Subgroups — #15861808

Order 3328.

1222 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#1055	C₂	1	2	yes	maximal
2	C₄⋊C₄×#1055	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₄×#1055	C₂	1	4	yes	maximal, complemented
5	C₄×C₂²⋊#1055	C₂	1	2	yes	maximal, complemented
6	#13114582	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718116	C₂	1	2	yes	maximal, complemented
8	C₄²×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂³×#1055	C₂²	1	4	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
16	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
19	D₄×#1055	C₄	1	16	yes	complemented
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄×#1055	C₂²	1	2	yes	—
22	C₂×C₄×#1055	C₂²	1	4	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	#11718116	C₂²	1	2	yes	—
25	#11718116	C₂²	1	4	yes	—
26	#11718116	C₂²	1	1	yes	characteristic
27	#11718116	C₂²	1	1	yes	characteristic
28	#11718116	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
32	C₄×D₄×C₁₃	C₈	1	16	yes	complemented
33	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	—	4	4	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×#1055	—	4	4	no	—
47	C₂²×#1055	—	4	4	no	—
48	C₂²×#1055	C₂×C₄	1	16	yes	complemented
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	4	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₁₃×#95	C₂×C₄	1	2	yes	—
53	C₁₃×#95	C₂×C₄	1	4	yes	—
54	C₂²⋊#1055	C₂×C₄	1	16	yes	complemented
55	C₄⋊#1055	C₂×C₄	1	8	yes	complemented
56	C₄×#1055	—	4	2	no	—
57	C₄×#1055	—	4	4	no	—
58	C₄×#1055	C₂×C₄	1	8	yes	complemented
59	#20555	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×D₄×C₁₃	C₄²	1	4	yes	—
62	C₂×D₄×C₁₃	C₂×C₈	1	8	yes	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	4	4	no	—
67	C₂²×C₄×C₁₃	—	4	4	no	—
68	C₂²×C₄×C₁₃	C₂×C₈	1	16	yes	complemented
69	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
78	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	complemented
79	C₁₃×C₂²⋊C₄	C₄²	1	8	yes	—
80	C₁₃×C₂²⋊C₄	C₂×C₈	1	16	yes	complemented
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	4	4	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₂×C₈	1	8	yes	complemented
88	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂×#1055	—	2	2	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	4	16	no	—
96	C₂×#1055	C₂²×C₄	1	4	yes	—
97	C₂×#1055	C₂²×C₄	1	8	yes	—
98	C₂×#1055	C₂²×C₄	1	4	yes	—
99	C₂×C₈×D₄	—	13	2	no	—
100	C₄²×D₄	D₁₃	1	1	yes	characteristic
101	C₈×C₄⋊C₄	—	13	2	no	—
102	C₄×C₄⋊C₈	—	13	1	no	—
103	C₈×C₂²⋊C₄	—	13	4	no	—
104	C₄×C₂²⋊C₈	—	13	2	no	—
105	#1744	—	13	1	no	—
106	C₂×#336	—	13	2	no	—
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	4	no	—
110	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
111	C₂³×C₁₃	C₂²×C₈	1	8	yes	—
112	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	D₄×C₁₃	C₄×C₈	1	16	yes	complemented
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	8	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	4	16	no	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
131	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
132	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
134	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
137	#1055	—	2	4	no	—
138	#1055	—	2	8	no	—
139	#1055	—	2	4	no	—
140	C₂³×C₈	—	13	4	no	—
141	C₂×C₄×D₄	Dic₁₃	1	2	yes	—
142	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
143	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
144	C₈×D₄	—	13	16	no	—
145	C₂×C₄⋊C₈	—	13	2	no	—
146	C₂×C₂²⋊C₈	—	13	4	no	—
147	C₂×C₄×C₈	—	13	2	no	—
148	C₂×C₄×C₈	—	13	4	no	—
149	C₂×C₄×C₈	—	13	2	no	—
150	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
151	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
152	C₄×C₂²⋊C₄	Dic₁₃	1	4	yes	—
153	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
154	C₄³	C₂×D₁₃	1	1	yes	characteristic
155	#336	—	13	1	no	—
156	#336	—	13	2	no	—
157	#336	—	13	4	no	—
158	#336	—	13	1	no	—
159	#336	—	13	2	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	8	no	—
162	C₂²×C₁₃	—	2	8	no	—
163	C₂²×C₁₃	—	2	4	no	—
164	C₂²×C₁₃	C₂×C₄×C₈	1	8	yes	—
165	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₁₃	#518	1	1	yes	characteristic
167	C₂²×C₁₃	#520	1	1	yes	characteristic
168	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
169	C₂²×C₁₃	#568	1	2	yes	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	2	8	no	—
173	C₄×C₁₃	—	2	4	no	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
176	C₄×C₁₃	#518	1	2	yes	—
177	C₄×C₁₃	#520	1	2	yes	—
178	C₂²×D₄	C₂×Dic₁₃	1	1	yes	characteristic
