Subgroups — #15881199

Order 3328.

1122 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718813	C₂	1	2	yes	maximal
2	D₄×C₄⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
3	#13115575	C₂	1	2	yes	maximal, complemented
4	#13115099	C₂	1	4	yes	maximal, complemented
5	#13114993	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114836	C₂	1	2	yes	maximal, complemented
7	#13114703	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂×D₄×Dic₁₃	C₂²	1	4	yes	—
12	C₂²×C₄⋊Dic₁₃	—	2	2	no	—
13	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#102	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88072	C₂²	1	4	yes	—
18	#11718873	—	2	2	no	—
19	#11718813	—	2	8	no	—
20	C₂×#88057	C₂²	1	2	yes	—
21	C₂×#88057	C₂²	1	2	yes	—
22	C₂×#88055	C₂²	1	2	yes	—
23	#11718646	C₂²	1	4	yes	—
24	#11718611	—	2	4	no	—
25	#11718596	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	#11718142	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	—	2	2	no	—
32	C₂³×Dic₁₃	—	2	4	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×D₈	D₄	1	4	yes	—
35	C₂×C₁₃×D₈	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	complemented
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
44	D₄×Dic₁₃	—	2	8	no	—
45	D₄×Dic₁₃	—	4	4	no	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	—	2	4	no	—
48	C₂×C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×C₄⋊Dic₁₃	—	4	8	no	—
50	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₁₃×#107	C₂×C₄	1	4	yes	complemented
55	C₁₃×#102	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
57	#88072	—	2	8	no	—
58	#88071	—	2	4	no	—
59	#88071	D₄	1	2	yes	—
60	#88057	—	2	4	no	—
61	#88057	—	4	2	no	—
62	#88055	—	2	4	no	—
63	C₄⋊#1055	—	2	2	no	—
64	#22670	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
71	C₂×D₄×C₁₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	—	4	4	no	—
75	C₂²×C₄×C₁₃	—	4	4	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	4	no	—
80	C₂²×Dic₁₃	—	4	8	no	—
81	C₂²×Dic₁₃	—	4	8	no	—
82	C₂²×Dic₁₃	—	8	4	no	—
83	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
84	C₁₃×D₈	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₁₃×C₂²⋊C₄	—	2	8	no	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	#41	1	2	yes	—
96	C₂²⋊Dic₁₃	—	4	8	no	—
97	C₄⋊Dic₁₃	—	2	4	no	—
98	C₄⋊Dic₁₃	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	4	8	no	—
101	C₄⋊Dic₁₃	—	4	4	no	—
102	C₄⋊Dic₁₃	—	8	2	no	—
103	C₄×Dic₁₃	—	2	2	no	—
104	C₄×Dic₁₃	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#442	—	13	2	no	—
109	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
110	D₄×C₄⋊C₄	—	13	2	no	—
111	#1943	—	13	2	no	—
112	#1864	—	13	4	no	—
113	#1844	—	13	1	no	—
114	#1802	—	13	2	no	—
115	#1770	—	13	1	no	—
116	C₂³×C₁₃	—	2	4	no	—
117	C₂³×C₁₃	—	2	4	no	—
118	C₂³×C₁₃	—	2	4	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	—	4	2	no	—
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	D₄×C₁₃	—	2	4	no	—
125	D₄×C₁₃	—	2	4	no	—
126	D₄×C₁₃	—	2	4	no	—
127	D₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	16	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	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	4	no	—
143	C₂×Dic₁₃	—	2	4	no	—
