Subgroups — C₄×C₄⋊Q₈×C₂₃₃ #44448339

Order 29824.

588 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃₃	C₂	1	2	yes	maximal
3	#391×C₂₃₃	C₂	1	2	yes	maximal
4	#389×C₂₃₃	C₂	1	4	yes	maximal
5	#384×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₂₃₃	C₂	1	4	yes	maximal
7	C₄³×C₂₃₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₂₃₃	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₂₃₃	C₄	1	8	yes	direct product factor
10	C₄×Q₈×C₂₃₃	—	2	8	no	—
11	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₃₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₃₃	C₂²	1	4	yes	—
18	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃₃	C₂²	1	2	yes	—
21	#95×C₂₃₃	C₂²	1	8	yes	—
22	C₂×Q₈×C₂₃₃	—	2	4	no	—
23	C₂×Q₈×C₂₃₃	—	2	4	no	—
24	C₂×Q₈×C₂₃₃	C₂×C₄	1	8	yes	—
25	C₂²×C₄×C₂₃₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₃₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₂₃₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₃₃	—	2	8	no	—
33	C₄⋊C₄×C₂₃₃	—	2	4	no	—
34	C₄⋊C₄×C₂₃₃	—	2	8	no	—
35	C₄⋊C₄×C₂₃₃	C₂×C₄	1	16	yes	—
36	C₄²×C₂₃₃	—	2	8	no	—
37	C₄²×C₂₃₃	—	2	2	no	—
38	C₄²×C₂₃₃	—	2	4	no	—
39	C₄²×C₂₃₃	C₂×C₄	1	4	yes	—
40	C₄²×C₂₃₃	D₄	1	4	yes	—
41	C₄²×C₂₃₃	Q₈	1	8	yes	complemented
42	C₂³×C₂₃₃	C₂⁴	1	1	yes	characteristic
43	Q₈×C₂₃₃	—	2	16	no	—
44	C₂×C₄×C₂₃₃	—	2	4	no	—
45	C₂×C₄×C₂₃₃	—	2	2	no	—
46	C₂×C₄×C₂₃₃	—	2	2	no	—
47	C₂×C₄×C₂₃₃	—	2	2	no	—
48	C₂×C₄×C₂₃₃	—	2	4	no	—
49	C₂×C₄×C₂₃₃	—	2	4	no	—
50	C₂×C₄×C₂₃₃	—	2	8	no	—
51	C₂×C₄×C₂₃₃	C₂²×C₄	1	8	yes	—
52	C₂×C₄×C₂₃₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₃₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₂₃₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₃₃	C₂×Q₈	1	4	yes	—
58	C₂×C₄×C₂₃₃	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₂₃₃	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₂₃₃	#41	1	2	yes	—
61	C₂×C₄×C₂₃₃	#41	1	4	yes	—
62	C₂²×C₂₃₃	C₂³×C₄	1	1	yes	characteristic
63	C₂²×C₂₃₃	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₂₃₃	C₂²×Q₈	1	2	yes	—
65	C₂²×C₂₃₃	C₂×#41	1	1	yes	characteristic
66	C₂²×C₂₃₃	C₂×#41	1	2	yes	—
67	C₄×C₂₃₃	—	2	8	no	—
68	C₄×C₂₃₃	—	2	2	no	—
69	C₄×C₂₃₃	—	2	4	no	—
70	C₄×C₂₃₃	—	4	4	no	—
71	C₄×C₂₃₃	C₄×D₄	1	4	yes	—
72	C₄×C₂₃₃	C₄×Q₈	1	8	yes	complemented
73	C₄×C₂₃₃	C₄⋊Q₈	1	4	yes	direct product factor
74	C₂×C₂₃₃	C₂×C₄×D₄	1	1	yes	characteristic
75	C₂×C₂₃₃	C₂×C₄×Q₈	1	2	yes	—
76	C₂×C₂₃₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₂₃₃	#532	1	1	yes	characteristic
78	C₂×C₂₃₃	#533	1	2	yes	—
79	C₂₃₃	C₄×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
80	C₄×C₄⋊Q₈	C₂₃₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₄⋊Q₈	C₂×C₂₃₃	1	1	yes	characteristic
82	C₂×C₄×Q₈	C₂×C₂₃₃	1	2	yes	—
83	#391	C₂×C₂₃₃	1	2	yes	—
84	#389	C₂×C₂₃₃	1	4	yes	—
85	#384	C₂×C₂₃₃	1	1	yes	characteristic
86	C₄×C₄⋊C₄	C₂×C₂₃₃	1	4	yes	—
87	C₄³	C₂×C₂₃₃	1	1	yes	characteristic
88	C₂²×Q₈	C₂²×C₂₃₃	1	2	yes	—
