Subgroups — #15834426

Order 3328.

644 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	#13123856	C₂	1	2	yes	maximal, complemented
2	#13113750	C₂	1	2	yes	maximal, complemented
3	#13113715	C₂	1	4	yes	maximal, complemented
4	#13113655	C₂	1	2	yes	maximal, complemented
5	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113270	C₂	1	2	yes	maximal, complemented
7	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
10	C₄×C₁₃×#41	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×SD₁₆	C₄	1	2	yes	—
14	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
16	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₄	1	1	yes	characteristic
18	C₁₃×#416	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	Q₈×#1055	C₂²	1	4	yes	—
21	#11718673	—	2	4	no	—
22	#11718673	C₂²	1	4	yes	—
23	#11718587	—	2	2	no	—
24	#11718587	C₂²	1	2	yes	—
25	#11718586	C₂²	1	1	yes	characteristic
26	C₄×#6099	—	2	2	no	—
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	D₄⋊#1055	C₂²	1	4	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	#11718047	C₂²	1	2	yes	—
33	#11718046	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
35	C₁₃×#135	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
43	C₄×D₄×C₁₃	—	2	2	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₃	C₂³	1	2	yes	—
47	C₁₃×#117	—	2	2	no	—
48	C₁₃×#117	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×#6099	—	2	4	no	—
52	C₂×#6099	D₄	1	2	yes	—
53	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
54	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₃×#106	C₂×C₄	1	2	yes	—
56	C₁₃×#103	C₂×C₄	1	2	yes	—
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#102	C₂×C₄	1	2	yes	—
59	C₁₃×#102	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²⋊#1055	—	2	2	no	—
65	C₂²⋊#1055	—	2	1	no	—
66	C₂²⋊#1055	—	2	4	no	—
67	C₄⋊#1055	—	2	1	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	C₂³	1	4	yes	—
70	#88043	—	2	2	no	—
71	#88043	D₄	1	2	yes	—
72	C₄×#1055	—	2	4	no	—
73	C₄×#1055	C₂³	1	2	yes	—
74	C₄×#1055	C₂³	1	2	yes	—
75	#18043	—	13	1	no	maximal, Sylow
76	C₁₃×#41	—	2	4	no	—
77	C₁₃×#41	—	4	2	no	—
78	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
87	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
88	C₁₃×SD₁₆	—	2	4	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
99	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	C₁₃×C₂²⋊C₄	—	2	2	no	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6099	—	4	2	no	—
111	#6099	—	4	8	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂×D₄	1	4	yes	—
115	#2971	—	13	2	no	—
116	C₄×#135	D₁₃	1	1	yes	characteristic
117	#1594	—	13	2	no	—
118	#1588	—	13	4	no	—
119	#1579	—	13	2	no	—
120	#1568	—	13	1	no	—
121	#1488	—	13	2	no	—
122	#1451	—	13	1	no	—
123	C₂³×C₁₃	—	2	2	no	—
124	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
127	D₄×C₁₃	—	2	2	no	—
128	D₄×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	4	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	8	2	no	—
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
150	#1055	—	2	4	no	—
151	#1055	—	4	2	no	—
152	#1055	—	4	4	no	—
153	C₂×#135	Dic₁₃	1	1	yes	characteristic
154	C₄×#41	C₂×D₁₃	1	2	yes	—
155	C₈×Q₈	—	13	4	no	—
156	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
157	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
158	C₄×SD₁₆	Dic₁₃	1	2	yes	—
159	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
160	C₄×D₈	Dic₁₃	1	1	yes	characteristic
161	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
162	#436	—	13	4	no	—
163	#436	—	26	4	no	—
164	#432	—	13	2	no	—
165	#432	—	26	2	no	—
166	#431	—	13	1	no	—
167	#427	Dic₁₃	1	1	yes	characteristic
168	#416	Dic₁₃	1	2	yes	—
169	C₄×M₁₆	—	13	1	no	—
170	C₄×M₁₆	—	26	2	no	—
171	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
