Subgroups — #16017766

Order 3328.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124485	C₂	1	1	yes	maximal, characteristic
2	#13123433	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117030	C₂	1	2	yes	maximal, complemented
4	#13117029	C₂	1	2	yes	maximal, complemented
5	#13117023	C₂	1	2	yes	maximal, complemented
6	#13117018	C₂	1	2	yes	maximal, complemented
7	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116551	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718101	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116449	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	#11719230	—	2	1	no	—
15	#11719228	—	2	1	no	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#427	C₂²	1	1	yes	characteristic
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	1	yes	characteristic
21	#11718662	—	2	1	no	—
22	#11718656	—	2	1	no	—
23	#11718544	C₂²	1	2	yes	—
24	#11718544	C₂²	1	2	yes	—
25	#11718541	C₂²	1	2	yes	complemented
26	D₁₃×#107	C₂²	1	2	yes	complemented
27	#11718526	—	2	2	no	—
28	#11718522	—	2	2	no	—
29	C₁₃×#367	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×#367	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#366	C₂²	1	1	yes	characteristic, complemented
32	C₁₃×#366	C₂²	1	1	yes	characteristic, complemented
33	#11718101	C₂²	1	2	yes	—
34	#11718101	C₂²	1	2	yes	—
35	#11718099	C₂²	1	2	yes	complemented
36	#11718099	C₂²	1	2	yes	complemented
37	#11718083	C₂²	1	2	yes	—
38	#11718083	C₂²	1	2	yes	—
39	#11718082	C₂²	1	2	yes	complemented
40	#11718081	C₂²	1	2	yes	complemented
41	C₂×#6127	D₄	1	1	yes	characteristic
42	C₂²×D₅₂	—	2	1	no	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	—	2	1	no	—
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×#6099	D₄	1	1	yes	characteristic
48	C₂×D₁₀₄	—	2	2	no	—
49	C₂×D₁₀₄	—	2	2	no	—
50	C₂×D₁₀₄	—	2	1	no	—
51	C₂×D₁₀₄	C₂³	1	2	yes	—
52	#88155	—	2	2	no	—
53	#88155	—	2	2	no	—
54	C₂×#6094	D₄	1	2	yes	—
55	C₂×C₈×D₁₃	C₂³	1	2	yes	—
56	#88149	—	2	2	no	—
57	#88149	—	2	2	no	—
58	#88146	—	2	2	no	—
59	C₄⋊C₄×D₁₃	—	2	2	no	—
60	#88136	—	4	2	no	—
61	#88135	—	4	2	no	—
62	#88129	—	4	1	no	—
63	#88128	—	4	1	no	—
64	C₄×D₅₂	—	4	1	no	—
65	C₄×Dic₂₆	—	4	1	no	—
66	C₂×C₁₃×C₁₆	—	2	2	no	—
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×#107	C₂³	1	1	yes	characteristic
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#106	—	2	1	no	—
75	#88058	—	2	1	no	—
76	#88058	C₂³	1	2	yes	—
77	#88052	—	2	2	no	—
78	C₂×#6090	C₂³	1	2	yes	—
79	#88049	—	2	2	no	—
80	#88049	—	2	2	no	—
81	#88048	—	2	2	no	—
82	#88047	—	2	2	no	—
83	#33650	—	13	1	no	maximal, Sylow
84	C₂³×D₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₁₃	D₈	1	2	yes	—
87	C₂²×Dic₁₃	—	4	1	no	—
88	#6127	—	2	2	no	—
89	#6127	—	2	2	no	—
90	C₂×D₅₂	—	2	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×D₅₂	—	4	1	no	—
94	C₂×D₅₂	—	4	2	no	—
95	C₂×D₅₂	—	4	2	no	—
96	C₂×D₅₂	D₈	1	1	yes	characteristic
97	C₂×C₄×D₁₃	—	4	2	no	—
98	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
99	C₂×Dic₂₆	D₈	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	2	yes	—
102	C₂×C₈×C₁₃	#41	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	1	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₁₃×C₂²⋊C₄	—	4	1	no	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	—	4	1	no	—
112	#6103	—	4	2	no	—
113	#6103	—	4	2	no	—
114	#6103	—	4	2	no	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	C₄⋊Dic₁₃	—	4	1	no	—
117	#6101	—	4	2	no	—
118	C₄×Dic₁₃	—	4	2	no	—
119	#6099	—	2	2	no	—
120	C₂×#1055	C₂×D₄	1	2	yes	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	—	2	2	no	—
123	D₁₀₄	—	4	2	no	—
124	#6094	—	2	4	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₈×D₁₃	—	2	2	no	—
127	C₁₃×C₁₆	—	2	1	no	—
