Subgroups — C₂³×C₁₁×#104 #14637283

Order 2816.

5758 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₁×#104	C₂	1	28	yes	maximal, direct product factor
5	C₂²×C₁₁×#41	—	2	7	no	—
6	C₂²×C₁₁×#41	—	2	7	no	—
7	C₂²×C₁₁×#41	C₄	1	14	yes	complemented
8	C₂²×C₁₁×#41	C₂²	1	14	yes	—
9	C₂³×Q₈×C₁₁	C₄	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₁₁	—	2	1	no	—
11	C₂³×D₄×C₁₁	C₄	1	1	yes	characteristic, complemented
12	C₂⁴×C₄×C₁₁	—	2	1	no	—
13	C₂⁴×C₄×C₁₁	—	2	1	no	—
14	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁×M₁₆	—	2	7	no	—
16	C₂²×C₁₁×M₁₆	C₂²	1	14	yes	complemented
17	C₂³×C₈×C₁₁	—	2	1	no	—
18	C₂²×C₄²×C₁₁	—	2	7	no	—
19	C₂²×C₄²×C₁₁	C₂²	1	14	yes	complemented
20	C₂×C₁₁×#104	C₂²	1	112	yes	direct product factor
21	C₂⁵×C₁₁	—	2	1	no	—
22	C₂⁵×C₁₁	D₄	1	1	yes	characteristic
23	C₂×C₁₁×#41	—	2	84	no	—
24	C₂×C₁₁×#41	—	2	42	no	—
25	C₂×C₁₁×#41	—	2	42	no	—
26	C₂×C₁₁×#41	C₂×C₄	1	84	yes	complemented
27	C₂×C₁₁×#41	C₂³	1	28	yes	—
28	C₂²×Q₈×C₁₁	—	2	7	no	—
29	C₂²×Q₈×C₁₁	C₂×C₄	1	14	yes	complemented
30	C₂²×D₄×C₁₁	—	2	14	no	—
31	C₂²×D₄×C₁₁	—	2	14	no	—
32	C₂²×D₄×C₁₁	—	2	7	no	—
33	C₂²×D₄×C₁₁	C₂×C₄	1	14	yes	complemented
34	C₂³×C₄×C₁₁	—	2	16	no	—
35	C₂³×C₄×C₁₁	—	2	1	no	—
36	C₂³×C₄×C₁₁	—	2	14	no	—
37	C₂³×C₄×C₁₁	—	2	14	no	—
38	C₂³×C₄×C₁₁	—	2	14	no	—
39	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₁₁	C₂×C₄	1	7	yes	—
41	C₂³×C₄×C₁₁	D₄	1	7	yes	—
42	C₂³×C₄×C₁₁	D₄	1	7	yes	—
43	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₁₁	C₂³	1	7	yes	—
45	C₂×C₁₁×M₁₆	—	2	42	no	—
46	C₂×C₁₁×M₁₆	C₂³	1	28	yes	complemented
47	C₂²×C₈×C₁₁	—	2	14	no	—
48	C₂×C₄²×C₁₁	—	2	42	no	—
49	C₂×C₄²×C₁₁	C₂³	1	28	yes	complemented
50	C₁₁×#104	C₂³	1	64	yes	direct product factor
51	C₂³×#104	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₁	—	2	8	no	—
53	C₂⁴×C₁₁	—	2	14	no	—
54	C₂⁴×C₁₁	—	4	8	no	—
55	C₂⁴×C₁₁	C₂²⋊C₄	1	7	yes	—
56	C₂⁴×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂⁴×C₁₁	C₂×D₄	1	7	yes	—
58	C₁₁×#41	—	2	168	no	—
59	C₁₁×#41	—	2	28	no	—
60	C₁₁×#41	—	2	28	no	—
61	C₁₁×#41	C₂²×C₄	1	56	yes	complemented
62	C₁₁×#41	C₂⁴	1	8	yes	—
63	C₂×Q₈×C₁₁	—	2	42	no	—
64	C₂×Q₈×C₁₁	C₂²×C₄	1	28	yes	complemented
65	C₂×D₄×C₁₁	—	2	84	no	—
66	C₂×D₄×C₁₁	—	2	28	no	—
67	C₂×D₄×C₁₁	—	2	42	no	—
68	C₂×D₄×C₁₁	C₂²×C₄	1	28	yes	complemented
