Subgroups — #2625×C₂₃₃ #44448657

Order 29824.

464 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₃₃	C₂	1	1	yes	maximal, characteristic
3	#400×C₂₃₃	C₂	1	2	yes	maximal, complemented
4	#398×C₂₃₃	C₂	1	2	yes	maximal, complemented
5	#397×C₂₃₃	C₂	1	4	yes	maximal
6	#391×C₂₃₃	C₂	1	2	yes	maximal, complemented
7	#389×C₂₃₃	C₂	1	2	yes	maximal, complemented
8	#380×C₂₃₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₂₃₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Q₈×C₂₃₃	—	2	4	no	—
12	#117×C₂₃₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
23	#95×C₂₃₃	C₂²	1	2	yes	—
24	#95×C₂₃₃	C₂²	1	4	yes	—
25	#95×C₂₃₃	C₂²	1	4	yes	—
26	#95×C₂₃₃	C₂²	1	4	yes	—
27	C₂⁴×C₂₃₃	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₂₃₃	—	2	2	no	—
29	C₂×Q₈×C₂₃₃	—	2	4	no	—
30	C₂²×C₄×C₂₃₃	—	2	4	no	—
31	C₂²×C₄×C₂₃₃	D₄	1	4	yes	—
32	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₂₃₃	—	2	2	no	—
40	C₄⋊C₄×C₂₃₃	—	2	2	no	—
41	C₄⋊C₄×C₂₃₃	—	2	2	no	—
42	C₄⋊C₄×C₂₃₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃₃	—	4	4	no	—
45	C₂²⋊C₄×C₂₃₃	—	2	4	no	—
46	C₂²⋊C₄×C₂₃₃	—	2	4	no	—
47	C₂²⋊C₄×C₂₃₃	—	4	2	no	—
48	C₄²×C₂₃₃	—	2	4	no	—
49	C₂³×C₂₃₃	—	2	2	no	—
50	C₂³×C₂₃₃	—	2	1	no	—
51	C₂³×C₂₃₃	—	2	2	no	—
52	C₂³×C₂₃₃	—	2	1	no	—
53	C₂³×C₂₃₃	C₂×D₄	1	2	yes	—
54	C₂³×C₂₃₃	C₂⁴	1	1	yes	characteristic
55	Q₈×C₂₃₃	—	4	4	no	—
56	C₂×C₄×C₂₃₃	—	2	2	no	—
57	C₂×C₄×C₂₃₃	—	2	8	no	—
58	C₂×C₄×C₂₃₃	—	2	2	no	—
59	C₂×C₄×C₂₃₃	—	2	2	no	—
60	C₂×C₄×C₂₃₃	—	2	2	no	—
61	C₂×C₄×C₂₃₃	—	2	2	no	—
62	C₂×C₄×C₂₃₃	—	2	4	no	—
63	C₂×C₄×C₂₃₃	—	2	4	no	—
64	C₂×C₄×C₂₃₃	—	2	4	no	—
65	C₂×C₄×C₂₃₃	—	2	2	no	—
66	C₂×C₄×C₂₃₃	—	2	2	no	—
67	C₂×C₄×C₂₃₃	—	2	4	no	—
68	C₂×C₄×C₂₃₃	—	2	2	no	—
69	C₂×C₄×C₂₃₃	—	2	2	no	—
70	C₂×C₄×C₂₃₃	—	2	4	no	—
71	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₂₃₃	#41	1	4	yes	—
74	C₂²×C₂₃₃	—	2	2	no	—
75	C₂²×C₂₃₃	—	2	2	no	—
76	C₂²×C₂₃₃	—	2	2	no	—
77	C₂²×C₂₃₃	—	4	2	no	—
78	C₂²×C₂₃₃	—	4	2	no	—
79	C₂²×C₂₃₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₂₃₃	C₂×#41	1	2	yes	—
81	C₂²×C₂₃₃	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₂₃₃	2₋¹⁺⁴	1	2	yes	—
83	C₂²×C₂₃₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₂₃₃	—	2	2	no	—
85	C₄×C₂₃₃	—	4	2	no	—
86	C₄×C₂₃₃	—	4	2	no	—
87	C₄×C₂₃₃	—	4	2	no	—
88	C₄×C₂₃₃	—	4	2	no	—
89	C₄×C₂₃₃	—	4	4	no	—
90	C₄×C₂₃₃	#123	1	4	yes	—
91	C₂×C₂₃₃	—	4	2	no	—
92	C₂×C₂₃₃	C₂×#123	1	1	yes	characteristic
93	C₂×C₂₃₃	#536	1	1	yes	characteristic
94	C₂×C₂₃₃	#537	1	1	yes	characteristic
95	C₂×C₂₃₃	#541	1	2	yes	—
96	C₂×C₂₃₃	#542	1	2	yes	—