128	C₁₃×C₁₆	—	2	1	no	—
129	#6090	—	2	2	no	—
130	#3058	—	13	1	no	—
131	#2913	—	13	1	no	—
132	#2234	—	13	2	no	—
133	#2234	—	13	2	no	—
134	#2233	—	13	2	no	—
135	#2232	—	13	2	no	—
136	#2159	—	13	1	no	—
137	#2158	D₁₃	1	1	yes	characteristic, complemented
138	#2140	—	13	1	no	—
139	C₂×#357	—	13	1	no	—
140	#2113	—	13	1	no	—
141	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂²×D₁₃	—	4	2	no	—
143	C₂²×D₁₃	—	8	2	no	—
144	C₂²×D₁₃	—	16	1	no	—
145	C₂²×D₁₃	—	16	1	no	—
146	C₂²×D₁₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₁₃	—	4	1	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	#123	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	#137	1	2	yes	—
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂²⋊D₁₃	—	2	4	no	—
156	C₂×Dic₁₃	—	4	2	no	—
157	C₂×Dic₁₃	—	16	1	no	—
158	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
159	D₅₂	—	2	2	no	—
160	D₅₂	—	4	2	no	—
161	D₅₂	—	4	2	no	—
162	D₅₂	—	8	2	no	—
163	D₅₂	—	16	2	no	—
164	D₅₂	—	16	2	no	—
165	C₄×D₁₃	—	2	2	no	—
166	C₄×D₁₃	—	2	2	no	—
167	C₄×D₁₃	—	16	2	no	—
168	Dic₂₆	—	2	2	no	—
169	C₈×C₁₃	#123	1	2	yes	—
170	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
171	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
172	#1055	—	2	2	no	—
173	C₂²×D₈	—	13	1	no	—
174	C₂×#131	—	13	1	no	—
175	#535	—	26	1	no	—
176	#520	—	26	1	no	—
177	C₂×M₃₂	—	13	1	no	—
178	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
179	#469	—	26	2	no	—
180	#466	—	13	2	no	—
181	#466	—	13	2	no	—
182	#428	—	26	2	no	—
183	#428	—	26	1	no	—
184	#427	—	13	2	no	—
185	#427	C₂×D₁₃	1	1	yes	characteristic
186	#427	C₂×D₁₃	1	1	yes	characteristic
187	C₂×#107	—	13	1	no	—
188	C₂×#107	—	13	2	no	—
189	#418	—	26	1	no	—
190	#367	—	13	2	no	—
191	#367	C₂×D₁₃	1	1	yes	characteristic, complemented
192	#367	C₂×D₁₃	1	1	yes	characteristic, complemented
193	#366	—	13	2	no	—
194	#366	C₂×D₁₃	1	1	yes	characteristic, complemented
195	#366	C₂×D₁₃	1	1	yes	characteristic, complemented
196	#357	—	13	2	no	—
197	#357	—	13	2	no	—
198	#357	—	13	2	no	—
199	#357	—	13	2	no	—
200	C₂²⋊C₁₆	—	13	2	no	—
201	C₂²⋊C₁₆	—	13	2	no	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	#480	1	2	yes	—
205	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
206	C₂×D₁₃	—	2	2	no	—
207	C₂×D₁₃	—	2	2	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₂×D₁₃	—	8	2	no	—
210	C₂×D₁₃	—	8	2	no	—
211	C₂×D₁₃	—	8	2	no	—
212	C₂×D₁₃	—	16	4	no	—
213	C₄×C₁₃	—	8	1	no	—
214	C₄×C₁₃	—	8	1	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	#480	1	2	yes	—
218	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
220	Dic₁₃	—	2	2	no	—
221	Dic₁₃	—	8	2	no	—
222	C₂×#41	—	13	1	no	—
223	C₂²×D₄	—	26	1	no	—
224	C₂×D₈	—	13	2	no	—
225	C₂×D₈	—	26	2	no	—
226	C₂×D₈	—	26	1	no	—
227	C₂×D₈	—	26	2	no	—
228	#131	—	26	2	no	—
229	#131	—	26	2	no	—
230	C₂×M₁₆	—	13	1	no	—
231	C₂×M₁₆	—	13	2	no	—
232	C₂²×C₈	—	13	2	no	—
233	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	#127	—	52	1	no	—
235	#124	—	52	1	no	—
236	#121	—	26	2	no	—
237	#121	—	26	2	no	—
238	#120	—	52	2	no	—
239	C₄×Q₈	—	52	1	no	—
240	C₄×D₄	—	52	1	no	—
241	C₄×D₄	—	52	2	no	—
242	#117	—	2	1	no	—
243	#117	—	2	1	no	—
244	#117	—	26	2	no	—
245	C₂×C₄⋊C₄	—	26	2	no	—
246	C₂×C₄⋊C₄	—	26	1	no	—
247	M₃₂	—	26	2	no	—
248	C₂×C₁₆	—	2	2	no	—
249	C₂×C₁₆	—	13	2	no	—
250	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
252	#107	—	13	2	no	—
253	#107	—	26	2	no	—
254	#107	—	26	1	no	—
255	#107	C₂²×D₁₃	1	1	yes	characteristic
256	#107	C₂²×D₁₃	1	1	yes	characteristic
257	#107	C₂²×D₁₃	1	1	yes	characteristic
258	#107	C₂²×D₁₃	1	1	yes	characteristic
259	#106	—	2	1	no	—
260	#106	—	2	1	no	—
261	#106	—	26	2	no	—
262	#102	—	26	2	no	—
263	#102	—	26	2	no	—
264	C₂×C₁₃	—	2	2	no	—
