Subgroups — C₂²×Q₈×#10596842 #34035041

Order 18688.

2478 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596842	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊#10596842	C₂	1	3	yes	maximal
3	C₂²×C₄×#10596842	C₂	1	3	yes	maximal
4	C₂²×Q₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10596842	C₂²	1	3	yes	—
6	C₂³×#10596842	C₂²	1	1	yes	characteristic
7	Q₈×#10596842	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊#10596842	C₂²	1	36	yes	—
9	C₂×C₄×#10596842	C₂²	1	36	yes	—
10	C₂×Q₈×#64408	C₄	1	12	yes	—
11	C₂×Q₈×#64408	C₂²	1	12	yes	—
12	C₂²×C₄⋊#64408	C₄	1	3	yes	—
13	C₂²×C₄×#64408	C₂²	1	3	yes	—
14	C₂²×Q₈×D₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596842	—	2	12	no	—
16	C₂²×#10596842	Q₈	1	8	yes	direct product factor
17	C₂²×#10596842	C₂³	1	18	yes	—
18	C₂²×#10596842	C₂³	1	6	yes	—
19	C₄⋊#10596842	C₂³	1	48	yes	—
20	C₄×#10596842	C₂³	1	48	yes	—
21	C₂³×#64408	C₂×C₄	1	3	yes	—
22	C₂³×#64408	C₂³	1	1	yes	characteristic
23	Q₈×#64408	C₂×C₄	1	48	yes	—
24	Q₈×#64408	C₂³	1	16	yes	—
25	C₂×C₄⋊#64408	C₂×C₄	1	36	yes	—
26	C₂×C₄×#64408	C₂×C₄	1	18	yes	—
27	C₂×C₄×#64408	C₂³	1	18	yes	—
28	C₂²×Dic₁₄₆	C₈	1	3	yes	complemented
29	C₂×Q₈×D₇₃	C₈	1	12	yes	complemented
30	C₂×Q₈×D₇₃	C₂×C₄	1	12	yes	—
31	C₂²×C₄×D₇₃	C₂×C₄	1	3	yes	—
32	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic, complemented
33	C₂×#10596842	—	2	36	no	—
34	C₂×#10596842	C₂×Q₈	1	24	yes	direct product factor
35	C₂×#10596842	C₂⁴	1	12	yes	—
36	C₂×#10596842	C₂⁴	1	4	yes	—
37	C₂²×#64408	—	2	12	no	—
38	C₂²×#64408	C₂²×C₄	1	18	yes	—
39	C₂²×#64408	C₂²×C₄	1	3	yes	—
40	C₂²×#64408	C₂×Q₈	1	4	yes	—
41	C₂²×#64408	#41	1	4	yes	—
42	C₂²×#64408	C₂⁴	1	3	yes	—
43	C₄⋊#64408	C₂²×C₄	1	48	yes	—
44	C₄×#64408	C₂²×C₄	1	36	yes	—
45	C₄×#64408	C₂⁴	1	12	yes	—
46	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₄₆	C₂×C₈	1	36	yes	complemented
48	Q₈×D₇₃	C₂×C₈	1	48	yes	complemented
49	Q₈×D₇₃	C₂²×C₄	1	16	yes	—
50	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
51	C₂×C₄×D₇₃	C₂×C₈	1	18	yes	—
52	C₂×C₄×D₇₃	C₂²×C₄	1	18	yes	—
53	C₂×Q₈×C₇₃	C₂×C₈	1	12	yes	complemented
54	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
55	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	C₂²×D₇₃	C₄×Q₈	1	4	yes	—
57	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
58	C₂²×D₇₃	#131	1	4	yes	—
59	C₂²×D₇₃	C₂³×C₄	1	3	yes	—
60	C₂×#64408	—	2	36	no	—
61	C₂×#64408	C₄×Q₈	1	12	yes	—
62	C₂×#64408	C₂³×C₄	1	12	yes	—
63	C₂×#64408	C₂³×C₄	1	3	yes	—
