Subgroups — #2374×C₂₃₃ #44448406

Order 29824.

552 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₂₃₃	C₂	1	1	yes	maximal, characteristic
2	#390×C₂₃₃	C₂	1	1	yes	maximal, characteristic
3	#388×C₂₃₃	C₂	1	2	yes	maximal, complemented
4	#388×C₂₃₃	C₂	1	2	yes	maximal, complemented
5	#387×C₂₃₃	C₂	1	2	yes	maximal, complemented
6	#386×C₂₃₃	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄×C₂₃₃	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₂₃₃	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₂₃₃	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₂₃₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
13	#123×C₂₃₃	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
27	#95×C₂₃₃	C₂²	1	2	yes	—
28	#95×C₂₃₃	C₂²	1	2	yes	—
29	#95×C₂₃₃	C₂²	1	2	yes	—
30	#95×C₂₃₃	C₂²	1	2	yes	—
31	#95×C₂₃₃	C₂²	1	2	yes	—
32	C₂⁴×C₂₃₃	C₂³	1	1	yes	characteristic
33	C₂⁴×C₂₃₃	C₂³	1	1	yes	characteristic
34	C₂×D₄×C₂₃₃	—	4	1	no	—
35	C₂×D₄×C₂₃₃	—	4	1	no	—
36	C₂×D₄×C₂₃₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₂₃₃	—	2	4	no	—
38	C₂²×C₄×C₂₃₃	—	4	1	no	—
39	C₂²×C₄×C₂₃₃	—	4	1	no	—
40	C₂²×C₄×C₂₃₃	—	4	1	no	—
41	C₂²×C₄×C₂₃₃	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
49	C₄⋊C₄×C₂₃₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃₃	—	4	1	no	—
51	C₄⋊C₄×C₂₃₃	C₂×C₄	1	8	yes	complemented
52	C₂²⋊C₄×C₂₃₃	—	2	4	no	—
53	C₂²⋊C₄×C₂₃₃	—	2	4	no	—
54	C₂²⋊C₄×C₂₃₃	—	2	4	no	—
55	C₂²⋊C₄×C₂₃₃	—	4	1	no	—
56	C₂²⋊C₄×C₂₃₃	C₂×C₄	1	8	yes	—
57	C₂²⋊C₄×C₂₃₃	C₂×C₄	1	4	yes	complemented
58	C₄²×C₂₃₃	—	2	4	no	—
59	C₄²×C₂₃₃	—	4	1	no	—
60	C₄²×C₂₃₃	—	4	2	no	—
61	C₄²×C₂₃₃	—	4	2	no	—
62	C₂³×C₂₃₃	—	2	2	no	—
63	C₂³×C₂₃₃	—	2	2	no	—
64	C₂³×C₂₃₃	—	2	2	no	—
65	C₂³×C₂₃₃	—	2	2	no	—
66	C₂³×C₂₃₃	—	2	1	no	—
67	C₂³×C₂₃₃	—	2	1	no	—
68	C₂³×C₂₃₃	—	2	1	no	—
69	C₂³×C₂₃₃	—	2	1	no	—
70	C₂³×C₂₃₃	C₂²×C₄	1	2	yes	—
71	C₂³×C₂₃₃	C₂²×C₄	1	2	yes	—
72	C₂³×C₂₃₃	C₂⁴	1	1	yes	characteristic
73	D₄×C₂₃₃	—	4	4	no	—
74	C₂×C₄×C₂₃₃	—	2	2	no	—
75	C₂×C₄×C₂₃₃	—	2	2	no	—
76	C₂×C₄×C₂₃₃	—	2	2	no	—
77	C₂×C₄×C₂₃₃	—	2	2	no	—
78	C₂×C₄×C₂₃₃	—	2	2	no	—
79	C₂×C₄×C₂₃₃	—	2	2	no	—
80	C₂×C₄×C₂₃₃	—	2	2	no	—
81	C₂×C₄×C₂₃₃	—	2	2	no	—
82	C₂×C₄×C₂₃₃	—	2	1	no	—
83	C₂×C₄×C₂₃₃	—	2	1	no	—
84	C₂×C₄×C₂₃₃	—	2	2	no	—
85	C₂×C₄×C₂₃₃	—	2	2	no	—
86	C₂×C₄×C₂₃₃	—	2	2	no	—
87	C₂×C₄×C₂₃₃	—	2	2	no	—
88	C₂×C₄×C₂₃₃	—	2	2	no	—
89	C₂×C₄×C₂₃₃	—	2	2	no	—
90	C₂×C₄×C₂₃₃	—	4	4	no	—
91	C₂×C₄×C₂₃₃	—	4	4	no	—
