Subgroups — #16018826

Order 3328.

636 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125264	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117091	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117085	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117065	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117061	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116891	C₂	1	4	yes	maximal, complemented
7	#13116757	C₂	1	4	yes	maximal, complemented
8	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#532	—	2	1	no	—
11	C₂×#88158	—	2	1	no	—
12	C₂×#88158	C₂²	1	2	yes	—
13	#11719182	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₁₃×#486	—	2	1	no	—
17	C₁₃×#466	—	2	1	no	—
18	C₁₃×#466	C₂²	1	2	yes	—
19	C₄×C₁₃×D₈	—	2	1	no	—
20	C₄×C₁₃×D₈	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×#88067	C₂²	1	4	yes	—
23	C₂×#88066	C₂²	1	4	yes	—
24	#11718360	C₂²	1	1	yes	characteristic
25	#11718357	—	2	1	no	—
26	#11718357	C₂²	1	2	yes	—
27	#11718356	C₂²	1	1	yes	characteristic
28	#11718354	—	2	1	no	—
29	C₂²⋊#6090	C₂²	1	2	yes	—
30	#11718084	C₂²	1	4	yes	—
31	#11718083	C₂²	1	4	yes	—
32	C₄⋊#6090	C₂²	1	2	yes	—
33	C₄×#6090	C₂²	1	1	yes	characteristic
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×#41	—	4	1	no	—
36	C₂×#6127	—	2	2	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×#127	—	2	1	no	—
43	C₁₃×#124	—	4	1	no	—
44	C₁₃×#121	—	2	2	no	—
45	C₁₃×#121	—	4	1	no	—
46	C₄×D₄×C₁₃	—	2	2	no	—
47	C₄×D₄×C₁₃	—	4	1	no	—
48	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
49	#88184	—	4	1	no	—
50	#88184	—	4	1	no	—
51	C₂×Dic₅₂	—	2	1	no	—
52	C₂×Dic₅₂	C₂³	1	2	yes	—
53	#88158	—	4	4	no	—
54	#88158	—	4	4	no	—
55	C₂×D₁₀₄	—	2	1	no	—
56	C₂×D₁₀₄	C₂³	1	2	yes	—
57	C₂×#6095	—	2	2	no	—
58	C₂×#6095	—	2	2	no	—
59	#88129	—	2	2	no	—
60	#88128	—	2	1	no	—
61	C₄×D₅₂	—	4	1	no	—
62	C₄×D₅₂	—	4	1	no	—
63	#88123	—	2	1	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#102	—	2	2	no	—
67	C₁₃×#102	—	2	2	no	—
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88067	—	4	4	no	—
73	#88066	—	4	4	no	—
74	C₂×#6090	C₂³	1	2	yes	—
75	C₂×#6090	C₂³	1	2	yes	—
76	#33773	—	13	1	no	maximal, Sylow
77	C₁₃×#41	—	8	4	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₁₃	—	4	2	no	—
88	#6127	—	4	4	no	—
89	#6127	—	4	4	no	—
90	C₂×D₅₂	—	2	2	no	—
91	C₂×D₅₂	—	4	1	no	—
92	C₂×D₅₂	—	4	1	no	—
93	C₂×C₄×D₁₃	—	4	2	no	—
94	C₂×Dic₂₆	—	2	2	no	—
95	C₁₃×D₈	—	2	4	no	—
96	C₁₃×D₈	—	4	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₁₃	D₈	1	1	yes	characteristic
108	C₄²×C₁₃	D₈	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	#6103	—	4	2	no	—
112	#6103	—	4	2	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	Dic₅₂	—	2	4	no	—
116	Dic₅₂	—	4	2	no	—
117	D₁₀₄	—	2	4	no	—
118	D₁₀₄	—	4	2	no	—
119	#6095	—	4	4	no	—
120	#6095	—	4	4	no	—
121	#6090	—	2	2	no	—
122	#6090	—	2	2	no	—
123	#3169	D₁₃	1	1	yes	characteristic, complemented
124	#3147	—	13	1	no	—
125	#2248	—	13	1	no	—
126	#2246	—	13	1	no	—
127	#2241	—	13	1	no	—
128	#2240	—	13	1	no	—
129	#2210	—	13	4	no	—
130	#2186	—	13	4	no	—
131	#2162	—	13	1	no	—
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	8	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	8	4	no	—
139	D₄×C₁₃	—	8	2	no	—
140	D₄×C₁₃	—	8	2	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	8	4	no	—
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₁₃	—	8	4	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	D₅₂	—	4	2	no	—
156	D₅₂	—	4	2	no	—
157	D₅₂	—	8	2	no	—
158	D₅₂	—	8	2	no	—
159	C₄×D₁₃	—	8	4	no	—
160	Dic₂₆	—	4	2	no	—
161	Dic₂₆	—	4	2	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
165	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
166	C₂×#135	—	13	2	no	—
167	C₂×#135	—	26	1	no	—
168	#532	—	2	1	no	—
169	#532	—	26	1	no	—
170	C₂×SD₃₂	—	13	4	no	—
171	C₂×D₁₆	—	13	4	no	—
172	#500	C₂×D₁₃	1	1	yes	characteristic
173	#495	—	13	2	no	—
174	#495	—	26	1	no	—
175	#494	—	13	1	no	—
176	#493	—	13	1	no	—
177	#493	C₂×D₁₃	1	1	yes	characteristic
178	#492	—	26	1	no	—
179	#486	—	2	1	no	—
180	#466	—	2	1	no	—
181	#466	C₂×D₁₃	1	2	yes	—
182	C₄×D₈	—	2	1	no	—
183	C₄×D₈	C₂×D₁₃	1	2	yes	—
184	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
185	C₄⋊C₁₆	—	13	2	no	—
186	#358	—	13	4	no	—
187	#357	—	13	4	no	—
