Subgroups — C₂²×#11718895 #16726354

Order 3328.

5326 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6099	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718895	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂²⋊#1055	C₂	1	4	yes	maximal, complemented
5	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	—	2	8	no	—
7	C₂²×C₁₃×#41	—	2	8	no	—
8	C₂²×C₁₃×#41	C₄	1	12	yes	—
9	C₂²×C₁₃×#41	C₂²	1	12	yes	—
10	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂³×D₄×C₁₃	C₄	1	2	yes	—
12	C₂³×D₄×C₁₃	C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6099	—	2	8	no	—
16	C₂²×#6099	C₂²	1	12	yes	complemented
17	C₂³×#1055	—	2	8	no	—
18	C₂³×#1055	C₂²	1	2	yes	—
19	C₂³×#1055	C₂²	1	2	yes	—
20	C₂³×#1055	C₂²	1	12	yes	complemented
21	#11718895	C₂²	1	64	yes	direct product factor
22	C₂×C₂²⋊#1055	C₂²	1	48	yes	complemented
23	C₂⁵×C₁₃	C₂×C₄	1	2	yes	—
24	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×#41	—	2	96	no	—
26	C₂×C₁₃×#41	—	2	48	no	—
27	C₂×C₁₃×#41	—	2	48	no	—
28	C₂×C₁₃×#41	C₂×C₄	1	48	yes	—
29	C₂×C₁₃×#41	C₂³	1	16	yes	—
30	C₂²×Q₈×C₁₃	—	2	8	no	—
31	C₂²×Q₈×C₁₃	C₂×C₄	1	12	yes	—
32	C₂²×D₄×C₁₃	—	2	16	no	—
33	C₂²×D₄×C₁₃	—	2	8	no	—
34	C₂²×D₄×C₁₃	C₂×C₄	1	24	yes	—
35	C₂²×D₄×C₁₃	C₂×C₄	1	12	yes	—
36	C₂³×C₄×C₁₃	—	2	8	no	—
37	C₂³×C₄×C₁₃	—	2	8	no	—
38	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₁₃	C₂×C₄	1	12	yes	—
40	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
42	C₂³×C₄×C₁₃	D₄	1	8	yes	—
43	C₂³×C₄×C₁₃	D₄	1	8	yes	—
44	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
45	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
46	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×#6099	—	2	48	no	—
48	C₂×#6099	C₂³	1	16	yes	complemented
49	C₂²×#1055	—	2	8	no	—
50	C₂²×#1055	—	2	48	no	—
51	C₂²×#1055	—	2	8	no	—
52	C₂²×#1055	C₂³	1	12	yes	—
53	C₂²×#1055	C₂³	1	12	yes	—
54	C₂²×#1055	C₂³	1	16	yes	complemented
55	C₂²⋊#1055	C₂³	1	64	yes	complemented
56	C₂²×#408	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	—	2	8	no	—
58	C₂⁴×C₁₃	—	2	8	no	—
59	C₂⁴×C₁₃	—	2	16	no	—
60	C₂⁴×C₁₃	C₂²⋊C₄	1	8	yes	—
61	C₂⁴×C₁₃	C₂²×C₄	1	12	yes	—
62	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
63	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₁₃×#41	—	2	192	no	—
65	C₁₃×#41	—	2	32	no	—
66	C₁₃×#41	—	2	32	no	—
67	C₂×Q₈×C₁₃	—	2	48	no	—
68	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
69	C₂×D₄×C₁₃	—	2	96	no	—
70	C₂×D₄×C₁₃	—	2	48	no	—
71	C₂×D₄×C₁₃	C₂²×C₄	1	32	yes	—
72	C₂×D₄×C₁₃	C₂²×C₄	1	16	yes	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	—	2	48	no	—
75	C₂²×C₄×C₁₃	—	2	24	no	—
76	C₂²×C₄×C₁₃	—	2	24	no	—
77	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
78	C₂²×C₄×C₁₃	C₂²⋊C₄	1	48	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
88	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
90	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
91	#6099	—	2	32	no	—
92	C₂×#1055	—	2	24	no	—
93	C₂×#1055	—	2	32	no	—
94	C₂×#1055	—	2	24	no	—
95	C₂×#1055	C₂⁴	1	8	yes	—
96	C₂×#1055	C₂⁴	1	8	yes	—
97	C₂³×#41	D₁₃	1	1	yes	characteristic
98	C₂³×M₁₆	—	13	1	no	—
99	C₂⁴×C₈	—	13	1	no	—
100	C₂×#408	—	13	24	no	—
101	C₂²×C₂²⋊C₈	—	13	4	no	—
102	C₂³×C₁₃	—	2	24	no	—
103	C₂³×C₁₃	—	2	32	no	—
104	C₂³×C₁₃	—	2	24	no	—
105	C₂³×C₁₃	—	2	48	no	—
106	C₂³×C₁₃	—	4	32	no	—
107	C₂³×C₁₃	C₂×C₂²⋊C₄	1	24	yes	—
108	C₂³×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂³×C₁₃	#131	1	8	yes	—
110	C₂³×C₁₃	C₂³×C₄	1	8	yes	—