92	C₂×C₄×C₂₃₃	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₂₃₃	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₂₃₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₃₃	#41	1	4	yes	—
96	C₂×C₄×C₂₃₃	#41	1	4	yes	—
97	C₂²×C₂₃₃	—	2	4	no	—
98	C₂²×C₂₃₃	—	2	4	no	—
99	C₂²×C₂₃₃	—	2	2	no	—
100	C₂²×C₂₃₃	—	2	2	no	—
101	C₂²×C₂₃₃	—	4	1	no	—
102	C₂²×C₂₃₃	—	4	1	no	—
103	C₂²×C₂₃₃	—	4	2	no	—
104	C₂²×C₂₃₃	—	4	2	no	—
105	C₂²×C₂₃₃	—	4	1	no	—
106	C₂²×C₂₃₃	—	4	1	no	—
107	C₂²×C₂₃₃	C₂³×C₄	1	1	yes	characteristic
108	C₂²×C₂₃₃	C₂×#41	1	2	yes	—
109	C₂²×C₂₃₃	C₂×#41	1	2	yes	—
110	C₂²×C₂₃₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₂₃₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₂₃₃	—	2	4	no	—
113	C₄×C₂₃₃	—	2	4	no	—
114	C₄×C₂₃₃	—	2	2	no	—
115	C₄×C₂₃₃	—	2	4	no	—
116	C₄×C₂₃₃	—	2	4	no	—
117	C₄×C₂₃₃	—	4	2	no	—
118	C₄×C₂₃₃	—	4	2	no	—
119	C₂×C₂₃₃	—	4	2	no	—
120	C₂×C₂₃₃	—	4	2	no	—
121	C₂×C₂₃₃	C₄×#41	1	2	yes	—
122	C₂×C₂₃₃	#518	1	1	yes	characteristic
123	C₂×C₂₃₃	#551	1	1	yes	characteristic
124	C₂×C₂₃₃	#553	1	2	yes	—
125	C₂×C₂₃₃	#559	1	1	yes	characteristic
126	C₂₃₃	#2374	1	1	yes	characteristic, direct product factor, Sylow
127	#2374	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×#123	C₂×C₂₃₃	1	1	yes	characteristic
129	#390	C₂×C₂₃₃	1	1	yes	characteristic
130	#388	C₂×C₂₃₃	1	2	yes	complemented
131	#388	C₂×C₂₃₃	1	2	yes	complemented
132	#387	C₂×C₂₃₃	1	2	yes	complemented
133	#386	C₂×C₂₃₃	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂×C₂₃₃	1	2	yes	complemented
135	C₄×C₂²⋊C₄	C₂×C₂₃₃	1	2	yes	complemented
136	C₄×C₂²⋊C₄	C₂×C₂₃₃	1	2	yes	complemented
137	C₂²×D₄	C₂²×C₂₃₃	1	1	yes	characteristic
138	C₂³×C₄	C₂²×C₂₃₃	1	1	yes	characteristic
139	C₂³×C₄	C₂²×C₂₃₃	1	1	yes	characteristic
140	#123	C₄×C₂₃₃	1	8	yes	complemented
141	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
150	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
151	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
152	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
153	C₂×C₄²	C₂²×C₂₃₃	1	1	yes	characteristic
154	#95	C₂²×C₂₃₃	1	2	yes	—
155	#95	C₂²×C₂₃₃	1	2	yes	—
156	#95	C₂²×C₂₃₃	1	2	yes	—
157	#95	C₂²×C₂₃₃	1	2	yes	—
158	#95	C₂²×C₂₃₃	1	2	yes	—
159	C₂⁴	C₂³×C₂₃₃	1	1	yes	characteristic
160	C₂⁴	C₂³×C₂₃₃	1	1	yes	characteristic
161	C₂×D₄	—	4	1	no	—
162	C₂×D₄	—	4	1	no	—
163	C₂×D₄	C₂×C₄×C₂₃₃	1	4	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	C₂×C₄×C₂₃₃	1	4	yes	complemented
