Subgroups — #15881311

Order 3328.

1134 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718533	C₂	1	2	yes	maximal
3	#13117736	C₂	1	2	yes	maximal, complemented
4	#13115567	C₂	1	2	yes	maximal, complemented
5	#13115080	C₂	1	4	yes	maximal, complemented
6	#13114833	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
10	C₂×#88147	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×D₁₃	—	2	2	no	—
12	C₂×C₄×D₅₂	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#384	C₂²	1	2	yes	—
17	C₂×#88061	C₂²	1	2	yes	—
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
20	#11718650	—	2	4	no	—
21	#11718648	—	2	2	no	—
22	#11718634	C₂²	1	4	yes	—
23	C₂×#88049	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
26	#11718533	—	2	8	no	—
27	C₄×D₁₀₄	C₄	1	8	yes	complemented
28	#11718142	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×D₅₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₁₃	—	2	2	no	—
32	C₂²×C₄×D₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6103	—	2	2	no	—
40	C₂×#6103	—	2	4	no	—
41	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
44	C₂×#6101	—	2	4	no	—
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×#1055	C₂³	1	2	yes	—
47	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
48	C₂×D₁₀₄	D₄	1	4	yes	—
49	C₂×D₁₀₄	Q₈	1	4	yes	—
50	#88147	—	2	8	no	—
51	#88147	—	4	4	no	—
52	C₄⋊C₄×D₁₃	—	4	8	no	—
53	C₄×D₅₂	—	2	4	no	—
54	C₄×D₅₂	C₂×C₄	1	8	yes	complemented
55	C₁₃×#107	—	2	4	no	—
56	C₁₃×#107	—	4	2	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
58	#88071	—	2	4	no	—
59	#88071	D₄	1	2	yes	—
60	#88061	C₂×C₄	1	8	yes	complemented
61	#88058	C₂×C₄	1	4	yes	complemented
62	C₈×Dic₁₃	—	2	4	no	—
63	#88049	—	2	8	no	—
64	C₄⋊#1055	—	2	2	no	—
65	C₄⋊C₄×D₈	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	2	2	no	—
72	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
73	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
74	C₂×D₅₂	—	2	2	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	C₂²×C₄	1	4	yes	—
78	C₂×D₅₂	C₂×D₄	1	4	yes	—
79	C₂×D₅₂	C₂×Q₈	1	4	yes	—
80	C₂×C₄×D₁₃	—	2	8	no	—
81	C₂×C₄×D₁₃	—	4	4	no	—
82	C₂×C₄×D₁₃	—	4	4	no	—
83	C₂×C₄×D₁₃	—	4	8	no	—
84	C₂×C₄×D₁₃	—	4	8	no	—
85	C₂×C₄×D₁₃	—	8	4	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₄⋊C₄	—	8	2	no	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	#41	1	2	yes	—
98	#6103	—	2	8	no	—
99	#6103	—	4	8	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	D₈	1	4	yes	complemented
103	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
104	#6101	—	4	8	no	—
105	C₄×Dic₁₃	—	2	2	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	2	no	—
109	C₂×#1055	—	2	2	no	—
110	D₁₀₄	C₄⋊C₄	1	16	yes	complemented
111	C₂×C₄×D₈	—	13	2	no	—
112	C₂×C₄×D₈	—	13	1	no	—
113	D₄×C₄⋊C₄	—	13	2	no	—
114	#1942	—	13	2	no	—
115	#1862	—	13	4	no	—
116	#1845	D₁₃	1	1	yes	characteristic, complemented
117	#1802	—	13	2	no	—
118	C₈×C₄⋊C₄	—	13	1	no	—
119	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂²×D₁₃	—	2	4	no	—
121	C₂²×D₁₃	—	2	4	no	—
122	C₂²×D₁₃	—	2	4	no	—
123	C₂²×D₁₃	—	4	2	no	—
124	C₂²×D₁₃	—	4	2	no	—
125	C₂²×D₁₃	—	4	2	no	—
126	C₂²×D₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	#135	1	2	yes	—
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	4	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
151	D₅₂	—	2	4	no	—
152	D₅₂	—	2	4	no	—
153	D₅₂	—	2	4	no	—
