Subgroups — #3137×C₂₃₃ #44449169

Order 29824.

372 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#536×C₂₃₃	C₂	1	1	yes	maximal, characteristic
3	#491×C₂₃₃	C₂	1	2	yes	maximal, complemented
4	#489×C₂₃₃	C₂	1	2	yes	maximal, complemented
5	#488×C₂₃₃	C₂	1	2	yes	maximal, complemented
6	#487×C₂₃₃	C₂	1	2	yes	maximal, complemented
7	#419×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#417×C₂₃₃	C₂	1	1	yes	maximal, characteristic
9	#416×C₂₃₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#103×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₃₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₂₃₃	—	2	1	no	—
17	C₄⋊Q₈×C₂₃₃	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₂₃₃	C₂²	1	2	yes	—
19	#125×C₂₃₃	—	2	1	no	—
20	#125×C₂₃₃	C₂²	1	2	yes	—
21	#124×C₂₃₃	C₂²	1	2	yes	—
22	#123×C₂₃₃	—	2	2	no	—
23	C₂²⋊Q₈×C₂₃₃	—	2	1	no	—
24	C₂²⋊Q₈×C₂₃₃	—	2	1	no	—
25	C₂²⋊Q₈×C₂₃₃	—	2	2	no	—
26	#117×C₂₃₃	C₂²	1	1	yes	characteristic
27	#117×C₂₃₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
29	#103×C₂₃₃	C₂²	1	1	yes	characteristic
30	#103×C₂₃₃	C₂²	1	2	yes	—
31	#103×C₂₃₃	C₂²	1	2	yes	—
32	#103×C₂₃₃	C₂²	1	2	yes	—
33	#103×C₂₃₃	C₂²	1	2	yes	—
34	#103×C₂₃₃	C₂²	1	2	yes	—
35	#103×C₂₃₃	C₂²	1	1	yes	characteristic
36	#102×C₂₃₃	C₂²	1	1	yes	characteristic
37	#102×C₂₃₃	C₂²	1	2	yes	—
38	#102×C₂₃₃	C₂²	1	1	yes	characteristic
39	#97×C₂₃₃	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃₃	C₂²	1	2	yes	—
41	#41×C₂₃₃	—	2	2	no	—
42	#41×C₂₃₃	—	2	2	no	—
43	C₂×Q₈×C₂₃₃	—	2	1	no	—
44	C₂×Q₈×C₂₃₃	—	2	2	no	—
45	C₂×Q₈×C₂₃₃	—	2	2	no	—
46	C₂×Q₈×C₂₃₃	—	2	2	no	—
47	C₂×Q₈×C₂₃₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₂₃₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃₃	—	2	1	no	—
50	C₂×D₄×C₂₃₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃₃	—	2	1	no	—
52	C₂²×C₄×C₂₃₃	—	2	1	no	—
53	C₂²×C₄×C₂₃₃	—	2	1	no	—
54	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
55	M₁₆×C₂₃₃	—	2	2	no	—
56	C₂×C₈×C₂₃₃	—	2	2	no	—
57	C₂×C₈×C₂₃₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₃₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃₃	—	2	2	no	—
65	C₄⋊C₄×C₂₃₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃₃	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃₃	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₂₃₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₃₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₂₃₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃₃	—	2	2	no	—
74	C₂²⋊C₄×C₂₃₃	D₄	1	2	yes	—
75	C₄²×C₂₃₃	—	2	1	no	—
76	C₄²×C₂₃₃	C₂³	1	2	yes	—
77	C₂³×C₂₃₃	—	2	1	no	—
78	C₂³×C₂₃₃	C₂×D₄	1	1	yes	characteristic
79	Q₈×C₂₃₃	—	2	2	no	—
80	Q₈×C₂₃₃	—	2	2	no	—
81	Q₈×C₂₃₃	—	2	1	no	—
82	Q₈×C₂₃₃	—	2	1	no	—
83	Q₈×C₂₃₃	—	4	2	no	—
84	Q₈×C₂₃₃	—	8	2	no	—