69	C₂²×C₄×C₁₁	—	2	112	no	—
70	C₂²×C₄×C₁₁	—	2	14	no	—
71	C₂²×C₄×C₁₁	—	2	28	no	—
72	C₂²×C₄×C₁₁	—	2	84	no	—
73	C₂²×C₄×C₁₁	—	2	28	no	—
74	C₂²×C₄×C₁₁	C₂²⋊C₄	1	7	yes	—
75	C₂²×C₄×C₁₁	C₂²⋊C₄	1	7	yes	—
76	C₂²×C₄×C₁₁	C₂²⋊C₄	1	42	yes	—
77	C₂²×C₄×C₁₁	C₂²×C₄	1	7	yes	—
78	C₂²×C₄×C₁₁	C₂²×C₄	1	21	yes	—
79	C₂²×C₄×C₁₁	C₂×D₄	1	21	yes	—
80	C₂²×C₄×C₁₁	C₂×D₄	1	7	yes	—
81	C₂²×C₄×C₁₁	C₂×D₄	1	21	yes	—
82	C₂²×C₄×C₁₁	C₂⁴	1	7	yes	—
83	C₁₁×M₁₆	—	2	28	no	—
84	C₁₁×M₁₆	C₂⁴	1	8	yes	complemented
85	C₂×C₈×C₁₁	—	2	28	no	—
86	C₄²×C₁₁	—	2	28	no	—
87	C₄²×C₁₁	C₂⁴	1	8	yes	complemented
88	C₂³×#41	C₂×C₁₁	1	1	yes	characteristic
89	C₂³×M₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
90	C₂³×C₄²	C₂×C₁₁	1	1	yes	characteristic, complemented
91	C₂²×#104	C₂×C₁₁	1	28	yes	direct product factor
92	C₂³×C₁₁	—	2	28	no	—
93	C₂³×C₁₁	—	2	28	no	—
94	C₂³×C₁₁	—	2	28	no	—
95	C₂³×C₁₁	—	4	56	no	—
96	C₂³×C₁₁	#104	1	8	yes	direct product factor
97	C₂³×C₁₁	C₂×C₂²⋊C₄	1	21	yes	—
98	C₂³×C₁₁	C₂×C₂²⋊C₄	1	7	yes	—
99	C₂³×C₁₁	C₂²×D₄	1	7	yes	—
100	Q₈×C₁₁	—	2	28	no	—
101	Q₈×C₁₁	C₂³×C₄	1	8	yes	complemented
102	D₄×C₁₁	—	2	56	no	—
103	D₄×C₁₁	—	2	8	no	—
104	D₄×C₁₁	—	2	28	no	—
105	D₄×C₁₁	C₂³×C₄	1	8	yes	complemented
106	C₂×C₄×C₁₁	—	2	112	no	—
107	C₂×C₄×C₁₁	—	2	28	no	—
108	C₂×C₄×C₁₁	—	2	8	no	—
109	C₂×C₄×C₁₁	—	2	56	no	—
110	C₂×C₄×C₁₁	—	2	8	no	—
111	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	21	yes	—
112	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	21	yes	—
113	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	42	yes	—
114	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
115	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
116	C₂×C₄×C₁₁	C₂²×D₄	1	7	yes	—
117	C₂×C₄×C₁₁	C₂²×D₄	1	7	yes	—
118	C₂×C₄×C₁₁	C₂²×D₄	1	7	yes	—
119	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
120	C₈×C₁₁	—	2	8	no	—
121	C₂²×#41	—	2	7	no	—
122	C₂²×#41	—	2	7	no	—
123	C₂²×#41	C₄×C₁₁	1	14	yes	complemented
124	C₂²×#41	C₂²×C₁₁	1	14	yes	—
125	C₂³×Q₈	C₄×C₁₁	1	1	yes	characteristic, complemented
126	C₂³×D₄	—	2	1	no	—
127	C₂³×D₄	C₄×C₁₁	1	1	yes	characteristic, complemented
128	C₂⁴×C₄	—	2	1	no	—
129	C₂⁴×C₄	—	2	1	no	—
130	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
131	C₂²×M₁₆	—	2	7	no	—
132	C₂²×M₁₆	C₂²×C₁₁	1	14	yes	complemented