169	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
171	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
173	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	C₂×C₄×C₂₃₃	1	8	yes	complemented
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	C₂×C₄×C₂₃₃	1	8	yes	—
184	C₂²⋊C₄	C₂×C₄×C₂₃₃	1	4	yes	complemented
185	C₄²	—	2	4	no	—
186	C₄²	—	4	1	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	4	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂²×C₄×C₂₃₃	1	2	yes	—
198	C₂³	C₂²×C₄×C₂₃₃	1	2	yes	—
199	C₂³	C₂⁴×C₂₃₃	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	C₂²×C₄×C₂₃₃	1	4	yes	—
220	C₂×C₄	C₂²×C₄×C₂₃₃	1	4	yes	—
221	C₂×C₄	C₂²×C₄×C₂₃₃	1	2	yes	—
222	C₂×C₄	#41×C₂₃₃	1	4	yes	—
223	C₂×C₄	#41×C₂₃₃	1	4	yes	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	C₂³×C₄×C₂₃₃	1	1	yes	characteristic
235	C₂²	C₂×#41×C₂₃₃	1	2	yes	—
236	C₂²	C₂×#41×C₂₃₃	1	2	yes	—
237	C₂²	H(5,𝔽₂)×C₂₃₃	1	1	yes	characteristic
238	C₂²	H(5,𝔽₂)×C₂₃₃	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	2	4	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₂	—	4	2	no	—
247	C₂	—	4	2	no	—
248	C₂	C₄×#41×C₂₃₃	1	2	yes	—
249	C₂	#518×C₂₃₃	1	1	yes	characteristic
250	C₂	#551×C₂₃₃	1	1	yes	characteristic
251	C₂	#553×C₂₃₃	1	2	yes	—
252	C₂	#559×C₂₃₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃₃
prime radical	C₂²×D₄×C₂₃₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #388×C₂₃₃	C₂	semidirect
4. #388×C₂₃₃	C₂	semidirect
5. #387×C₂₃₃	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₃₃	C₂	semidirect
8. C₄×C₂²⋊C₄×C₂₃₃	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₃₃	C₂	semidirect
13. #123×C₂₃₃	C₄	semidirect
41. C₂²×C₄×C₂₃₃	C₂×C₄	semidirect
51. C₄⋊C₄×C₂₃₃	C₂×C₄	semidirect
57. C₂²⋊C₄×C₂₃₃	C₂×C₄	semidirect
126. C₂₃₃	#2374	direct (N × H)
127. #2374	C₂₃₃	direct (N × H)
130. #388	C₂×C₂₃₃	semidirect
131. #388	C₂×C₂₃₃	semidirect
132. #387	C₂×C₂₃₃	semidirect
134. C₄×C₄⋊C₄	C₂×C₂₃₃	semidirect
135. C₄×C₂²⋊C₄	C₂×C₂₃₃	semidirect
136. C₄×C₂²⋊C₄	C₂×C₂₃₃	semidirect
140. #123	C₄×C₂₃₃	semidirect
168. C₂²×C₄	C₂×C₄×C₂₃₃	semidirect
178. C₄⋊C₄	C₂×C₄×C₂₃₃	semidirect
184. C₂²⋊C₄	C₂×C₄×C₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₂₃₃	2
#390×C₂₃₃	2
#388×C₂₃₃	2
#387×C₂₃₃	2
#386×C₂₃₃	2
C₄×C₄⋊C₄×C₂₃₃	2
C₄×C₂²⋊C₄×C₂₃₃	2
#2374	233

Subgroups — #2374×C233 #44448406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2374×C₂₃₃ #44448406