Subgroups — #15867698

Order 3328.

1030 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	#13115555	C₂	1	4	yes	maximal, complemented
4	#13115016	C₂	1	4	yes	maximal
5	C₄×C₄⋊#1055	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	2	yes	—
11	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₁₃×#376	C₄	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
18	#11718586	—	2	4	no	—
19	C₂×C₄⋊#1055	C₂²	1	8	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄×#1055	—	2	4	no	—
22	C₂×C₄×#1055	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718116	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#117	—	2	4	no	—
30	C₁₃×#117	C₈	1	8	yes	complemented
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	D₄	1	4	yes	—
41	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×#1055	—	2	4	no	—
46	C₂²×#1055	—	2	8	no	—
47	C₂²×#1055	C₂³	1	4	yes	—
48	C₂²×#1055	C₂³	1	4	yes	—
49	C₁₃×#95	C₂×C₄	1	4	yes	—
50	C₁₃×#95	C₂×C₄	1	4	yes	—
51	C₂²⋊#1055	—	2	4	no	—
52	C₄⋊#1055	—	2	4	no	—
53	C₄⋊#1055	—	2	16	no	—
54	C₄×#1055	—	2	4	no	—
55	C₄×#1055	—	2	4	no	—
56	C₄×#1055	—	2	4	no	—
57	#21349	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	complemented
77	C₁₃×C₂²⋊C₄	—	2	4	no	—
78	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
79	C₄²×C₁₃	—	2	4	no	—
80	C₄²×C₁₃	—	2	8	no	—
81	C₄²×C₁₃	—	2	8	no	—
82	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
83	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
84	C₄²×C₁₃	C₂×C₈	1	8	yes	complemented
85	C₄²×C₁₃	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
89	C₂×#1055	—	2	16	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	8	no	—
94	C₂×#1055	#41	1	8	yes	—
95	C₂×#431	—	13	1	no	—
96	C₂²×C₄×C₈	—	13	1	no	—
97	C₄×#117	D₁₃	1	1	yes	characteristic
98	#1939	—	13	4	no	—
99	#1850	—	13	4	no	—
100	C₄×C₄⋊C₈	—	13	4	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
105	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
106	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	8	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	C₄⋊C₈	1	16	yes	complemented
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
124	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₁₃	#131	1	4	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
132	#1055	—	2	8	no	—
133	#1055	—	4	4	no	—
134	C₂³×C₈	—	13	2	no	—
135	C₂×#117	Dic₁₃	1	1	yes	characteristic
136	C₂×#117	Dic₁₃	1	1	yes	characteristic
137	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
138	#431	—	26	4	no	—
139	C₂×C₄⋊C₈	—	13	8	no	—
140	C₂×C₄⋊C₈	—	13	2	no	—
141	C₂×C₂²⋊C₈	—	13	2	no	—
142	C₂×C₄×C₈	—	13	2	no	—
143	C₂×C₄×C₈	—	13	2	no	—
144	C₂×C₄×C₈	—	13	2	no	—
145	C₂×C₄×C₈	—	26	4	no	—
146	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
147	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
148	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
149	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
150	#376	Dic₁₃	1	2	yes	—
151	C₄³	C₂×D₁₃	1	2	yes	—
152	#336	—	13	8	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	4	no	—
155	C₂²×C₁₃	—	2	4	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
158	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃	C₄×#41	1	2	yes	—
160	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
161	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#131	1	2	yes	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	8	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	4	no	—
167	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
168	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
169	C₄×C₁₃	#424	1	4	yes	—
170	C₄×C₁₃	#525	1	2	yes	—
171	C₄×C₁₃	#533	1	2	yes	—
172	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
173	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	C₂²×C₈	—	13	4	no	—