144	C₂×Dic₁₃	—	2	4	no	—
145	C₂×Dic₁₃	—	4	4	no	—
146	C₂×Dic₁₃	—	4	4	no	—
147	C₂×Dic₁₃	—	4	4	no	—
148	C₂×Dic₁₃	—	8	16	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	complemented
151	#1055	—	4	2	no	—
152	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₄×D₄	—	13	4	no	—
154	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
155	C₂²×C₄⋊C₄	—	26	2	no	—
156	#442	—	26	8	no	—
157	C₄×D₈	Dic₁₃	1	8	yes	complemented
158	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
159	C₂×#106	—	13	2	no	—
160	C₂×#106	—	13	2	no	—
161	C₂×C₄⋊C₈	—	13	1	no	—
162	C₂×#102	—	13	4	no	—
163	C₂×#102	C₂×D₁₃	1	2	yes	—
164	C₂×#97	—	13	2	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#389	—	13	4	no	—
167	#385	—	26	4	no	—
168	#384	—	13	2	no	—
169	#380	—	26	2	no	—
170	C₄×C₄⋊C₄	—	13	1	no	—
171	#340	—	13	4	no	—
172	#336	—	13	2	no	—
173	C₂²×C₁₃	—	2	8	no	—
174	C₂²×C₁₃	—	4	4	no	—
175	C₂²×C₁₃	—	4	4	no	—
176	C₂²×C₁₃	—	4	4	no	—
177	C₂²×C₁₃	—	4	4	no	—
178	C₂²×C₁₃	—	4	4	no	—
179	C₂²×C₁₃	—	4	4	no	—
180	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
181	C₂²×C₁₃	D₄²	1	1	yes	characteristic
182	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
183	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
184	C₂²×C₁₃	#577	1	2	yes	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	4	4	no	—
187	C₄×C₁₃	#442	1	4	yes	complemented
188	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₁₃	#520	1	1	yes	characteristic
190	C₄×C₁₃	#547	1	1	yes	characteristic
191	C₄×C₁₃	#554	1	1	yes	characteristic
192	Dic₁₃	—	4	4	no	—
193	Dic₁₃	—	8	4	no	—
194	C₂²×D₄	C₂²×D₁₃	1	2	yes	—
195	C₂³×C₄	—	2	2	no	—
196	C₂³×C₄	—	26	4	no	—
197	C₂×D₈	Dic₂₆	1	4	yes	—
198	C₂×D₈	D₅₂	1	4	yes	—
199	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	26	8	no	—
205	C₄×D₄	—	52	4	no	—
206	C₄×D₄	C₂×Dic₁₃	1	8	yes	complemented
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	13	4	no	—
209	C₂×C₄⋊C₄	—	26	4	no	—
210	C₂×C₄⋊C₄	—	26	2	no	—
211	C₂×C₄⋊C₄	—	26	2	no	—
212	C₂×C₄⋊C₄	—	52	8	no	—
213	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	—	26	4	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
218	#107	C₂×Dic₁₃	1	4	yes	complemented
219	#106	—	26	4	no	—
220	#106	—	52	2	no	—
221	C₄⋊C₈	—	26	2	no	—
222	#102	—	26	8	no	—
223	#102	C₂×Dic₁₃	1	8	yes	complemented
224	#97	—	26	4	no	—
225	C₄×C₈	C₂×Dic₁₃	1	4	yes	complemented
226	#95	—	13	2	no	—
227	#95	—	26	4	no	—
228	C₂×C₁₃	—	4	8	no	—
229	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
231	C₂×C₁₃	#2952	1	1	yes	characteristic
232	C₂×C₁₃	#3272	1	2	yes	—
233	C₂×C₁₃	#3384	1	2	yes	—
234	C₂⁴	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	C₂×Dic₂₆	1	4	yes	—
239	C₂×D₄	C₂×D₅₂	1	4	yes	—
240	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	13	4	no	—
246	C₂²×C₄	—	26	4	no	—
247	C₂²×C₄	—	52	8	no	—
248	C₂²×C₄	—	52	8	no	—
249	C₂²×C₄	—	104	4	no	—
250	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
252	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
253	D₈	C₄⋊Dic₁₃	1	16	yes	complemented
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
260	C₂×C₈	C₂×D₅₂	1	2	yes	—
261	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