85	D₄×C₂₃₃	—	2	1	no	—
86	D₄×C₂₃₃	—	2	1	no	—
87	D₄×C₂₃₃	—	4	2	no	—
88	C₂×C₄×C₂₃₃	—	2	1	no	—
89	C₂×C₄×C₂₃₃	—	2	1	no	—
90	C₂×C₄×C₂₃₃	—	2	2	no	—
91	C₂×C₄×C₂₃₃	—	2	2	no	—
92	C₂×C₄×C₂₃₃	—	2	1	no	—
93	C₂×C₄×C₂₃₃	—	4	2	no	—
94	C₂×C₄×C₂₃₃	—	4	2	no	—
95	C₂×C₄×C₂₃₃	—	8	1	no	—
96	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃₃	#41	1	2	yes	—
100	C₂×C₄×C₂₃₃	#41	1	2	yes	—
101	C₂×C₄×C₂₃₃	C₂⁴	1	1	yes	characteristic
102	C₈×C₂₃₃	—	2	1	no	—
103	C₈×C₂₃₃	—	2	1	no	—
104	C₈×C₂₃₃	—	2	2	no	—
105	C₂²×C₂₃₃	—	2	1	no	—
106	C₂²×C₂₃₃	—	2	1	no	—
107	C₂²×C₂₃₃	—	4	1	no	—
108	C₂²×C₂₃₃	—	4	1	no	—
109	C₂²×C₂₃₃	—	4	1	no	—
110	C₂²×C₂₃₃	#124	1	2	yes	—
111	C₂²×C₂₃₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₂₃₃	—	2	2	no	—
113	C₄×C₂₃₃	—	2	2	no	—
114	C₄×C₂₃₃	—	4	1	no	—
115	C₄×C₂₃₃	—	4	2	no	—
116	C₄×C₂₃₃	—	4	2	no	—
117	C₄×C₂₃₃	—	4	1	no	—
118	C₄×C₂₃₃	—	4	1	no	—
119	C₄×C₂₃₃	#124	1	2	yes	—
120	C₄×C₂₃₃	C₂×#41	1	1	yes	characteristic
121	C₄×C₂₃₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₂₃₃	—	2	2	no	—
123	C₂×C₂₃₃	—	8	1	no	—
124	C₂×C₂₃₃	C₂×#124	1	1	yes	characteristic
125	C₂×C₂₃₃	#577	1	1	yes	characteristic
126	C₂×C₂₃₃	#578	1	1	yes	characteristic
127	C₂₃₃	#3137	1	1	yes	characteristic, direct product factor, Sylow
128	#3137	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	#544	C₂×C₂₃₃	1	1	yes	characteristic, complemented
130	#536	C₂×C₂₃₃	1	1	yes	characteristic
131	#491	C₂×C₂₃₃	1	2	yes	complemented
132	#489	C₂×C₂₃₃	1	2	yes	complemented
133	#488	C₂×C₂₃₃	1	2	yes	complemented
134	#487	C₂×C₂₃₃	1	2	yes	complemented
135	#419	C₂×C₂₃₃	1	1	yes	characteristic, complemented
136	#417	C₂×C₂₃₃	1	1	yes	characteristic
137	#416	C₂×C₂₃₃	1	1	yes	characteristic
138	C₂×#103	C₂×C₂₃₃	1	1	yes	characteristic, complemented
139	#405	C₂×C₂₃₃	1	1	yes	characteristic, complemented
140	C₂×#41	C₂²×C₂₃₃	1	1	yes	characteristic
141	C₂²×Q₈	C₂²×C₂₃₃	1	1	yes	characteristic
142	C₂×M₁₆	C₂²×C₂₃₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×C₂₃₃	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	1	no	—
145	C₄⋊Q₈	C₂²×C₂₃₃	1	2	yes	—
146	C₄⋊Q₈	C₂²×C₂₃₃	1	2	yes	—
147	#125	—	2	1	no	—
148	#125	C₂²×C₂₃₃	1	2	yes	—
149	#124	C₂²×C₂₃₃	1	2	yes	—
150	#123	—	2	2	no	—
151	C₂²⋊Q₈	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	2	no	—
154	#117	C₂²×C₂₃₃	1	1	yes	characteristic
155	#117	C₂²×C₂₃₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
157	#103	C₂²×C₂₃₃	1	2	yes	—
158	#103	C₂²×C₂₃₃	1	2	yes	—
159	#103	C₂²×C₂₃₃	1	1	yes	characteristic
160	#103	C₂²×C₂₃₃	1	1	yes	characteristic
161	#103	C₂²×C₂₃₃	1	2	yes	—
162	#103	C₂²×C₂₃₃	1	2	yes	—
163	#103	C₂²×C₂₃₃	1	2	yes	—
164	#102	C₂²×C₂₃₃	1	1	yes	characteristic
165	#102	C₂²×C₂₃₃	1	1	yes	characteristic
166	#102	C₂²×C₂₃₃	1	2	yes	—
167	#97	C₂²×C₂₃₃	1	2	yes	—
