Subgroups — #2345×C₂₃₃ #44448377

Order 29824.

548 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic
2	#389×C₂₃₃	C₂	1	2	yes	maximal, complemented
3	#387×C₂₃₃	C₂	1	4	yes	maximal, complemented
4	#380×C₂₃₃	C₂	1	2	yes	maximal
5	#380×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₂₃₃	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₃₃	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊C₄×C₂₃₃	C₂	1	2	yes	maximal
9	C₂³×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₃₃	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃₃	C₂²	1	4	yes	—
22	#95×C₂₃₃	C₂²	1	2	yes	—
23	#95×C₂₃₃	C₂²	1	4	yes	—
24	#95×C₂₃₃	C₂²	1	4	yes	—
25	#95×C₂₃₃	C₂²	1	2	yes	—
26	C₂⁴×C₂₃₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₂₃₃	—	2	4	no	—
28	C₂²×C₄×C₂₃₃	—	4	4	no	—
29	C₂²×C₄×C₂₃₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃₃	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₂₃₃	—	2	4	no	—
39	C₄⋊C₄×C₂₃₃	—	4	4	no	—
40	C₄⋊C₄×C₂₃₃	C₂×C₄	1	8	yes	—
41	C₄⋊C₄×C₂₃₃	C₂×C₄	1	4	yes	—
42	C₄⋊C₄×C₂₃₃	C₂×C₄	1	4	yes	—
43	C₂²⋊C₄×C₂₃₃	—	2	2	no	—
44	C₂²⋊C₄×C₂₃₃	—	2	4	no	—
45	C₂²⋊C₄×C₂₃₃	Q₈	1	4	yes	—
46	C₄²×C₂₃₃	—	2	4	no	—
47	C₄²×C₂₃₃	—	4	4	no	—
48	C₂³×C₂₃₃	—	2	2	no	—
49	C₂³×C₂₃₃	—	2	2	no	—
50	C₂³×C₂₃₃	C₂²×C₄	1	2	yes	—
51	C₂³×C₂₃₃	C₂×Q₈	1	2	yes	—
52	C₂³×C₂₃₃	#41	1	2	yes	—
53	C₂³×C₂₃₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₂₃₃	—	2	2	no	—
55	C₂×C₄×C₂₃₃	—	2	2	no	—
56	C₂×C₄×C₂₃₃	—	2	2	no	—
57	C₂×C₄×C₂₃₃	—	2	4	no	—
58	C₂×C₄×C₂₃₃	—	2	4	no	—
59	C₂×C₄×C₂₃₃	—	2	8	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	—	4	8	no	—
67	C₂×C₄×C₂₃₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃₃	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₂₃₃	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₂₃₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₃₃	C₂×Q₈	1	4	yes	—
72	C₂×C₄×C₂₃₃	#41	1	4	yes	—
73	C₂²×C₂₃₃	—	2	2	no	—
74	C₂²×C₂₃₃	—	2	2	no	—
75	C₂²×C₂₃₃	—	2	4	no	—
76	C₂²×C₂₃₃	—	2	4	no	—
77	C₂²×C₂₃₃	C₄×Q₈	1	4	yes	—
78	C₂²×C₂₃₃	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₂₃₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₂₃₃	C₂×#41	1	2	yes	—
81	C₂²×C₂₃₃	C₂×#41	1	1	yes	characteristic
82	C₂²×C₂₃₃	H(5,𝔽₂)	1	2	yes	—
83	C₄×C₂₃₃	—	2	2	no	—
84	C₄×C₂₃₃	—	2	4	no	—
85	C₄×C₂₃₃	—	2	4	no	—
86	C₄×C₂₃₃	—	2	4	no	—
87	C₄×C₂₃₃	—	4	2	no	—
88	C₄×C₂₃₃	—	4	4	no	—
89	C₄×C₂₃₃	#117	1	4	yes	—
90	C₂×C₂₃₃	—	2	4	no	—
91	C₂×C₂₃₃	C₂×#117	1	1	yes	characteristic
92	C₂×C₂₃₃	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₂×C₂₃₃	#518	1	1	yes	characteristic
94	C₂×C₂₃₃	#553	1	2	yes	—
95	C₂×C₂₃₃	#554	1	2	yes	—
96	C₂₃₃	#2345	1	1	yes	characteristic, direct product factor, Sylow
97	#2345	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₄⋊C₄	C₂×C₂₃₃	1	1	yes	characteristic
99	#389	C₂×C₂₃₃	1	2	yes	complemented
100	#387	C₂×C₂₃₃	1	4	yes	complemented
101	#380	C₂×C₂₃₃	1	2	yes	—
