Subgroups — #16015520

Order 3328.

822 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718537	C₂	1	2	yes	maximal
2	C₂×#11718517	C₂	1	2	yes	maximal
3	#13123212	C₂	1	1	yes	maximal, characteristic
4	#13123198	C₂	1	1	yes	maximal, characteristic
5	#13115003	C₂	1	2	yes	maximal
6	#13115001	C₂	1	2	yes	maximal
7	#13115000	C₂	1	1	yes	maximal, characteristic
8	#13114998	C₂	1	1	yes	maximal, characteristic
9	#13114998	C₂	1	1	yes	maximal, characteristic
10	C₄×#88055	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
12	C₂×#88144	C₂²	1	2	yes	—
13	C₂×#88142	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#382	C₂²	1	1	yes	characteristic
18	C₁₃×#382	C₂²	1	1	yes	characteristic
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88057	C₂²	1	2	yes	—
21	C₂×#88055	C₂²	1	1	yes	characteristic
22	C₂×#88055	C₂²	1	1	yes	characteristic
23	C₂×#88055	C₂²	1	1	yes	characteristic
24	C₂×#88055	C₂²	1	1	yes	characteristic
25	#11718638	C₂²	1	2	yes	—
26	#11718637	C₂²	1	2	yes	—
27	C₂×#88048	C₂²	1	2	yes	—
28	C₂×#88048	C₂²	1	2	yes	—
29	C₂×#88047	C₂²	1	2	yes	—
30	C₂×#88047	C₂²	1	2	yes	—
31	#11718597	C₂²	1	1	yes	characteristic
32	#11718597	C₂²	1	1	yes	characteristic
33	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
35	#11718537	—	2	8	no	—
36	#11718517	—	2	8	no	—
37	#11718308	—	2	4	no	—
38	#11718307	—	2	4	no	—
39	C₂²×Dic₂₆	—	2	2	no	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₃	—	2	2	no	—
46	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
48	C₂×#6101	—	2	2	no	—
49	C₂×#6101	—	2	2	no	—
50	C₂×#6101	—	2	4	no	—
51	C₂×C₄×Dic₁₃	D₄	1	4	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	#88144	—	2	4	no	—
57	#88144	—	2	4	no	—
58	#88142	—	2	4	no	—
59	#88142	—	2	4	no	—
60	C₁₃×#107	—	2	4	no	—
61	C₁₃×#106	—	2	4	no	—
62	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	#88071	—	2	2	no	—
66	#88071	—	2	2	no	—
67	#88071	—	2	4	no	—
68	#88058	—	2	4	no	—
69	#88057	—	2	4	no	—
70	#88055	D₄	1	4	yes	—
71	#88055	D₄	1	4	yes	—
72	#88055	Q₈	1	4	yes	complemented
73	#88055	Q₈	1	4	yes	complemented
74	#88048	—	2	4	no	—
75	#88048	—	2	4	no	—
76	#88047	—	2	4	no	—
77	#88047	—	2	4	no	—
78	C₄×#1055	Q₈	1	4	yes	complemented
79	#33317	—	13	1	no	maximal, Sylow
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₁₃	—	2	2	no	—
85	C₂²×Dic₁₃	—	2	2	no	—
86	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
87	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
88	C₂×Dic₂₆	—	4	8	no	—
89	C₂×Dic₂₆	—	4	4	no	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
101	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	4	4	no	—
107	#6101	—	4	8	no	—
108	#6101	—	4	4	no	—
109	#6101	—	4	4	no	—
110	C₄×Dic₁₃	—	4	4	no	—
111	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
112	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
113	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
114	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
115	C₂×#1055	C₂×D₄	1	2	yes	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	C₂×#1055	C₂×Q₈	1	2	yes	—
118	C₂×#1055	C₂×Q₈	1	2	yes	—
119	C₂×#1055	C₂×Q₈	1	2	yes	—
120	C₂×#1055	C₂×Q₈	1	2	yes	—
121	C₂×#501	—	13	2	no	—
122	C₂×#501	—	13	2	no	—
123	#2862	—	13	1	no	—
124	#2859	—	13	1	no	—
125	#1847	—	13	2	no	—
126	#1847	—	13	2	no	—
127	#1846	—	13	1	no	—
128	#1846	—	13	1	no	—
129	#1846	—	13	1	no	—
130	#1846	D₁₃	1	1	yes	characteristic
131	C₄×#97	—	13	1	no	—
132	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	4	4	no	—
138	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×Dic₁₃	—	2	4	no	—
146	C₂×Dic₁₃	—	2	4	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	4	no	—
150	C₂×Dic₁₃	—	4	4	no	—
151	C₂×Dic₁₃	#127	1	4	yes	—
152	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
153	Dic₂₆	—	8	8	no	—
154	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
155	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
156	#1055	C₄⋊Q₈	1	4	yes	—
157	#1055	C₄⋊Q₈	1	4	yes	—
158	C₂×C₄⋊Q₈	—	13	2	no	—
159	C₂×#125	—	13	2	no	—
160	#501	—	26	8	no	—
161	#501	—	26	8	no	—
162	C₂×#107	—	13	2	no	—
163	C₂×#107	—	13	2	no	—
164	C₂×#107	—	13	2	no	—
165	C₂×#107	C₂×D₁₃	1	2	yes	—
166	C₂×#106	—	13	2	no	—
167	C₂×#106	—	13	2	no	—
168	C₂×#106	—	13	2	no	—
