Subgroups — #15845161

Order 3328.

802 conjugacy classes of subgroups.

The subgroup lattice has 396 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	#13124836	C₂	1	2	yes	maximal, complemented
3	#13114201	C₂	1	2	yes	maximal, complemented
4	#13114191	C₂	1	2	yes	maximal, complemented
5	#13114177	C₂	1	4	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113486	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃	—	2	1	no	—
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88158	—	2	1	no	—
12	C₂×#88158	—	2	1	no	—
13	#11719182	C₂²	1	2	yes	—
14	C₈×D₄×C₁₃	—	2	1	no	—
15	C₈×D₄×C₁₃	—	2	1	no	—
16	C₈×D₄×C₁₃	C₂²	1	2	yes	—
17	C₈×D₄×C₁₃	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718878	—	2	2	no	—
22	#11718676	C₂²	1	4	yes	—
23	C₄×#6104	—	2	4	no	—
24	#11718668	C₂²	1	4	yes	—
25	#11718628	—	2	2	no	—
26	#11718586	C₂²	1	2	yes	—
27	#11718360	C₂²	1	1	yes	characteristic, complemented
28	#11718357	—	2	1	no	—
29	#11718357	—	2	1	no	—
30	#11718356	C₂²	1	1	yes	characteristic, complemented
31	#11718354	C₂²	1	2	yes	complemented
32	#11718077	C₂²	1	4	yes	complemented
33	#11718074	C₂²	1	4	yes	complemented
34	#11718050	C₂²	1	2	yes	complemented
35	#11718049	C₂²	1	2	yes	complemented
36	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₃	—	2	1	no	—
38	C₂×#6127	—	2	2	no	—
39	C₂²×C₈×C₁₃	—	2	1	no	—
40	C₂²×C₈×C₁₃	—	2	2	no	—
41	C₂²×C₈×C₁₃	—	2	2	no	—
42	C₂²×C₈×C₁₃	—	4	4	no	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×D₄×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	—	2	1	no	—
47	C₄×D₄×C₁₃	D₄	1	2	yes	—
48	C₄×D₄×C₁₃	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×#6104	—	2	4	no	—
53	#88184	—	2	2	no	—
54	#88184	—	2	2	no	—
55	C₂×Dic₅₂	—	2	1	no	—
56	C₂×Dic₅₂	—	2	1	no	—
57	#88158	—	4	4	no	—
58	#88158	—	4	4	no	—
59	C₂×D₁₀₄	—	2	1	no	—
60	C₂×D₁₀₄	—	2	1	no	—
61	C₂×#6095	—	2	2	no	—
62	C₂×#6095	—	2	2	no	—
63	#88129	—	2	2	no	—
64	#88128	C₂³	1	2	yes	—
65	C₄×D₅₂	—	2	2	no	—
66	C₄×D₅₂	—	2	2	no	—
67	#88123	C₂³	1	2	yes	—
68	C₁₃×C₄⋊C₈	—	2	1	no	—
69	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
70	C₁₃×C₂²⋊C₈	—	2	2	no	—
71	C₁₃×C₂²⋊C₈	—	2	2	no	—
72	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88072	—	2	4	no	—
77	C₂²⋊#1055	—	2	2	no	—
78	#88049	—	2	4	no	—
79	#88047	—	2	4	no	—
80	C₄⋊#1055	C₂³	1	4	yes	—
81	C₄×#1055	—	2	2	no	—
82	#19101	—	13	1	no	maximal, Sylow
83	C₂⁴×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	2	4	no	—
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	—	2	1	no	—
88	C₂²×C₄×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₁₃	—	4	2	no	—
96	#6127	—	4	4	no	—
97	#6127	—	4	4	no	—
98	C₂×D₅₂	—	2	2	no	—
99	C₂×D₅₂	—	2	2	no	—
100	C₂×D₅₂	—	2	2	no	—
101	C₂×C₄×D₁₃	—	4	2	no	—
102	C₂×Dic₂₆	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	4	2	no	—
106	C₂×C₈×C₁₃	—	4	4	no	—
107	C₂×C₈×C₁₃	—	4	4	no	—
108	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
115	C₁₃×C₂²⋊C₄	—	2	2	no	—
116	C₁₃×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	#41	1	1	yes	characteristic
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	#6104	—	4	8	no	—
122	#6103	—	4	2	no	—
123	#6103	—	4	2	no	—
124	C₄⋊Dic₁₃	—	2	2	no	—
125	C₄⋊Dic₁₃	—	2	2	no	—
126	C₂×#1055	—	2	4	no	—
127	Dic₅₂	—	4	2	no	—
128	Dic₅₂	—	4	2	no	—
129	D₁₀₄	—	4	2	no	—
130	D₁₀₄	—	4	2	no	—
131	#6095	—	4	4	no	—
132	#6095	—	4	4	no	—
133	#3147	—	13	1	no	—
134	#3100	—	13	2	no	—
135	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic, complemented
136	#1669	—	13	2	no	—
137	#1667	—	13	2	no	—
138	#1665	—	13	4	no	—
139	#1564	—	13	1	no	—
