
292	C₂×C₄	—	104	1	no	—
293	C₂×C₄	—	104	1	no	—
294	C₂×C₄	—	104	1	no	—
295	C₂×C₄	—	104	1	no	—
296	C₂×C₄	#88040	1	2	yes	—
297	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
298	C₂×C₄	#88136	1	1	yes	characteristic
299	C₂×C₄	#88151	1	1	yes	characteristic
300	C₂×C₄	#88158	1	1	yes	characteristic
301	C₂×C₄	#88162	1	1	yes	characteristic
302	C₂×C₄	#88163	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	#88135	1	1	yes	characteristic
305	C₈	#88137	1	1	yes	characteristic
306	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
307	C₈	#88150	1	1	yes	characteristic
308	C₂²	—	52	4	no	—
309	C₂²	—	104	2	no	—
310	C₂²	—	104	2	no	—
311	C₂²	—	104	2	no	—
312	C₂²	—	104	2	no	—
313	C₂²	—	104	2	no	—
314	C₂²	—	104	2	no	—
315	C₂²	C₄×D₁₀₄	1	1	yes	characteristic
316	C₂²	#11718393	1	1	yes	characteristic
317	C₂²	#11718507	1	1	yes	characteristic
318	C₂²	C₂×#88040	1	1	yes	characteristic
319	C₂²	#11718565	1	1	yes	characteristic
320	C₂²	#11718569	1	1	yes	characteristic
321	C₂²	#11718569	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	104	2	no	—
325	C₄	—	104	2	no	—
326	C₄	—	208	1	no	—
327	C₄	#11718329	1	1	yes	characteristic
328	C₄	#11718371	1	1	yes	characteristic
329	C₄	#11718394	1	1	yes	characteristic
330	C₄	#11718508	1	1	yes	characteristic
331	C₂	—	104	4	no	—
332	C₂	#13115086	1	1	yes	characteristic
333	C₂	C₄×#88040	1	1	yes	characteristic
334	C₂	#13116713	1	1	yes	characteristic
335	C₂	#13116761	1	1	yes	characteristic
336	C₂	#13116767	1	1	yes	characteristic
337	C₂	#13116984	1	1	yes	characteristic
338	C₂	#13116994	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1374	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1374	D₁₃
hypocenter	C₁₃	#12937
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#11718097	C₂	semidirect
5. #13115569	C₂	semidirect
6. #13112928	C₂	semidirect
7. C₁₃×#1374	C₂	semidirect
22. #11718101	C₄	semidirect
47. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
56. #88058	C₂×C₄	semidirect
97. D₁₀₄	C₄⋊C₄	semidirect
106. #1374	D₁₃	semidirect
208. C₂×C₁₆	C₄×D₁₃	semidirect
268. C₁₃	#12937	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
C₂×#11718097	2
C₂×#11718101	2
#13115569	2
#13112928	2
C₁₃×#1374	2
#12937	13