154	D₅₂	C₂×C₄⋊C₄	1	8	yes	complemented
155	C₄×D₁₃	—	4	16	no	—
156	C₄×D₁₃	—	8	16	no	—
157	C₈×C₁₃	—	2	2	no	—
158	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	complemented
159	#1055	—	4	2	no	—
160	C₂²×D₈	—	13	1	no	—
161	C₂×C₄×D₄	—	13	4	no	—
162	C₂×C₄×D₄	—	13	2	no	—
163	C₂²×C₄⋊C₄	—	26	2	no	—
164	C₄×D₈	—	13	8	no	—
165	C₄×D₈	—	26	8	no	—
166	C₂×#107	—	13	1	no	—
167	C₂×#107	C₂×D₁₃	1	2	yes	—
168	C₂×#107	C₂×D₁₃	1	2	yes	—
169	C₂×C₄⋊C₈	—	13	1	no	—
170	C₂×#102	—	13	2	no	—
171	C₂×#102	—	13	4	no	—
172	C₂×C₄×C₈	—	13	2	no	—
173	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
174	#389	—	13	4	no	—
175	#385	—	26	4	no	—
176	#384	C₂×D₁₃	1	2	yes	—
177	#380	—	26	2	no	—
178	C₄×C₄⋊C₄	—	13	1	no	—
179	#340	—	13	4	no	—
180	#336	—	13	2	no	—
181	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂²×C₁₃	D₄²	1	1	yes	characteristic
183	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
186	C₂²×C₁₃	#576	1	1	yes	characteristic
187	C₂²×C₁₃	#578	1	1	yes	characteristic
188	C₂×D₁₃	—	2	8	no	—
189	C₂×D₁₃	—	4	4	no	—
190	C₂×D₁₃	—	4	4	no	—
191	C₂×D₁₃	—	4	4	no	—
192	C₂×D₁₃	—	4	4	no	—
193	C₂×D₁₃	—	4	4	no	—
194	C₂×D₁₃	—	4	4	no	—
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	8	4	no	—
197	C₄×C₁₃	C₄×D₈	1	4	yes	complemented
198	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
199	C₄×C₁₃	#520	1	1	yes	characteristic
200	C₄×C₁₃	#547	1	1	yes	characteristic
201	C₄×C₁₃	#554	1	1	yes	characteristic
202	Dic₁₃	—	2	4	no	—
203	Dic₁₃	—	4	4	no	—
204	Dic₁₃	—	4	2	no	—
205	C₂²×D₄	—	13	2	no	—
206	C₂³×C₄	—	26	4	no	—
207	C₂³×C₄	—	26	2	no	—
208	C₂×D₈	—	13	4	no	—
209	C₂×D₈	—	13	4	no	—
210	C₂×D₈	—	13	4	no	—
211	C₂²×C₈	—	13	2	no	—
212	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₄×D₄	—	13	8	no	—
215	C₄×D₄	—	26	8	no	—
216	C₄×D₄	—	26	4	no	—
217	C₄×D₄	—	52	4	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	13	1	no	—
221	C₂×C₄⋊C₄	—	13	2	no	—
222	C₂×C₄⋊C₄	—	13	1	no	—
223	C₂×C₄⋊C₄	—	26	4	no	—
224	C₂×C₄⋊C₄	—	52	8	no	—
225	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
226	C₂×C₂²⋊C₄	—	26	2	no	—
227	C₂×C₂²⋊C₄	—	26	4	no	—
228	C₂×C₄²	—	13	2	no	—
229	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
230	#107	—	2	4	no	—
231	#107	—	4	2	no	—
232	#107	—	13	4	no	—
233	C₄⋊C₈	—	26	2	no	—
234	#102	—	13	8	no	—
235	#102	—	26	8	no	—
236	C₄×C₈	—	26	4	no	—
237	C₄×C₈	C₄×D₁₃	1	4	yes	complemented
238	#95	—	13	2	no	—
239	#95	—	26	4	no	—
240	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
242	C₂×C₁₃	#2951	1	1	yes	characteristic
243	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
244	C₂×C₁₃	#3283	1	1	yes	characteristic
245	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
246	C₂×C₁₃	#3380	1	1	yes	characteristic
247	D₁₃	—	4	8	no	—
248	C₂⁴	—	26	2	no	—
249	C₂×D₄	—	13	4	no	—
250	C₂×D₄	—	13	4	no	—
251	C₂×D₄	—	13	4	no	—
252	C₂×D₄	—	26	2	no	—
253	C₂×D₄	—	26	2	no	—
254	C₂×D₄	—	26	2	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	26	2	no	—
259	C₂²×C₄	—	26	8	no	—
260	C₂²×C₄	—	52	8	no	—
261	C₂²×C₄	—	52	8	no	—
262	C₂²×C₄	—	52	4	no	—
263	C₂²×C₄	—	52	4	no	—
264	C₂²×C₄	—	104	4	no	—
265	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
268	D₈	—	13	16	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
275	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
276	C₂×C₈	D₄×D₁₃	1	2	yes	—
277	C₂×C₈	Q₈×D₁₃	1	2	yes	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	8	2	no	—