102	#380	C₂×C₂₃₃	1	1	yes	characteristic
103	C₄×C₄⋊C₄	C₂×C₂₃₃	1	2	yes	complemented
104	C₄×C₂²⋊C₄	C₂×C₂₃₃	1	1	yes	characteristic
105	C₄×C₂²⋊C₄	C₂×C₂₃₃	1	2	yes	—
106	C₂³×C₄	C₂²×C₂₃₃	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₂₃₃	1	2	yes	—
108	C₂×C₄⋊C₄	C₄×C₂₃₃	1	8	yes	complemented
109	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	4	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
115	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	2	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₂₃₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
118	C₂×C₄²	C₂²×C₂₃₃	1	4	yes	—
119	#95	C₂²×C₂₃₃	1	2	yes	—
120	#95	C₂²×C₂₃₃	1	4	yes	—
121	#95	C₂²×C₂₃₃	1	4	yes	—
122	#95	C₂²×C₂₃₃	1	2	yes	—
123	C₂⁴	C₂³×C₂₃₃	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	C₂×C₄×C₂₃₃	1	4	yes	—
127	C₂²×C₄	C₂×C₄×C₂₃₃	1	8	yes	—
128	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₃₃	1	4	yes	—
130	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	C₂×C₄×C₂₃₃	1	8	yes	—
138	C₄⋊C₄	C₂×C₄×C₂₃₃	1	4	yes	—
139	C₄⋊C₄	C₂×C₄×C₂₃₃	1	4	yes	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	Q₈×C₂₃₃	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	4	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	C₂²×C₄×C₂₃₃	1	2	yes	—
148	C₂³	C₂×Q₈×C₂₃₃	1	2	yes	—
149	C₂³	#41×C₂₃₃	1	2	yes	—
150	C₂³	C₂⁴×C₂₃₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	C₂²×C₄×C₂₃₃	1	2	yes	—
165	C₂×C₄	C₂²×C₄×C₂₃₃	1	4	yes	—
166	C₂×C₄	C₂²×C₄×C₂₃₃	1	4	yes	—
167	C₂×C₄	C₂²×C₄×C₂₃₃	1	2	yes	—
168	C₂×C₄	C₂×Q₈×C₂₃₃	1	4	yes	—
169	C₂×C₄	#41×C₂₃₃	1	4	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	4	no	—
174	C₂²	C₄×Q₈×C₂₃₃	1	4	yes	—
175	C₂²	C₂³×C₄×C₂₃₃	1	1	yes	characteristic
176	C₂²	C₂²×Q₈×C₂₃₃	1	1	yes	characteristic
177	C₂²	C₂×#41×C₂₃₃	1	2	yes	—
178	C₂²	C₂×#41×C₂₃₃	1	1	yes	characteristic
179	C₂²	H(5,𝔽₂)×C₂₃₃	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	4	no	—
186	C₄	#117×C₂₃₃	1	4	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₂×#117×C₂₃₃	1	1	yes	characteristic
189	C₂	C₂×C₄×Q₈×C₂₃₃	1	1	yes	characteristic
190	C₂	#518×C₂₃₃	1	1	yes	characteristic
191	C₂	#553×C₂₃₃	1	2	yes	—
192	C₂	#554×C₂₃₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃₃
prime radical	C₂⁴×C₂₃₃	C₂³
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
2. #389×C₂₃₃	C₂	semidirect
3. #387×C₂₃₃	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₃₃	C₂	semidirect
11. C₂×C₄⋊C₄×C₂₃₃	C₄	semidirect
96. C₂₃₃	#2345	direct (N × H)
97. #2345	C₂₃₃	direct (N × H)
99. #389	C₂×C₂₃₃	semidirect
100. #387	C₂×C₂₃₃	semidirect
103. C₄×C₄⋊C₄	C₂×C₂₃₃	semidirect
108. C₂×C₄⋊C₄	C₄×C₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₄×C₂₃₃	2
#389×C₂₃₃	2
#387×C₂₃₃	2
#380×C₂₃₃	2
C₄×C₄⋊C₄×C₂₃₃	2
C₄×C₂²⋊C₄×C₂₃₃	2
#2345	233

Subgroups — #2345×C233 #44448377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2345×C₂₃₃ #44448377