262	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	26	4	no	—
265	C₄⋊C₄	—	26	4	no	—
266	C₄⋊C₄	—	26	4	no	—
267	C₄⋊C₄	—	52	8	no	—
268	C₄⋊C₄	—	52	4	no	—
269	C₄⋊C₄	—	104	2	no	—
270	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
271	C₂²⋊C₄	—	2	8	no	—
272	C₂²⋊C₄	—	52	8	no	—
273	C₄²	—	26	2	no	—
274	C₄²	—	26	2	no	—
275	C₄²	#6129	1	2	yes	—
276	C₄²	C₂²×Dic₁₃	1	2	yes	—
277	C₁₃	#22670	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	2	4	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	4	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
286	D₄	—	2	4	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	4	no	—
289	D₄	C₂×C₄⋊Dic₁₃	1	8	yes	complemented
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	26	4	no	—
296	C₂×C₄	—	26	4	no	—
297	C₂×C₄	—	26	4	no	—
298	C₂×C₄	—	52	4	no	—
299	C₂×C₄	—	52	4	no	—
300	C₂×C₄	—	52	4	no	—
301	C₂×C₄	—	104	16	no	—
302	C₂×C₄	#88165	1	4	yes	—
303	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
304	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
305	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
307	C₂×C₄	C₂×#6129	1	1	yes	characteristic
308	C₂×C₄	#88256	1	1	yes	characteristic
309	C₂×C₄	#88257	1	1	yes	characteristic
310	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	52	2	no	—
313	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	complemented
314	C₂²	—	2	8	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	4	4	no	—
321	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
322	C₂²	D₄×D₅₂	1	1	yes	characteristic
323	C₂²	C₂×#88165	1	2	yes	—
324	C₂²	#11719475	1	2	yes	—
325	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	52	4	no	—
329	C₄	—	104	4	no	—
330	C₄	#11718813	1	4	yes	complemented
331	C₄	#11719248	1	1	yes	characteristic
332	C₄	#11719260	1	1	yes	characteristic
333	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
334	C₄	#11719532	1	1	yes	characteristic
335	C₂	—	4	8	no	—
336	C₂	D₄×C₄⋊Dic₁₃	1	1	yes	characteristic
337	C₂	C₂×#11718813	1	1	yes	characteristic
338	C₂	#13123720	1	1	yes	characteristic
339	C₂	#13126437	1	2	yes	—
340	C₂	#13127733	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×C₁₃×D₈	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	#22670
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×C₄⋊Dic₁₃	C₂	semidirect
3. #13115575	C₂	semidirect
4. #13115099	C₂	semidirect
5. #13114993	C₂	semidirect
6. #13114836	C₂	semidirect
7. #13114703	C₂	semidirect
13. C₄×C₁₃×D₈	C₄	semidirect
39. C₄×D₄×C₁₃	C₂×C₄	semidirect
54. C₁₃×#107	C₂×C₄	semidirect
55. C₁₃×#102	C₂×C₄	semidirect
56. C₄×C₈×C₁₃	C₂×C₄	semidirect
84. C₁₃×D₈	C₄⋊C₄	semidirect
127. D₄×C₁₃	C₂×C₄⋊C₄	semidirect
150. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
157. C₄×D₈	Dic₁₃	semidirect
187. C₄×C₁₃	#442	semidirect
206. C₄×D₄	C₂×Dic₁₃	semidirect
218. #107	C₂×Dic₁₃	semidirect
223. #102	C₂×Dic₁₃	semidirect
225. C₄×C₈	C₂×Dic₁₃	semidirect
253. D₈	C₄⋊Dic₁₃	semidirect
277. C₁₃	#22670	semidirect
289. D₄	C₂×C₄⋊Dic₁₃	semidirect
313. C₈	C₂×C₄⋊Dic₁₃	semidirect
330. C₄	#11718813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718813	2
D₄×C₄⋊Dic₁₃	2
#13115575	2
#13115099	2
#13114993	2
#13114836	2
#13114703	2
C₂×C₄×C₁₃×D₈	2
#22670	13