188	C₂²⋊C₁₆	—	13	2	no	—
189	C₄×C₁₆	—	13	1	no	—
190	C₄×C₁₆	—	13	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
198	C₂×D₁₃	—	8	2	no	—
199	C₂×D₁₃	—	8	2	no	—
200	C₂×D₁₃	—	8	2	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	#508	1	2	yes	—
206	C₄×C₁₃	#509	1	2	yes	—
207	C₄×C₁₃	#535	1	1	yes	characteristic
208	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	Dic₁₃	—	8	2	no	—
210	C₂×#41	—	4	1	no	—
211	C₂×#41	—	26	2	no	—
212	#135	—	52	4	no	—
213	#135	—	52	4	no	—
214	C₂×Q₁₆	—	13	2	no	—
215	C₂×Q₁₆	—	26	1	no	—
216	C₂×SD₁₆	—	26	2	no	—
217	C₂×SD₁₆	—	26	2	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	13	2	no	—
220	C₂×D₈	—	26	1	no	—
221	C₂×D₈	C₂²×D₁₃	1	2	yes	—
222	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	26	1	no	—
226	#127	—	2	1	no	—
227	#127	—	26	1	no	—
228	#124	—	4	1	no	—
229	#124	—	26	2	no	—
230	#121	—	2	2	no	—
231	#121	—	4	1	no	—
232	#121	—	52	1	no	—
233	#121	—	52	1	no	—
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	—	4	1	no	—
236	C₄×D₄	—	52	1	no	—
237	C₄×D₄	—	52	1	no	—
238	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
239	SD₃₂	—	52	4	no	—
240	D₁₆	—	52	4	no	—
241	C₂×C₁₆	—	13	2	no	—
242	C₂×C₁₆	—	13	2	no	—
243	#107	—	2	1	no	—
244	#107	C₂²×D₁₃	1	2	yes	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₁₃	—	4	1	no	—
252	C₂×C₁₃	—	16	2	no	—
253	C₂×C₁₃	#3205	1	1	yes	characteristic
254	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
255	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
256	D₁₃	—	16	2	no	—
257	#41	—	8	4	no	—
258	#41	—	52	4	no	—
259	#41	—	52	4	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	26	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	26	2	no	—
267	C₂×D₄	—	52	2	no	—
268	C₂×D₄	—	52	1	no	—
269	C₂×D₄	—	52	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	52	2	no	—
272	C₂²×C₄	#6104	1	1	yes	characteristic
273	C₂²×C₄	#6104	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	Q₁₆	—	26	4	no	—
276	Q₁₆	—	52	2	no	—
277	SD₁₆	—	52	4	no	—
278	SD₁₆	—	52	4	no	—
279	D₈	—	2	4	no	—
280	D₈	—	4	2	no	—
281	D₈	—	26	4	no	—
282	D₈	—	52	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	52	1	no	—
292	C₄⋊C₄	—	52	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	52	2	no	—
296	C₂²⋊C₄	—	52	2	no	—
297	C₄²	#6104	1	1	yes	characteristic
298	C₄²	#6104	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₁₆	—	26	2	no	—
302	C₁₆	—	26	2	no	—
303	C₁₃	#33773	1	1	yes	characteristic, complemented, Sylow
304	C₂³	—	4	2	no	—
305	C₂³	—	52	2	no	—
306	C₂³	C₂×#6104	1	1	yes	characteristic
307	Q₈	—	8	2	no	—
308	Q₈	—	52	2	no	—
309	Q₈	—	52	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	D₄	—	8	2	no	—
315	D₄	—	52	2	no	—
316	D₄	—	52	2	no	—
317	D₄	—	104	4	no	—
318	D₄	—	104	2	no	—
319	D₄	—	104	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	104	4	no	—
328	C₂×C₄	C₂×#6104	1	1	yes	characteristic
329	C₂×C₄	C₂×#6104	1	1	yes	characteristic
330	C₂×C₄	C₂×#6104	1	1	yes	characteristic
331	C₂×C₄	C₂×#6104	1	1	yes	characteristic
332	C₂×C₄	C₂×#6104	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	#88203	1	2	yes	—
337	C₈	#88256	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	104	2	no	—
345	C₂²	—	104	2	no	—
346	C₂²	—	104	2	no	—
347	C₂²	C₂²×#6104	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	8	2	no	—
352	C₄	—	104	2	no	—
353	C₄	#11718808	1	2	yes	—
354	C₄	#11718855	1	2	yes	—
355	C₄	C₂×#88203	1	1	yes	characteristic
356	C₄	#11719535	1	1	yes	characteristic
357	C₂	—	4	1	no	—
358	C₂	—	16	2	no	—
359	C₂	—	208	2	no	—
360	C₂	#13125751	1	1	yes	characteristic
361	C₂	C₂×#11718855	1	1	yes	characteristic
362	C₂	C₂×#11718808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#33773
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13125264	C₂	semidirect
2. #13117091	C₂	semidirect
3. #13117085	C₂	semidirect
4. #13117065	C₂	semidirect
5. #13117061	C₂	semidirect
6. #13116891	C₂	semidirect
7. #13116757	C₂	semidirect
8. #13116635	C₂	semidirect
9. C₁₃×#3169	C₂	semidirect
123. #3169	D₁₃	semidirect
303. C₁₃	#33773	semidirect

Maximal subgroups

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

subgroup	index
#13125264	2
#13117091	2
#13117085	2
#13117065	2
#13117061	2
#13116891	2
#13116757	2
#13116635	2
C₁₃×#3169	2
#33773	13