140	#1540	—	13	2	no	—
141	C₂³×C₁₃	—	2	4	no	—
142	C₂³×C₁₃	—	2	2	no	—
143	C₂³×C₁₃	—	4	2	no	—
144	C₂³×C₁₃	—	4	2	no	—
145	C₂³×C₁₃	#121	1	1	yes	characteristic
146	C₂²×D₁₃	—	4	2	no	—
147	D₄×C₁₃	—	2	4	no	—
148	D₄×C₁₃	—	2	2	no	—
149	D₄×C₁₃	#135	1	4	yes	complemented
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	4	no	—
157	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₂²⋊D₁₃	—	8	4	no	—
166	C₂×Dic₁₃	—	4	2	no	—
167	D₅₂	—	4	2	no	—
168	D₅₂	—	4	4	no	—
169	D₅₂	—	4	2	no	—
170	D₅₂	—	8	2	no	—
171	C₄×D₁₃	—	8	4	no	—
172	Dic₂₆	—	4	2	no	—
173	Dic₂₆	—	4	2	no	—
174	C₈×C₁₃	—	2	1	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	4	2	no	—
177	C₈×C₁₃	C₂×D₈	1	2	yes	—
178	C₈×C₁₃	C₂×D₈	1	2	yes	—
179	#1055	—	4	4	no	—
180	C₂×#135	—	26	1	no	—
181	C₂×#135	—	26	1	no	—
182	C₂³×C₈	—	2	1	no	—
183	#532	—	13	2	no	—
184	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
185	#495	—	26	1	no	—
186	#495	—	26	1	no	—
187	#494	—	13	1	no	—
188	#493	—	13	1	no	—
189	#492	—	13	2	no	—
190	#480	—	26	2	no	—
191	#474	—	13	4	no	—
192	#459	—	13	4	no	—
193	#447	—	26	2	no	—
194	C₄×D₈	—	26	4	no	—
195	C₈×D₄	—	2	1	no	—
196	C₈×D₄	—	2	1	no	—
197	C₈×D₄	C₂×D₁₃	1	2	yes	—
198	C₈×D₄	C₂×D₁₃	1	2	yes	—
199	#431	—	13	2	no	—
200	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
201	C₂×C₂²⋊C₈	—	2	1	no	—
202	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
203	#335	—	13	2	no	—
204	#334	—	13	2	no	—
205	#332	—	13	4	no	—
206	#331	—	13	4	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	—	2	4	no	—
210	C₂²×C₁₃	—	4	8	no	—
211	C₂²×C₁₃	—	4	2	no	—
212	C₂²×C₁₃	—	4	2	no	—
213	C₂²×C₁₃	#459	1	2	yes	—
214	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
215	C₂×D₁₃	—	8	2	no	—
216	C₂×D₁₃	—	8	2	no	—
217	C₂×D₁₃	—	8	2	no	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	4	2	no	—
222	C₄×C₁₃	#459	1	2	yes	—
223	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
224	C₄×C₁₃	C₂×#135	1	2	yes	—
225	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	Dic₁₃	—	8	2	no	—
227	C₂×#41	—	26	2	no	—
228	C₂²×D₄	D₅₂	1	1	yes	characteristic
229	C₂³×C₄	—	2	1	no	—
230	#135	—	52	4	no	—
231	#135	—	52	4	no	—
232	C₂×Q₁₆	—	26	1	no	—
233	C₂×Q₁₆	—	26	1	no	—
234	C₂×SD₁₆	—	26	2	no	—
235	C₂×SD₁₆	—	26	2	no	—
236	C₂×D₈	—	26	4	no	—
237	C₂×D₈	—	26	1	no	—
238	C₂×D₈	—	26	1	no	—
239	C₂²×C₈	—	2	1	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	—	4	4	no	—
243	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
244	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
245	C₄⋊Q₈	—	13	2	no	—
246	#127	—	13	2	no	—
247	#124	—	26	2	no	—
248	#121	—	26	2	no	—
249	#121	—	26	2	no	—
250	C₄×D₄	—	2	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	26	2	no	—
253	C₄×D₄	—	26	2	no	—
254	C₄×D₄	D₅₂	1	2	yes	—
255	C₄×D₄	C₂²×D₁₃	1	2	yes	—
256	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
257	C₂×C₂²⋊C₄	—	2	1	no	—
258	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
259	#107	—	26	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	13	4	no	—
262	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
263	#102	—	26	4	no	—
264	#102	—	26	4	no	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	C₂²⋊C₈	—	26	2	no	—
268	C₄×C₈	—	26	2	no	—
269	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
270	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₂×C₁₃	—	2	2	no	—
274	C₂×C₁₃	—	4	4	no	—
275	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
276	C₂×C₁₃	#3050	1	1	yes	characteristic
277	C₂×C₁₃	#3067	1	1	yes	characteristic
278	D₁₃	—	16	2	no	—
279	C₂⁴	—	2	1	no	—