111	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
112	C₂³×C₁₃	C₂³×C₄	1	3	yes	—
113	Q₈×C₁₃	—	2	32	no	—
114	D₄×C₁₃	—	2	64	no	—
115	D₄×C₁₃	—	2	32	no	—
116	C₂×C₄×C₁₃	—	2	24	no	—
117	C₂×C₄×C₁₃	—	2	48	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	12	yes	—
121	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	24	yes	—
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	48	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
130	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
131	#1055	—	2	8	no	—
132	#1055	—	2	8	no	—
133	C₂²×#41	—	2	8	no	—
134	C₂²×#41	—	2	8	no	—
135	C₂²×#41	Dic₁₃	1	12	yes	—
136	C₂²×#41	C₂×D₁₃	1	12	yes	—
137	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
138	C₂³×D₄	Dic₁₃	1	2	yes	—
139	C₂³×D₄	Dic₁₃	1	1	yes	characteristic
140	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
141	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
142	C₂²×M₁₆	—	13	12	no	—
143	C₂²×M₁₆	—	26	8	no	—
144	C₂³×C₈	—	13	12	no	—
145	C₂³×C₈	—	13	2	no	—
146	C₂³×C₈	—	13	2	no	—
147	C₂³×C₈	—	26	8	no	—
148	#408	—	13	64	no	—
149	C₂×C₂²⋊C₈	—	13	48	no	—
150	C₂²×C₁₃	—	2	48	no	—
151	C₂²×C₁₃	—	2	8	no	—
152	C₂²×C₁₃	—	2	8	no	—
153	C₂²×C₁₃	—	2	16	no	—
154	C₂²×C₁₃	—	4	48	no	—
155	C₂²×C₁₃	#408	1	16	yes	complemented
156	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	8	yes	—
157	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	6	yes	—
158	C₂²×C₁₃	C₂×#131	1	12	yes	—
159	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
160	C₄×C₁₃	—	2	8	no	—
161	C₄×C₁₃	C₂²×C₂²⋊C₄	1	6	yes	—
162	C₄×C₁₃	C₂²×C₂²⋊C₄	1	8	yes	—
163	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
164	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
165	C₂⁵	C₂×Dic₁₃	1	2	yes	—
166	C₂⁵	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂×#41	—	2	96	no	—
168	C₂×#41	—	2	48	no	—
169	C₂×#41	—	2	48	no	—
170	C₂×#41	C₂×Dic₁₃	1	48	yes	—
171	C₂×#41	C₂²×D₁₃	1	16	yes	—
172	C₂²×Q₈	—	2	8	no	—
173	C₂²×Q₈	C₂×Dic₁₃	1	12	yes	—
174	C₂²×D₄	—	2	16	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	C₂×Dic₁₃	1	24	yes	—
177	C₂²×D₄	C₂×Dic₁₃	1	12	yes	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	2	8	no	—
180	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
181	C₂³×C₄	C₂×Dic₁₃	1	12	yes	—
182	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
183	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
184	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
185	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
186	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
187	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
188	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
189	C₂×M₁₆	—	13	16	no	—
190	C₂×M₁₆	—	26	48	no	—
191	C₂²×C₈	—	13	12	no	—
192	C₂²×C₈	—	13	16	no	—
193	C₂²×C₈	—	13	12	no	—
194	C₂²×C₈	—	26	8	no	—
195	C₂²×C₈	—	26	48	no	—
196	C₂²×C₈	—	26	8	no	—
197	C₂²⋊C₈	—	13	64	no	—
198	C₂×C₁₃	—	2	8	no	—
199	C₂×C₁₃	—	4	8	no	—
200	C₂×C₁₃	C₂×#408	1	12	yes	complemented
201	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₁₃	C₂²×#131	1	2	yes	—
203	C₂⁴	—	2	8	no	—
204	C₂⁴	—	2	16	no	—
205	C₂⁴	—	2	8	no	—
206	C₂⁴	C₂²⋊Dic₁₃	1	8	yes	—
207	C₂⁴	C₂²×Dic₁₃	1	12	yes	—
208	C₂⁴	C₂²×Dic₁₃	1	6	yes	—
209	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
210	#41	—	2	192	no	—
211	#41	—	2	32	no	—
212	#41	—	2	32	no	—
213	C₂×Q₈	—	2	48	no	—
214	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
215	C₂×D₄	—	2	96	no	—
216	C₂×D₄	—	2	48	no	—