133	C₂³×C₈	—	2	1	no	—
134	C₂²×C₄²	—	2	7	no	—
135	C₂²×C₄²	C₂²×C₁₁	1	14	yes	complemented
136	C₂×#104	C₂²×C₁₁	1	112	yes	direct product factor
137	C₂²×C₁₁	—	2	14	no	—
138	C₂²×C₁₁	—	2	42	no	—
139	C₂²×C₁₁	—	2	8	no	—
140	C₂²×C₁₁	—	4	56	no	—
141	C₂²×C₁₁	C₂×#104	1	28	yes	direct product factor
142	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	7	yes	—
143	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	7	yes	—
144	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
145	C₄×C₁₁	—	2	16	no	—
146	C₄×C₁₁	—	2	8	no	—
147	C₄×C₁₁	C₂²×C₂²⋊C₄	1	7	yes	—
148	C₄×C₁₁	C₂²×C₂²⋊C₄	1	7	yes	—
149	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
150	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
151	C₂⁵	—	2	1	no	—
152	C₂⁵	D₄×C₁₁	1	1	yes	characteristic
153	C₂×#41	—	2	84	no	—
154	C₂×#41	—	2	42	no	—
155	C₂×#41	—	2	42	no	—
156	C₂×#41	C₂×C₄×C₁₁	1	84	yes	complemented
157	C₂×#41	C₂³×C₁₁	1	28	yes	—
158	C₂²×Q₈	—	2	7	no	—
159	C₂²×Q₈	C₂×C₄×C₁₁	1	14	yes	complemented
160	C₂²×D₄	—	2	14	no	—
161	C₂²×D₄	—	2	14	no	—
162	C₂²×D₄	—	2	7	no	—
163	C₂²×D₄	C₂×C₄×C₁₁	1	14	yes	complemented
164	C₂³×C₄	—	2	16	no	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	—	2	14	no	—
167	C₂³×C₄	—	2	14	no	—
168	C₂³×C₄	—	2	14	no	—
169	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₁₁	1	7	yes	—
171	C₂³×C₄	D₄×C₁₁	1	7	yes	—
172	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₁₁	1	7	yes	—
174	C₂³×C₄	C₂³×C₁₁	1	7	yes	—
175	C₂×M₁₆	—	2	42	no	—
176	C₂×M₁₆	C₂³×C₁₁	1	28	yes	complemented
177	C₂²×C₈	—	2	14	no	—
178	C₂×C₄²	—	2	42	no	—
179	C₂×C₄²	C₂³×C₁₁	1	28	yes	complemented
180	#104	C₂³×C₁₁	1	64	yes	direct product factor
181	C₂×C₁₁	—	2	1	no	—
182	C₂×C₁₁	—	2	7	no	—
183	C₂×C₁₁	—	4	8	no	—
184	C₂×C₁₁	C₂²×#104	1	14	yes	direct product factor
185	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
186	C₂⁴	—	2	8	no	—
187	C₂⁴	—	2	14	no	—
188	C₂⁴	—	4	8	no	—
189	C₂⁴	C₁₁×C₂²⋊C₄	1	7	yes	—
190	C₂⁴	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
191	C₂⁴	C₂×D₄×C₁₁	1	7	yes	—
192	#41	—	2	168	no	—
193	#41	—	2	28	no	—
194	#41	—	2	28	no	—
195	#41	C₂²×C₄×C₁₁	1	56	yes	complemented
196	#41	C₂⁴×C₁₁	1	8	yes	—
197	C₂×Q₈	—	2	42	no	—
198	C₂×Q₈	C₂²×C₄×C₁₁	1	28	yes	complemented
199	C₂×D₄	—	2	84	no	—