176	C₂²×C₈	—	13	4	no	—
177	C₂²×C₈	—	26	8	no	—
178	C₂²×C₈	—	26	4	no	—
179	#117	—	2	4	no	—
180	#117	#1055	1	8	yes	complemented
181	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
185	C₂×C₄²	Dic₂₆	1	4	yes	—
186	C₂×C₄²	D₅₂	1	4	yes	—
187	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
188	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
189	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
190	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
193	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊C₈	—	26	16	no	—
196	C₄⋊C₈	—	26	4	no	—
197	C₂²⋊C₈	—	26	4	no	—
198	C₄×C₈	—	26	4	no	—
199	C₄×C₈	—	26	4	no	—
200	C₄×C₈	—	26	4	no	—
201	#95	C₂×Dic₁₃	1	4	yes	—
202	#95	C₂×Dic₁₃	1	4	yes	—
203	C₂×C₁₃	—	2	4	no	—
204	C₂×C₁₃	#2296	1	1	yes	characteristic
205	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
207	C₂×C₁₃	C₈×#41	1	2	yes	—
208	C₂×C₁₃	#2965	1	2	yes	—
209	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×#1055	1	4	yes	—
213	C₂²×C₄	#6099	1	4	yes	—
214	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
215	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
216	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
217	C₂²×C₄	C₂×D₅₂	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
219	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂×C₈	—	13	8	no	—
227	C₂×C₈	—	26	16	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	—	26	8	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	C₂×#1055	1	8	yes	complemented
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	C₂×#1055	1	8	yes	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	8	no	—
239	C₄²	C₂×#1055	1	8	yes	complemented
240	C₄²	C₄⋊Dic₁₃	1	4	yes	—
241	C₄²	C₄⋊Dic₁₃	1	4	yes	—
242	C₄²	C₂×Dic₂₆	1	2	yes	—
243	C₄²	C₂×Dic₂₆	1	2	yes	—
244	C₄²	C₂×D₅₂	1	2	yes	—
245	C₄²	C₂×D₅₂	1	2	yes	—
246	C₁₃	#21349	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂²×#1055	1	2	yes	—
250	C₂³	C₂×#6099	1	2	yes	—
251	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
252	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	C₄⋊#1055	1	16	yes	complemented
264	C₂×C₄	C₂²×#1055	1	8	yes	—
265	C₂×C₄	C₂²×#1055	1	2	yes	—
266	C₂×C₄	C₂²×#1055	1	2	yes	—
267	C₂×C₄	C₂×#6099	1	2	yes	—
268	C₂×C₄	C₂×#6099	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
270	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
274	C₂×C₄	#88202	1	4	yes	—
275	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
277	C₂×C₄	#41×D₁₃	1	4	yes	—
278	C₈	—	26	8	no	—
279	C₈	—	52	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₂×C₄⋊#1055	1	4	yes	—
285	C₂²	C₂³×#1055	1	1	yes	characteristic
286	C₂²	C₂²×#6099	1	1	yes	characteristic
287	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
288	C₂²	C₂×#88202	1	2	yes	—
289	C₂²	#41×Dic₁₃	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	8	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₄⋊#1055	1	4	yes	—
295	C₄	C₂×C₄⋊#1055	1	4	yes	—
296	C₄	#11718660	1	4	yes	—
297	C₄	#11719222	1	2	yes	—
298	C₄	#11719229	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	#13117320	1	1	yes	characteristic
301	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
302	C₂	C₂×#11718660	1	1	yes	characteristic
303	C₂	#41×#1055	1	2	yes	—
304	C₂	#13123803	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#21349
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #13115555	C₂	semidirect
5. C₄×C₄⋊#1055	C₂	semidirect
30. C₁₃×#117	C₈	semidirect
76. C₁₃×C₄⋊C₄	C₂×C₈	semidirect
84. C₄²×C₁₃	C₂×C₈	semidirect
117. C₂×C₄×C₁₃	C₄⋊C₈	semidirect
180. #117	#1055	semidirect
233. C₄⋊C₄	C₂×#1055	semidirect
239. C₄²	C₂×#1055	semidirect
246. C₁₃	#21349	semidirect
263. C₂×C₄	C₄⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
C₂²×C₄×#1055	2
#13115555	2
#13115016	2
C₄×C₄⋊#1055	2
C₄×C₁₃×#117	2
#21349	13