168	C₄×C₈	C₂²×C₂₃₃	1	2	yes	—
169	#41	—	2	2	no	—
170	#41	—	2	2	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	C₂³×C₂₃₃	1	2	yes	—
176	C₂×Q₈	C₂³×C₂₃₃	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	C₂³×C₂₃₃	1	1	yes	characteristic
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
183	M₁₆	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	C₂³×C₂₃₃	1	1	yes	characteristic
186	C₂×C₈	C₂³×C₂₃₃	1	2	yes	—
187	C₂×C₈	C₂³×C₂₃₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	D₄×C₂₃₃	1	1	yes	characteristic
195	C₄⋊C₄	D₄×C₂₃₃	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₂₃₃	1	2	yes	—
197	C₄⋊C₄	C₂³×C₂₃₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₂₃₃	1	1	yes	characteristic
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	D₄×C₂₃₃	1	2	yes	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂³×C₂₃₃	1	2	yes	—
205	C₂³	—	2	1	no	—
206	C₂³	C₂×D₄×C₂₃₃	1	1	yes	characteristic
207	Q₈	—	2	2	no	—
208	Q₈	—	2	2	no	—
209	Q₈	—	2	1	no	—
210	Q₈	—	2	1	no	—
211	Q₈	—	4	2	no	—
212	Q₈	—	8	2	no	—
213	D₄	—	2	1	no	—
214	D₄	—	2	1	no	—
215	D₄	—	4	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
225	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₂₃₃	1	1	yes	characteristic
227	C₂×C₄	#41×C₂₃₃	1	2	yes	—
228	C₂×C₄	#41×C₂₃₃	1	2	yes	—
229	C₂×C₄	C₂⁴×C₂₃₃	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	#124×C₂₃₃	1	2	yes	—
239	C₂²	C₂²×D₄×C₂₃₃	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	#124×C₂₃₃	1	2	yes	—
248	C₄	C₂×#41×C₂₃₃	1	1	yes	characteristic
249	C₄	C₂×#41×C₂₃₃	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	8	1	no	—
252	C₂	C₂×#124×C₂₃₃	1	1	yes	characteristic
253	C₂	#577×C₂₃₃	1	1	yes	characteristic
254	C₂	#578×C₂₃₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃₃
prime radical	C₂×#41×C₂₃₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₃ ◁ #117×C₂₃₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #544×C₂₃₃	C₂	semidirect
3. #491×C₂₃₃	C₂	semidirect
4. #489×C₂₃₃	C₂	semidirect
5. #488×C₂₃₃	C₂	semidirect
6. #487×C₂₃₃	C₂	semidirect
7. #419×C₂₃₃	C₂	semidirect
10. C₂×#103×C₂₃₃	C₂	semidirect
11. #405×C₂₃₃	C₂	semidirect
127. C₂₃₃	#3137	direct (N × H)
128. #3137	C₂₃₃	direct (N × H)
129. #544	C₂×C₂₃₃	semidirect
131. #491	C₂×C₂₃₃	semidirect
132. #489	C₂×C₂₃₃	semidirect
133. #488	C₂×C₂₃₃	semidirect
134. #487	C₂×C₂₃₃	semidirect
135. #419	C₂×C₂₃₃	semidirect
138. C₂×#103	C₂×C₂₃₃	semidirect
139. #405	C₂×C₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
#544×C₂₃₃	2
#536×C₂₃₃	2
#491×C₂₃₃	2
#489×C₂₃₃	2
#488×C₂₃₃	2
#487×C₂₃₃	2
#419×C₂₃₃	2
#417×C₂₃₃	2
#416×C₂₃₃	2
C₂×#103×C₂₃₃	2
#405×C₂₃₃	2
#3137	233

Subgroups — #3137×C233 #44449169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3137×C₂₃₃ #44449169