200	C₂×D₄	—	2	28	no	—
201	C₂×D₄	—	2	42	no	—
202	C₂×D₄	C₂²×C₄×C₁₁	1	28	yes	complemented
203	C₂²×C₄	—	2	112	no	—
204	C₂²×C₄	—	2	14	no	—
205	C₂²×C₄	—	2	28	no	—
206	C₂²×C₄	—	2	84	no	—
207	C₂²×C₄	—	2	28	no	—
208	C₂²×C₄	C₁₁×C₂²⋊C₄	1	7	yes	—
209	C₂²×C₄	C₁₁×C₂²⋊C₄	1	7	yes	—
210	C₂²×C₄	C₁₁×C₂²⋊C₄	1	42	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₁	1	7	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₁	1	21	yes	—
213	C₂²×C₄	C₂×D₄×C₁₁	1	21	yes	—
214	C₂²×C₄	C₂×D₄×C₁₁	1	7	yes	—
215	C₂²×C₄	C₂×D₄×C₁₁	1	21	yes	—
216	C₂²×C₄	C₂⁴×C₁₁	1	7	yes	—
217	M₁₆	—	2	28	no	—
218	M₁₆	C₂⁴×C₁₁	1	8	yes	complemented
219	C₂×C₈	—	2	28	no	—
220	C₄²	—	2	28	no	—
221	C₄²	C₂⁴×C₁₁	1	8	yes	complemented
222	C₁₁	C₂³×#104	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	28	no	—
224	C₂³	—	2	28	no	—
225	C₂³	—	2	28	no	—
226	C₂³	—	4	56	no	—
227	C₂³	C₁₁×#104	1	8	yes	direct product factor
228	C₂³	C₂×C₁₁×C₂²⋊C₄	1	21	yes	—
229	C₂³	C₂×C₁₁×C₂²⋊C₄	1	7	yes	—
230	C₂³	C₂²×D₄×C₁₁	1	7	yes	—
231	Q₈	—	2	28	no	—
232	Q₈	C₂³×C₄×C₁₁	1	8	yes	complemented
233	D₄	—	2	56	no	—
234	D₄	—	2	8	no	—
235	D₄	—	2	28	no	—
236	D₄	C₂³×C₄×C₁₁	1	8	yes	complemented
237	C₂×C₄	—	2	112	no	—
238	C₂×C₄	—	2	28	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	56	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	21	yes	—
243	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	21	yes	—
244	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	42	yes	—
245	C₂×C₄	C₂³×C₄×C₁₁	1	7	yes	—
246	C₂×C₄	C₂³×C₄×C₁₁	1	7	yes	—
247	C₂×C₄	C₂²×D₄×C₁₁	1	7	yes	—
248	C₂×C₄	C₂²×D₄×C₁₁	1	7	yes	—
249	C₂×C₄	C₂²×D₄×C₁₁	1	7	yes	—
250	C₂×C₄	C₂⁵×C₁₁	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₂²	—	2	14	no	—
253	C₂²	—	2	42	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	56	no	—
256	C₂²	C₂×C₁₁×#104	1	28	yes	direct product factor
257	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	7	yes	—
258	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	7	yes	—
259	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	2	8	no	—
262	C₄	C₂²×C₁₁×C₂²⋊C₄	1	7	yes	—
263	C₄	C₂²×C₁₁×C₂²⋊C₄	1	7	yes	—
264	C₄	C₂⁴×C₄×C₁₁	1	1	yes	characteristic
265	C₄	C₂³×D₄×C₁₁	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	7	no	—
268	C₂	—	4	8	no	—