280	#41	—	52	4	no	—
281	#41	—	52	4	no	—
282	C₂×Q₈	—	26	2	no	—
283	C₂×D₄	—	2	4	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	26	2	no	—
286	C₂×D₄	—	26	2	no	—
287	C₂×D₄	—	26	2	no	—
288	C₂×D₄	—	52	2	no	—
289	C₂×D₄	C₂×D₅₂	1	2	yes	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	52	2	no	—
296	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
297	C₂²×C₄	#6127	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
299	Q₁₆	—	52	2	no	—
300	Q₁₆	—	52	2	no	—
301	SD₁₆	—	52	4	no	—
302	SD₁₆	—	52	4	no	—
303	D₈	—	52	2	no	—
304	D₈	—	52	8	no	—
305	D₈	—	52	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	4	4	no	—
310	C₂×C₈	—	4	4	no	—
311	C₂×C₈	—	26	4	no	—
312	C₂×C₈	#6104	1	4	yes	complemented
313	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	—	26	2	no	—
320	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	52	2	no	—
324	C₂²⋊C₄	—	52	2	no	—
325	C₄²	C₂×D₅₂	1	1	yes	characteristic
326	C₄²	#6127	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
328	C₄²	C₂³×D₁₃	1	1	yes	characteristic
329	C₁₃	#19101	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	2	4	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	52	2	no	—
335	C₂³	#88184	1	1	yes	characteristic
336	Q₈	—	52	2	no	—
337	Q₈	—	52	2	no	—
338	D₄	—	2	4	no	—
339	D₄	—	2	2	no	—
340	D₄	—	52	4	no	—
341	D₄	—	52	2	no	—
342	D₄	—	52	2	no	—
343	D₄	—	104	2	no	—
344	D₄	—	104	4	no	—
345	D₄	#88158	1	4	yes	complemented
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	52	2	no	—
354	C₂×C₄	—	104	4	no	—
355	C₂×C₄	#88184	1	1	yes	characteristic
356	C₂×C₄	#88184	1	1	yes	characteristic
357	C₂×C₄	#88184	1	1	yes	characteristic
358	C₂×C₄	C₂×#6104	1	2	yes	—
359	C₂×C₄	#88203	1	2	yes	—
360	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
361	C₂×C₄	C₂×#6127	1	1	yes	characteristic
362	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	—	4	2	no	—
366	C₈	—	52	4	no	—
367	C₈	C₂×#6104	1	2	yes	—
368	C₈	C₂×#6104	1	2	yes	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	4	no	—
372	C₂²	—	4	8	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	104	2	no	—
376	C₂²	—	104	2	no	—
377	C₂²	—	104	2	no	—
378	C₂²	#11718676	1	2	yes	—
379	C₂²	C₂×#88184	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	4	2	no	—
384	C₄	—	104	2	no	—
385	C₄	#11718676	1	2	yes	—
386	C₄	C₂×#88158	1	2	yes	—
387	C₄	C₂²×#6104	1	1	yes	characteristic
388	C₄	C₂×#88203	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	4	4	no	—
391	C₂	—	208	2	no	—
392	C₂	C₂×#11718676	1	1	yes	characteristic
393	C₂	#13124417	1	1	yes	characteristic
394	C₂	#13124556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	C₂×#121
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	#19101
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #13125264	C₂	semidirect
2. #13124836	C₂	semidirect
3. #13114201	C₂	semidirect
4. #13114191	C₂	semidirect
5. #13114177	C₂	semidirect
6. #13113584	C₂	semidirect
7. #13113486	C₂	semidirect
8. C₂×C₈×D₄×C₁₃	C₂	semidirect
27. #11718360	C₂²	semidirect
30. #11718356	C₂²	semidirect
31. #11718354	C₂²	semidirect
32. #11718077	C₂²	semidirect
33. #11718074	C₂²	semidirect
34. #11718050	C₂²	semidirect
35. #11718049	C₂²	semidirect
108. C₂×C₈×C₁₃	D₈	semidirect
135. C₂×C₈×D₄	D₁₃	semidirect
149. D₄×C₁₃	#135	semidirect
312. C₂×C₈	#6104	semidirect
329. C₁₃	#19101	semidirect
345. D₄	#88158	semidirect

Maximal subgroups

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

subgroup	index
#13125264	2
#13124836	2
#13114201	2
#13114191	2
#13114177	2
#13113584	2
#13113486	2
C₂×C₈×D₄×C₁₃	2
#19101	13