172	#335	—	13	2	no	—
173	#334	—	13	2	no	—
174	Q₈⋊C₈	—	13	4	no	—
175	D₄⋊C₈	—	13	4	no	—
176	#322	—	13	2	no	—
177	#322	—	13	2	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	—	4	2	no	—
184	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	4	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	—	4	2	no	—
192	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
193	C₄×C₁₃	#568	1	1	yes	characteristic
194	C₄×C₁₃	#575	1	2	yes	—
195	C₂×#41	C₂×Dic₁₃	1	2	yes	—
196	#135	—	2	4	no	—
197	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
198	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
199	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂×M₁₆	—	13	2	no	—
201	C₂×M₁₆	—	26	4	no	—
202	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
204	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
205	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
209	C₄×D₄	C₂²×D₁₃	1	2	yes	—
210	#117	—	2	2	no	—
211	#117	C₂×Dic₁₃	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
214	#107	C₂×Dic₁₃	1	1	yes	characteristic
215	#107	C₂×Dic₁₃	1	1	yes	characteristic
216	#106	C₂×Dic₁₃	1	2	yes	—
217	C₄⋊C₈	—	13	4	no	—
218	C₄⋊C₈	—	26	4	no	—
219	C₄⋊C₈	—	26	1	no	—
220	#103	C₂×Dic₁₃	1	2	yes	—
221	#103	C₂×Dic₁₃	1	2	yes	—
222	#102	C₂×Dic₁₃	1	2	yes	—
223	#102	C₂×Dic₁₃	1	2	yes	—
224	C₂²⋊C₈	—	26	4	no	—
225	C₂²⋊C₈	—	26	2	no	—
226	C₂²⋊C₈	—	26	1	no	—
227	#97	—	13	2	no	—
228	#97	—	26	2	no	—
229	C₄×C₈	—	13	2	no	—
230	C₄×C₈	—	13	2	no	—
231	C₄×C₈	—	26	4	no	—
232	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂×Dic₁₃	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	—	4	1	no	—
237	C₂×C₁₃	—	8	2	no	—
238	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
239	C₂×C₁₃	#2943	1	1	yes	characteristic
240	C₂×C₁₃	#2957	1	1	yes	characteristic
241	#41	—	2	4	no	—
242	#41	—	4	2	no	—
243	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	#6129	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
252	Q₁₆	#6099	1	4	yes	complemented
253	SD₁₆	—	2	4	no	—
254	D₈	—	2	2	no	—
255	M₁₆	—	52	2	no	—
256	M₁₆	—	52	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	13	4	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
269	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	D₄×D₁₃	1	1	yes	characteristic
277	C₄²	#6129	1	1	yes	characteristic
278	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
279	C₄²	C₂³×D₁₃	1	1	yes	characteristic
280	C₁₃	#18043	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	2	2	no	—
282	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	C₂×#6099	1	4	yes	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	4	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
299	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂×#6129	1	1	yes	characteristic
303	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	26	4	no	—
308	C₈	—	52	2	no	—
309	C₈	—	52	4	no	—
310	C₈	C₂×#6099	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	#11719473	1	2	yes	—
326	C₄	C₂²×#6099	1	1	yes	characteristic
327	C₄	#11719524	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	2	no	—
330	C₂	D₄×#6099	1	1	yes	characteristic
331	C₂	#13123637	1	1	yes	characteristic
332	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#18043
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13123856	C₂	semidirect
2. #13113750	C₂	semidirect
3. #13113715	C₂	semidirect
4. #13113655	C₂	semidirect
5. #13113599	C₂	semidirect
6. #13113270	C₂	semidirect
7. #13113125	C₂	semidirect
87. C₁₃×Q₁₆	M₁₆	semidirect
252. Q₁₆	#6099	semidirect
280. C₁₃	#18043	semidirect

Maximal subgroups

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

subgroup	index
#13123856	2
#13113750	2
#13113715	2
#13113655	2
#13113599	2
#13113270	2
#13113125	2
C₄×C₁₃×#135	2
#18043	13