269	C₂	C₂²×C₁₁×#104	1	14	yes	direct product factor
270	C₂	C₂³×C₁₁×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₁	D₄
commutator	C₄	C₂⁴×C₄×C₁₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₁
socle	C₂⁴×C₁₁	C₂²⋊C₄
prime radical	C₂³×C₁₁×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₁ ◁ C₂⁴×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₁×M₁₆	C₂	semidirect
3. C₂³×C₄²×C₁₁	C₂	semidirect
4. C₂²×C₁₁×#104	C₂	direct (N × H)
7. C₂²×C₁₁×#41	C₄	semidirect
9. C₂³×Q₈×C₁₁	C₄	semidirect
11. C₂³×D₄×C₁₁	C₄	semidirect
16. C₂²×C₁₁×M₁₆	C₂²	semidirect
19. C₂²×C₄²×C₁₁	C₂²	semidirect
20. C₂×C₁₁×#104	C₂²	direct (N × H)
26. C₂×C₁₁×#41	C₂×C₄	semidirect
29. C₂²×Q₈×C₁₁	C₂×C₄	semidirect
33. C₂²×D₄×C₁₁	C₂×C₄	semidirect
46. C₂×C₁₁×M₁₆	C₂³	semidirect
49. C₂×C₄²×C₁₁	C₂³	semidirect
50. C₁₁×#104	C₂³	direct (N × H)
51. C₂³×#104	C₁₁	direct (N × H)
61. C₁₁×#41	C₂²×C₄	semidirect
64. C₂×Q₈×C₁₁	C₂²×C₄	semidirect
68. C₂×D₄×C₁₁	C₂²×C₄	semidirect
84. C₁₁×M₁₆	C₂⁴	semidirect
87. C₄²×C₁₁	C₂⁴	semidirect
89. C₂³×M₁₆	C₂×C₁₁	semidirect
90. C₂³×C₄²	C₂×C₁₁	semidirect
91. C₂²×#104	C₂×C₁₁	direct (N × H)
96. C₂³×C₁₁	#104	direct (N × H)
101. Q₈×C₁₁	C₂³×C₄	semidirect
105. D₄×C₁₁	C₂³×C₄	semidirect
123. C₂²×#41	C₄×C₁₁	semidirect
125. C₂³×Q₈	C₄×C₁₁	semidirect
127. C₂³×D₄	C₄×C₁₁	semidirect
132. C₂²×M₁₆	C₂²×C₁₁	semidirect
135. C₂²×C₄²	C₂²×C₁₁	semidirect
136. C₂×#104	C₂²×C₁₁	direct (N × H)
141. C₂²×C₁₁	C₂×#104	direct (N × H)
156. C₂×#41	C₂×C₄×C₁₁	semidirect
159. C₂²×Q₈	C₂×C₄×C₁₁	semidirect
163. C₂²×D₄	C₂×C₄×C₁₁	semidirect
176. C₂×M₁₆	C₂³×C₁₁	semidirect
179. C₂×C₄²	C₂³×C₁₁	semidirect
180. #104	C₂³×C₁₁	direct (N × H)
184. C₂×C₁₁	C₂²×#104	direct (N × H)
195. #41	C₂²×C₄×C₁₁	semidirect
198. C₂×Q₈	C₂²×C₄×C₁₁	semidirect
202. C₂×D₄	C₂²×C₄×C₁₁	semidirect
218. M₁₆	C₂⁴×C₁₁	semidirect
221. C₄²	C₂⁴×C₁₁	semidirect
222. C₁₁	C₂³×#104	direct (N × H)
227. C₂³	C₁₁×#104	direct (N × H)
232. Q₈	C₂³×C₄×C₁₁	semidirect
236. D₄	C₂³×C₄×C₁₁	semidirect
256. C₂²	C₂×C₁₁×#104	direct (N × H)
269. C₂	C₂²×C₁₁×#104	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×C₁₁×#41	2
C₂³×C₁₁×M₁₆	2
C₂³×C₄²×C₁₁	2
C₂²×C₁₁×#104	2
C₂³×#104	11

Subgroups — C23×C11×#104 #14637283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×#104 #14637283