89	C₄⋊Q₈	C₄×C₂₃₃	1	8	yes	direct product factor
90	C₄×Q₈	—	2	8	no	—
91	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	4	yes	—
93	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	4	yes	—
94	C₂×C₄⋊C₄	C₂²×C₂₃₃	1	4	yes	—
95	C₂×C₄²	C₂²×C₂₃₃	1	4	yes	—
96	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
97	C₂×C₄²	C₂²×C₂₃₃	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₂₃₃	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₂₃₃	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₂₃₃	1	2	yes	—
101	#95	C₂²×C₂₃₃	1	8	yes	—
102	C₂×Q₈	—	2	4	no	—
103	C₂×Q₈	—	2	4	no	—
104	C₂×Q₈	C₂×C₄×C₂₃₃	1	8	yes	—
105	C₂²×C₄	C₂³×C₂₃₃	1	4	yes	—
106	C₂²×C₄	C₂³×C₂₃₃	1	4	yes	—
107	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
108	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
109	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
110	C₂²×C₄	C₂³×C₂₃₃	1	2	yes	—
111	C₂²×C₄	C₂³×C₂₃₃	1	1	yes	characteristic
112	C₄⋊C₄	—	2	8	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	8	no	—
115	C₄⋊C₄	C₂×C₄×C₂₃₃	1	16	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	4	no	—
119	C₄²	C₂×C₄×C₂₃₃	1	4	yes	—
120	C₄²	D₄×C₂₃₃	1	4	yes	—
121	C₄²	Q₈×C₂₃₃	1	8	yes	complemented
122	C₂³	C₂⁴×C₂₃₃	1	1	yes	characteristic
123	Q₈	—	2	16	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	C₂²×C₄×C₂₃₃	1	8	yes	—
132	C₂×C₄	C₂²×C₄×C₂₃₃	1	2	yes	—
133	C₂×C₄	C₂²×C₄×C₂₃₃	1	4	yes	—
134	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
135	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
136	C₂×C₄	C₂×D₄×C₂₃₃	1	2	yes	—
137	C₂×C₄	C₂×Q₈×C₂₃₃	1	4	yes	—
138	C₂×C₄	C₂×Q₈×C₂₃₃	1	4	yes	—
139	C₂×C₄	C₂×Q₈×C₂₃₃	1	4	yes	—
140	C₂×C₄	#41×C₂₃₃	1	2	yes	—
141	C₂×C₄	#41×C₂₃₃	1	4	yes	—
142	C₂²	C₂³×C₄×C₂₃₃	1	1	yes	characteristic
143	C₂²	C₂²×D₄×C₂₃₃	1	1	yes	characteristic
144	C₂²	C₂²×Q₈×C₂₃₃	1	2	yes	—
145	C₂²	C₂×#41×C₂₃₃	1	1	yes	characteristic
146	C₂²	C₂×#41×C₂₃₃	1	2	yes	—
147	C₄	—	2	8	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	4	no	—
151	C₄	C₄×D₄×C₂₃₃	1	4	yes	—
152	C₄	C₄×Q₈×C₂₃₃	1	8	yes	complemented
153	C₄	C₄⋊Q₈×C₂₃₃	1	4	yes	direct product factor
154	C₂	C₂×C₄×D₄×C₂₃₃	1	1	yes	characteristic
155	C₂	C₂×C₄×Q₈×C₂₃₃	1	2	yes	—
156	C₂	C₂×C₄⋊Q₈×C₂₃₃	1	1	yes	characteristic
157	C₂	#532×C₂₃₃	1	1	yes	characteristic
158	C₂	#533×C₂₃₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₄⋊Q₈×C₂₃₃	C₄	direct (N × H)
41. C₄²×C₂₃₃	Q₈	semidirect
72. C₄×C₂₃₃	C₄×Q₈	semidirect
73. C₄×C₂₃₃	C₄⋊Q₈	direct (N × H)
79. C₂₃₃	C₄×C₄⋊Q₈	direct (N × H)
80. C₄×C₄⋊Q₈	C₂₃₃	direct (N × H)
89. C₄⋊Q₈	C₄×C₂₃₃	direct (N × H)
121. C₄²	Q₈×C₂₃₃	semidirect
152. C₄	C₄×Q₈×C₂₃₃	semidirect
153. C₄	C₄⋊Q₈×C₂₃₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₂₃₃	2
C₂×C₄×Q₈×C₂₃₃	2
#391×C₂₃₃	2
#389×C₂₃₃	2
#384×C₂₃₃	2
C₄×C₄⋊C₄×C₂₃₃	2
C₄³×C₂₃₃	2
C₄×C₄⋊Q₈	233

Subgroups — C4×C4⋊Q8×C233 #44448339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊Q₈×C₂₃₃ #44448339