179	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
180	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	13	16	no	—
185	C₂²×C₈	—	52	4	no	—
186	C₂²×C₈	—	52	4	no	—
187	C₄×D₄	#1055	1	16	yes	complemented
188	C₄×D₄	C₄×D₁₃	1	8	yes	—
189	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
190	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
192	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
193	C₂×C₄²	—	4	4	no	—
194	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
195	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
196	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
197	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
200	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
201	C₄⋊C₈	—	13	8	no	—
202	C₂²⋊C₈	—	13	16	no	—
203	C₄×C₈	—	13	8	no	—
204	C₄×C₈	—	52	4	no	—
205	C₄×C₈	—	52	2	no	—
206	#95	C₂×Dic₁₃	1	2	yes	—
207	#95	C₂×Dic₁₃	1	4	yes	—
208	C₂×C₁₃	—	2	8	no	—
209	C₂×C₁₃	#2275	1	1	yes	characteristic
210	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
211	C₂×C₁₃	#2875	1	1	yes	characteristic
212	C₂×C₁₃	#2972	1	2	yes	—
213	C₂×C₁₃	#2988	1	2	yes	—
214	C₂⁴	C₂²×Dic₁₃	1	2	yes	—
215	C₂×D₄	C₂×#1055	1	8	yes	—
216	C₂×D₄	C₄×Dic₁₃	1	4	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	C₂×#1055	1	16	yes	complemented
223	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
224	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
231	C₂×C₈	—	13	4	no	—
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	13	8	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	26	2	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	2	no	—
239	C₂×C₈	—	26	2	no	—
240	C₂×C₈	—	52	16	no	—
241	C₄⋊C₄	C₂×#1055	1	8	yes	complemented
242	C₄⋊C₄	C₄×Dic₁₃	1	4	yes	—
243	C₂²⋊C₄	C₂×#1055	1	16	yes	complemented
244	C₂²⋊C₄	C₄×Dic₁₃	1	8	yes	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	4	4	no	—
250	C₄²	—	4	2	no	—
251	C₄²	C₂×#1055	1	8	yes	complemented
252	C₄²	C₂×C₄×D₁₃	1	4	yes	—
253	C₁₃	#20555	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂²×#1055	1	8	yes	—
258	C₂³	C₂×C₄×Dic₁₃	1	4	yes	—
259	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
260	D₄	C₄×#1055	1	16	yes	complemented
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	C₂²×#1055	1	4	yes	—
274	C₂×C₄	C₂²×#1055	1	4	yes	—
275	C₂×C₄	C₂²×#1055	1	8	yes	—
276	C₂×C₄	C₂²×#1055	1	4	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
278	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
279	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
281	C₂×C₄	#88251	1	2	yes	—
282	C₂×C₄	#88256	1	1	yes	characteristic
283	C₂×C₄	#88257	1	1	yes	characteristic
284	C₈	—	26	8	no	—
285	C₈	—	26	4	no	—
286	C₈	—	26	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×C₄×#1055	1	8	yes	—
292	C₂²	C₂³×#1055	1	2	yes	—
293	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
294	C₂²	#11719503	1	1	yes	characteristic
295	C₂²	#11719524	1	2	yes	—
296	C₂²	#11719532	1	1	yes	characteristic
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	8	no	—
300	C₄	—	2	4	no	—
301	C₄	—	2	2	no	—
302	C₄	C₂×C₄×#1055	1	4	yes	—
303	C₄	#11719250	1	2	yes	—
304	C₄	#11719251	1	2	yes	—
305	C₂	—	2	8	no	—
306	C₂	#13117197	1	1	yes	characteristic
307	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
308	C₂	#13123264	1	1	yes	characteristic
309	C₂	#13123860	1	2	yes	—
310	C₂	#13123947	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₄²×D₄×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×D₄×C₁₃	C₂×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×D₄	D₁₃
hypocenter	C₁₃	#20555
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×#1055	C₂	semidirect
3. C₄×C₄⋊#1055	C₂	semidirect
4. C₂²⋊C₄×#1055	C₂	semidirect
5. C₄×C₂²⋊#1055	C₂	semidirect
6. #13114582	C₂	semidirect
7. C₂×#11718116	C₂	semidirect
19. D₄×#1055	C₄	semidirect
32. C₄×D₄×C₁₃	C₈	semidirect
48. C₂²×#1055	C₂×C₄	semidirect
54. C₂²⋊#1055	C₂×C₄	semidirect
55. C₄⋊#1055	C₂×C₄	semidirect
58. C₄×#1055	C₂×C₄	semidirect
68. C₂²×C₄×C₁₃	C₂×C₈	semidirect
78. C₁₃×C₄⋊C₄	C₂×C₈	semidirect
80. C₁₃×C₂²⋊C₄	C₂×C₈	semidirect
87. C₄²×C₁₃	C₂×C₈	semidirect
113. D₄×C₁₃	C₄×C₈	semidirect
187. C₄×D₄	#1055	semidirect
222. C₂²×C₄	C₂×#1055	semidirect
241. C₄⋊C₄	C₂×#1055	semidirect
243. C₂²⋊C₄	C₂×#1055	semidirect
251. C₄²	C₂×#1055	semidirect
253. C₁₃	#20555	semidirect
260. D₄	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×D₄×#1055	2
C₄⋊C₄×#1055	2
C₄×C₄⋊#1055	2
C₂²⋊C₄×#1055	2
C₄×C₂²⋊#1055	2
#13114582	2
C₂×#11718116	2
C₄²×D₄×C₁₃	2
#20555	13