217	C₂×D₄	C₂²×Dic₁₃	1	32	yes	—
218	C₂×D₄	C₂²×Dic₁₃	1	16	yes	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	48	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	C₂²⋊Dic₁₃	1	8	yes	—
224	C₂²×C₄	C₂²⋊Dic₁₃	1	48	yes	—
225	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
226	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
227	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
228	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
229	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
234	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
235	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
236	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
237	M₁₆	—	26	32	no	—
238	C₂×C₈	—	13	8	no	—
239	C₂×C₈	—	13	8	no	—
240	C₂×C₈	—	26	24	no	—
241	C₂×C₈	—	26	32	no	—
242	C₂×C₈	—	26	24	no	—
243	C₁₃	C₂²×#408	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	2	24	no	—
245	C₂³	—	2	32	no	—
246	C₂³	—	2	48	no	—
247	C₂³	—	2	24	no	—
248	C₂³	—	4	32	no	—
249	C₂³	#88202	1	8	yes	—
250	C₂³	C₂×C₂²⋊Dic₁₃	1	24	yes	—
251	C₂³	C₂×C₂²⋊Dic₁₃	1	4	yes	—
252	C₂³	C₂³×Dic₁₃	1	8	yes	—
253	C₂³	C₂³×Dic₁₃	1	4	yes	—
254	C₂³	C₂³×Dic₁₃	1	3	yes	—
255	Q₈	—	2	32	no	—
256	D₄	—	2	64	no	—
257	D₄	—	2	32	no	—
258	C₂×C₄	—	2	24	no	—
259	C₂×C₄	—	2	48	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	12	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	24	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	48	yes	—
265	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
266	C₂×C₄	C₂³×Dic₁₃	1	3	yes	—
267	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
268	C₂×C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
269	C₂×C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
271	C₂×C₄	C₂²×C₂²⋊D₁₃	1	8	yes	—
272	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
273	C₈	—	26	8	no	—
274	C₈	—	26	8	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	2	48	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	16	no	—
279	C₂²	—	4	48	no	—
280	C₂²	#11718895	1	16	yes	direct product factor
281	C₂²	C₂×#88202	1	12	yes	—
282	C₂²	C₂²×C₂²⋊Dic₁₃	1	8	yes	—
283	C₂²	C₂²×C₂²⋊Dic₁₃	1	6	yes	—
284	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
285	C₄	—	2	8	no	—
286	C₄	C₂²×C₂²⋊Dic₁₃	1	6	yes	—
287	C₄	C₂²×C₂²⋊Dic₁₃	1	8	yes	—
288	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₂	—	2	8	no	—
291	C₂	—	4	8	no	—
292	C₂	C₂×#11718895	1	12	yes	direct product factor
293	C₂	C₂³×C₂²⋊Dic₁₃	1	1	yes	characteristic
294	C₂	C₂²×#88202	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting, prime radical	C₂³×C₁₃×#41	C₂
socle	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	C₂²×#408
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×#6099	C₂	semidirect
2. C₂⁴×#1055	C₂	semidirect
3. C₂×#11718895	C₂	direct (N × H)
4. C₂²×C₂²⋊#1055	C₂	semidirect
16. C₂²×#6099	C₂²	semidirect
20. C₂³×#1055	C₂²	semidirect
21. #11718895	C₂²	direct (N × H)
22. C₂×C₂²⋊#1055	C₂²	semidirect
48. C₂×#6099	C₂³	semidirect
54. C₂²×#1055	C₂³	semidirect
55. C₂²⋊#1055	C₂³	semidirect
155. C₂²×C₁₃	#408	semidirect
200. C₂×C₁₃	C₂×#408	semidirect
243. C₁₃	C₂²×#408	semidirect
280. C₂²	#11718895	direct (N × H)
292. C₂	C₂×#11718895	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6099	2
C₂⁴×#1055	2
C₂×#11718895	2
C₂²×C₂²⋊#1055	2
C₂³×C₁₃×#41	2
C₂²×#408	13

Subgroups — C22×#11718895 #16726354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718895 #16726354