97	C₂₃₃	#2625	1	1	yes	characteristic, direct product factor, Sylow
98	#2625	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₂²⋊Q₈	C₂×C₂₃₃	1	1	yes	characteristic
100	C₂×#117	C₂×C₂₃₃	1	1	yes	characteristic
101	#400	C₂×C₂₃₃	1	2	yes	complemented
102	#398	C₂×C₂₃₃	1	2	yes	complemented
103	#397	C₂×C₂₃₃	1	4	yes	—
104	#391	C₂×C₂₃₃	1	2	yes	complemented
105	#389	C₂×C₂₃₃	1	2	yes	complemented
106	#380	C₂×C₂₃₃	1	1	yes	characteristic
107	C₂²×Q₈	C₂²×C₂₃₃	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₂₃₃	1	1	yes	characteristic
109	C₂²⋊Q₈	—	2	4	no	—
110	#117	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	4	yes	—
117	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
120	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
121	#95	C₂²×C₂₃₃	1	2	yes	—
122	#95	C₂²×C₂₃₃	1	4	yes	—
123	#95	C₂²×C₂₃₃	1	4	yes	—
124	#95	C₂²×C₂₃₃	1	4	yes	—
125	C₂⁴	C₂³×C₂₃₃	1	1	yes	characteristic
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	D₄×C₂₃₃	1	4	yes	—
130	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₂₃₃	1	4	yes	—
132	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	2	4	no	—
147	C₂³	—	2	2	no	—
148	C₂³	—	2	1	no	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	1	no	—
151	C₂³	C₂×D₄×C₂₃₃	1	2	yes	—
152	C₂³	C₂⁴×C₂₃₃	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
170	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
171	C₂×C₄	#41×C₂₃₃	1	4	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	C₂²×D₄×C₂₃₃	1	1	yes	characteristic
178	C₂²	C₂×#41×C₂₃₃	1	2	yes	—
179	C₂²	H(5,𝔽₂)×C₂₃₃	1	1	yes	characteristic
180	C₂²	2₋¹⁺⁴×C₂₃₃	1	2	yes	—
181	C₂²	2₋¹⁺⁴×C₂₃₃	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	4	no	—
188	C₄	#123×C₂₃₃	1	4	yes	—
189	C₂	—	4	2	no	—
190	C₂	C₂×#123×C₂₃₃	1	1	yes	characteristic
191	C₂	#536×C₂₃₃	1	1	yes	characteristic
192	C₂	#537×C₂₃₃	1	1	yes	characteristic
193	C₂	#541×C₂₃₃	1	2	yes	—
194	C₂	#542×C₂₃₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #400×C₂₃₃	C₂	semidirect
4. #398×C₂₃₃	C₂	semidirect
6. #391×C₂₃₃	C₂	semidirect
7. #389×C₂₃₃	C₂	semidirect
97. C₂₃₃	#2625	direct (N × H)
98. #2625	C₂₃₃	direct (N × H)
101. #400	C₂×C₂₃₃	semidirect
102. #398	C₂×C₂₃₃	semidirect
104. #391	C₂×C₂₃₃	semidirect
105. #389	C₂×C₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₂₃₃	2
C₂×#117×C₂₃₃	2
#400×C₂₃₃	2
#398×C₂₃₃	2
#397×C₂₃₃	2
#391×C₂₃₃	2
#389×C₂₃₃	2
#380×C₂₃₃	2
#2625	233

Subgroups — #2625×C233 #44448657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2625×C₂₃₃ #44448657