265	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
266	C₂×C₁₃	#3415	1	1	yes	characteristic
267	C₂×C₁₃	#3416	1	1	yes	characteristic
268	D₁₃	—	4	2	no	—
269	D₁₃	—	16	2	no	—
270	C₂⁴	—	52	1	no	—
271	#41	—	26	2	no	—
272	#41	—	26	2	no	—
273	C₂×Q₈	—	13	1	no	—
274	C₂×D₄	—	13	2	no	—
275	C₂×D₄	—	13	1	no	—
276	C₂×D₄	—	26	2	no	—
277	C₂×D₄	—	52	2	no	—
278	C₂×D₄	—	52	2	no	—
279	C₂×D₄	—	52	2	no	—
280	C₂×D₄	—	52	1	no	—
281	C₂×D₄	—	52	2	no	—
282	C₂²×C₄	—	13	2	no	—
283	C₂²×C₄	—	52	2	no	—
284	C₂²×C₄	—	52	1	no	—
285	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
286	D₈	—	26	2	no	—
287	D₈	—	26	2	no	—
288	D₈	—	52	2	no	—
289	M₁₆	—	26	2	no	—
290	M₁₆	—	26	4	no	—
291	C₂×C₈	—	13	2	no	—
292	C₂×C₈	—	26	2	no	—
293	C₂×C₈	—	26	2	no	—
294	C₂×C₈	#6127	1	2	yes	—
295	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
296	C₂×C₈	#6131	1	2	yes	—
297	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	26	2	no	—
303	C₄⋊C₄	—	52	2	no	—
304	C₄⋊C₄	—	52	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	52	2	no	—
308	C₂²⋊C₄	—	52	2	no	—
309	C₂²⋊C₄	—	52	2	no	—
310	C₄²	—	4	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	52	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	26	2	no	—
316	C₁₃	#33650	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	13	2	no	—
318	C₂³	—	52	2	no	—
319	C₂³	—	104	2	no	—
320	C₂³	—	208	1	no	—
321	C₂³	—	208	1	no	—
322	C₂³	D₈×D₁₃	1	1	yes	characteristic
323	Q₈	—	26	2	no	—
324	D₄	—	26	4	no	—
325	D₄	—	26	2	no	—
326	D₄	—	52	2	no	—
327	D₄	—	52	2	no	—
328	D₄	—	104	2	no	—
329	D₄	—	208	2	no	—
330	D₄	—	208	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	13	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	208	2	no	—
340	C₂×C₄	—	208	1	no	—
341	C₂×C₄	#88148	1	2	yes	—
342	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
343	C₂×C₄	#88172	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
345	C₈	—	26	2	no	—
346	C₈	#88148	1	2	yes	—
347	C₈	C₂×#6127	1	1	yes	characteristic
348	C₈	C₂×#6131	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	26	2	no	—
352	C₂²	—	26	2	no	—
353	C₂²	—	26	2	no	—
354	C₂²	—	104	2	no	—
355	C₂²	—	104	2	no	—
356	C₂²	—	104	2	no	—
357	C₂²	—	208	4	no	—
358	C₂²	#11718543	1	2	yes	—
359	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	26	2	no	—
365	C₄	—	104	2	no	—
366	C₄	#11718543	1	2	yes	—
367	C₄	C₂×#88148	1	1	yes	characteristic
368	C₄	C₂×#88172	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	52	2	no	—
371	C₂	—	208	2	no	—
372	C₂	C₂×#11718543	1	1	yes	characteristic
373	C₂	#13127974	1	1	yes	characteristic
374	C₂	#13127979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13124485	C₂
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33650
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #13123433	C₂	semidirect
3. #13117030	C₂	semidirect
4. #13117029	C₂	semidirect
5. #13117023	C₂	semidirect
6. #13117018	C₂	semidirect
7. #13116625	C₂	semidirect
8. #13116551	C₂	semidirect
9. C₂×#11718101	C₂	semidirect
10. #13116449	C₂	semidirect
11. C₁₃×#2158	C₂	semidirect
25. #11718541	C₂²	semidirect
26. D₁₃×#107	C₂²	semidirect
29. C₁₃×#367	C₂²	semidirect
30. C₁₃×#367	C₂²	semidirect
31. C₁₃×#366	C₂²	semidirect
32. C₁₃×#366	C₂²	semidirect
35. #11718099	C₂²	semidirect
36. #11718099	C₂²	semidirect
39. #11718082	C₂²	semidirect
40. #11718081	C₂²	semidirect
137. #2158	D₁₃	semidirect
191. #367	C₂×D₁₃	semidirect
192. #367	C₂×D₁₃	semidirect
194. #366	C₂×D₁₃	semidirect
195. #366	C₂×D₁₃	semidirect
316. C₁₃	#33650	semidirect

Maximal subgroups

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

subgroup	index
#13124485	2
#13123433	2
#13117030	2
#13117029	2
#13117023	2
#13117018	2
#13116625	2
#13116551	2
C₂×#11718101	2
#13116449	2
C₁₃×#2158	2
#33650	13