169	C₂×#106	C₂×D₁₃	1	2	yes	—
170	C₂×#97	—	13	1	no	—
171	C₂×#97	—	13	1	no	—
172	C₂×#97	—	13	1	no	—
173	C₂×#97	—	13	1	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#391	—	26	4	no	—
177	#389	—	26	4	no	—
178	#384	—	13	2	no	—
179	#382	—	13	2	no	—
180	#382	—	13	1	no	—
181	#382	—	13	1	no	—
182	#382	C₂×D₁₃	1	1	yes	characteristic
183	#382	C₂×D₁₃	1	1	yes	characteristic
184	C₄³	—	13	1	no	—
185	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
188	C₂²×C₁₃	#575	1	4	yes	—
189	C₄×C₁₃	—	2	4	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
193	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
194	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
196	Dic₁₃	—	2	4	no	—
197	Dic₁₃	—	2	4	no	—
198	Dic₁₃	—	8	2	no	—
199	Dic₁₃	—	8	2	no	—
200	C₂²×Q₈	—	26	2	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
205	C₄⋊Q₈	—	26	4	no	—
206	C₄⋊Q₈	—	26	4	no	—
207	#125	—	26	4	no	—
208	#125	—	26	4	no	—
209	C₂×C₄⋊C₄	—	13	2	no	—
210	C₂×C₄⋊C₄	—	13	2	no	—
211	C₂×C₄⋊C₄	—	26	2	no	—
212	C₂×C₄⋊C₄	—	26	2	no	—
213	C₂×C₄⋊C₄	—	26	2	no	—
214	C₂×C₄⋊C₄	—	26	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	4	no	—
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	#107	—	2	4	no	—
222	#107	—	26	4	no	—
223	#107	—	26	4	no	—
224	#107	—	26	4	no	—
225	#106	—	2	4	no	—
226	#106	—	26	4	no	—
227	#106	—	26	4	no	—
228	#106	—	26	4	no	—
229	#97	—	13	4	no	—
230	#97	—	13	4	no	—
231	#97	—	13	4	no	—
232	#97	—	13	4	no	—
233	C₄×C₈	—	13	4	no	—
234	C₄×C₈	Dic₂₆	1	4	yes	complemented
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	26	4	no	—
238	#95	—	26	2	no	—
239	#95	—	26	2	no	—
240	C₂×C₁₃	#2868	1	1	yes	characteristic
241	C₂×C₁₃	#3153	1	2	yes	—
242	C₂×C₁₃	#3162	1	2	yes	—
243	C₂×C₁₃	#3166	1	2	yes	—
244	C₂×Q₈	—	52	4	no	—
245	C₂×Q₈	—	52	8	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	13	2	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	D₄×D₁₃	1	2	yes	—
253	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	2	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
261	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
262	C₂×C₈	D₄×D₁₃	1	2	yes	—
263	C₂×C₈	D₄×D₁₃	1	2	yes	—
264	C₂×C₈	Q₈×D₁₃	1	2	yes	—
265	C₂×C₈	Q₈×D₁₃	1	2	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	52	4	no	—
275	C₄⋊C₄	—	52	4	no	—
276	C₄⋊C₄	—	52	4	no	—
277	C₄⋊C₄	—	52	8	no	—
278	C₄²	—	13	2	no	—
279	C₄²	—	13	2	no	—
280	C₄²	—	13	2	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	52	4	no	—
283	C₄²	C₂×Dic₂₆	1	2	yes	—
284	C₄²	Q₈×D₁₃	1	2	yes	—
285	C₁₃	#33317	1	1	yes	characteristic, complemented, Sylow
286	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
287	Q₈	—	104	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	13	4	no	—
294	C₂×C₄	—	13	4	no	—
295	C₂×C₄	—	26	4	no	—
296	C₂×C₄	—	26	4	no	—
297	C₂×C₄	—	52	4	no	—
298	C₂×C₄	—	52	4	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	—	52	2	no	—
301	C₂×C₄	#88142	1	4	yes	—
302	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
308	C₈	—	13	4	no	—
309	C₈	—	13	4	no	—
310	C₈	#88142	1	4	yes	—
311	C₈	#88142	1	4	yes	—
312	C₂²	C₂×#88142	1	1	yes	characteristic
313	C₂²	D₁₃×#127	1	1	yes	characteristic
314	C₂²	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
315	C₂²	#11719473	1	4	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	26	4	no	—
320	C₄	—	26	4	no	—
321	C₄	—	104	2	no	—
322	C₄	—	104	2	no	—
323	C₄	C₂×#88142	1	1	yes	characteristic
324	C₄	C₂×#88142	1	1	yes	characteristic
325	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
326	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
327	C₂	#13123233	1	1	yes	characteristic
328	C₂	#13125297	1	2	yes	—
329	C₂	#13125359	1	2	yes	—
330	C₂	#13125384	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1846	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#1846	D₁₃
hypocenter	C₁₃	#33317
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
62. C₄×C₈×C₁₃	Q₈	semidirect
72. #88055	Q₈	semidirect
73. #88055	Q₈	semidirect
78. C₄×#1055	Q₈	semidirect
234. C₄×C₈	Dic₂₆	semidirect
285. C₁₃	#33317	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718537	2
C₂×#11718517	2
#13123212	2
#13123198	2
#13115003	2
#13115001	2
#13115000	2
#13114998	2
C₄×#88055	2
C₁₃×#1846	2
#33317	13