282	C₄⋊C₄	—	13	4	no	—
283	C₄⋊C₄	—	13	4	no	—
284	C₄⋊C₄	—	26	2	no	—
285	C₄⋊C₄	—	26	2	no	—
286	C₄⋊C₄	—	52	8	no	—
287	C₂²⋊C₄	—	26	8	no	—
288	C₂²⋊C₄	—	52	8	no	—
289	C₄²	—	26	2	no	—
290	C₄²	—	26	2	no	—
291	C₄²	C₂×C₄×D₁₃	1	2	yes	—
292	C₄²	#6131	1	2	yes	—
293	C₁₃	C₄⋊C₄×D₈	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	26	4	no	—
295	C₂³	—	26	4	no	—
296	C₂³	—	26	4	no	—
297	C₂³	—	52	2	no	—
298	C₂³	—	52	2	no	—
299	C₂³	—	52	2	no	—
300	C₂³	—	52	2	no	—
301	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
302	D₄	—	13	8	no	—
303	D₄	—	26	4	no	—
304	D₄	—	26	4	no	—
305	D₄	—	26	4	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	13	4	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	26	4	no	—
315	C₂×C₄	—	26	2	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	52	2	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	—	52	16	no	—
321	C₂×C₄	—	104	16	no	—
322	C₂×C₄	#88147	1	2	yes	—
323	C₂×C₄	#88147	1	2	yes	—
324	C₂×C₄	D₈×D₁₃	1	2	yes	—
325	C₂×C₄	#88173	1	2	yes	—
326	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
328	C₂×C₄	#88251	1	1	yes	characteristic
329	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#6131	1	1	yes	characteristic
331	C₂×C₄	#88254	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	52	2	no	—
334	C₈	C₄⋊C₄×D₁₃	1	4	yes	complemented
335	C₂²	—	26	8	no	—
336	C₂²	—	52	4	no	—
337	C₂²	—	52	4	no	—
338	C₂²	—	52	4	no	—
339	C₂²	—	52	4	no	—
340	C₂²	—	52	4	no	—
341	C₂²	—	52	4	no	—
342	C₂²	C₂×#88147	1	1	yes	characteristic
343	C₂²	#11719422	1	1	yes	characteristic
344	C₂²	#11719432	1	1	yes	characteristic
345	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
346	C₂²	#11719462	1	1	yes	characteristic
347	C₂²	C₂×#88173	1	1	yes	characteristic
348	C₂²	#11719471	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	8	4	no	—
351	C₄	—	26	4	no	—
352	C₄	—	52	4	no	—
353	C₄	—	52	2	no	—
354	C₄	#11718533	1	4	yes	complemented
355	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
356	C₄	#11719209	1	1	yes	characteristic
357	C₄	#11719431	1	1	yes	characteristic
358	C₄	#11719435	1	1	yes	characteristic
359	C₂	—	52	8	no	—
360	C₂	#13117736	1	1	yes	characteristic
361	C₂	C₂×#11718533	1	1	yes	characteristic
362	C₂	#13123710	1	1	yes	characteristic
363	C₂	#13126511	1	1	yes	characteristic
364	C₂	#13126556	1	1	yes	characteristic
365	C₂	#13127688	1	1	yes	characteristic
366	C₂	#13127705	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1845	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1845	D₁₃
hypocenter	C₁₃	C₄⋊C₄×D₈
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
3. #13117736	C₂	semidirect
4. #13115567	C₂	semidirect
5. #13115080	C₂	semidirect
6. #13114833	C₂	semidirect
7. C₈×C₄⋊Dic₁₃	C₂	semidirect
8. C₁₃×#1845	C₂	semidirect
27. C₄×D₁₀₄	C₄	semidirect
54. C₄×D₅₂	C₂×C₄	semidirect
57. C₄×C₈×C₁₃	C₂×C₄	semidirect
60. #88061	C₂×C₄	semidirect
61. #88058	C₂×C₄	semidirect
102. C₄⋊Dic₁₃	D₈	semidirect
110. D₁₀₄	C₄⋊C₄	semidirect
116. #1845	D₁₃	semidirect
154. D₅₂	C₂×C₄⋊C₄	semidirect
158. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
197. C₄×C₁₃	C₄×D₈	semidirect
237. C₄×C₈	C₄×D₁₃	semidirect
293. C₁₃	C₄⋊C₄×D₈	semidirect
334. C₈	C₄⋊C₄×D₁₃	semidirect
354. C₄	#11718533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
C₂×#11718533	2
#13117736	2
#13115567	2
#13115080	2
#13114833	2
C₈×C₄⋊Dic₁₃	2
C₁₃×#1845	2
C₄⋊C₄×D₈	13