64	C₂×#64408	C₂²×Q₈	1	6	yes	—
65	C₂×#64408	C₂×#41	1	6	yes	—
66	C₂×#64408	C₂⁵	1	1	yes	characteristic
67	#10596842	—	2	12	no	—
68	#10596842	C₂²×Q₈	1	8	yes	direct product factor
69	Q₈×C₇₃	C₂²×C₈	1	16	yes	complemented
70	C₂×C₄×C₇₃	C₂²×C₈	1	18	yes	—
71	C₂×Dic₇₃	C₂²×C₈	1	18	yes	—
72	C₄×D₇₃	C₂²×C₈	1	36	yes	—
73	C₄×D₇₃	C₂³×C₄	1	12	yes	—
74	Dic₁₄₆	C₂²×C₈	1	48	yes	complemented
75	C₂²×C₇₃	C₈×Q₈	1	4	yes	complemented
76	C₂²×C₇₃	C₂³×C₈	1	3	yes	—
77	C₂×D₇₃	C₈×Q₈	1	12	yes	complemented
78	C₂×D₇₃	C₂×C₄×Q₈	1	6	yes	—
79	C₂×D₇₃	C₂³×C₈	1	3	yes	—
80	C₂×D₇₃	C₂×#131	1	6	yes	—
81	C₂×D₇₃	C₂⁴×C₄	1	1	yes	characteristic
82	#64408	—	2	12	no	—
83	#64408	C₂×C₄×Q₈	1	6	yes	—
84	#64408	C₂³×Q₈	1	1	yes	characteristic
85	#64408	C₂²×#41	1	1	yes	characteristic
86	C₄×C₇₃	C₂³×C₈	1	12	yes	—
87	Dic₇₃	C₂³×C₈	1	12	yes	—
88	C₂²×C₈×Q₈	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	C₂×C₈×Q₈	1	6	yes	complemented
90	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
91	D₇₃	C₂×C₈×Q₈	1	6	yes	complemented
92	D₇₃	C₂²×C₄×Q₈	1	1	yes	characteristic
93	D₇₃	C₂²×#131	1	1	yes	characteristic
94	C₂²×C₄×Q₈	—	73	1	no	—
95	C₂×C₈×Q₈	—	73	24	no	—
96	C₂²×C₄⋊C₈	—	73	3	no	—
97	C₂²×C₄×C₈	—	73	3	no	—
98	C₇₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
99	C₂³×Q₈	—	73	1	no	—
100	C₂³×C₈	—	73	1	no	—
101	C₂³×C₈	—	73	3	no	—
102	C₂×C₄×Q₈	—	73	12	no	—
103	C₂×C₄×Q₈	—	73	12	no	—
104	C₂²×C₄⋊C₄	—	73	3	no	—
105	C₂²×C₄²	—	73	3	no	—
106	C₈×Q₈	—	73	64	no	—
107	C₂×C₄⋊C₈	—	73	36	no	—
108	C₂×C₄×C₈	—	73	36	no	—
109	C₂²×Q₈	—	73	12	no	—
110	C₂²×Q₈	—	73	12	no	—
111	C₂²×Q₈	—	73	3	no	—
112	C₂²×Q₈	#10596842	1	1	yes	characteristic, direct product factor
113	C₂³×C₄	—	73	1	no	—
114	C₂³×C₄	—	73	3	no	—
115	C₂³×C₄	—	73	3	no	—
116	C₂²×C₈	—	73	8	no	—
117	C₂²×C₈	—	73	6	no	—
118	C₂²×C₈	—	73	18	no	—
119	C₂²×C₈	—	146	12	no	—
120	C₄×Q₈	—	73	48	no	—
121	C₄×Q₈	—	73	16	no	—
122	C₂×C₄⋊C₄	—	73	36	no	—
123	C₂×C₄²	—	73	18	no	—
124	C₂×C₄²	—	73	18	no	—
125	C₄⋊C₈	—	73	48	no	—
126	C₄×C₈	—	73	48	no	—
127	C₂⁴	—	73	1	no	—
128	C₂×Q₈	—	73	48	no	—
129	C₂×Q₈	—	73	16	no	—
130	C₂×Q₈	—	73	36	no	—
131	C₂×Q₈	C₂×#10596842	1	12	yes	direct product factor
132	C₂²×C₄	—	73	3	no	—
133	C₂²×C₄	—	73	4	no	—
134	C₂²×C₄	—	73	4	no	—
135	C₂²×C₄	—	73	3	no	—
136	C₂²×C₄	—	73	3	no	—
137	C₂²×C₄	—	73	18	no	—
138	C₂²×C₄	—	73	18	no	—
139	C₂²×C₄	—	73	18	no	—
140	C₂²×C₄	—	146	12	no	—
141	C₂²×C₄	C₂×#10596842	1	3	yes	—
142	C₂×C₈	—	73	24	no	—
143	C₂×C₈	—	73	4	no	—
144	C₂×C₈	—	73	12	no	—
145	C₂×C₈	—	146	36	no	—
146	C₄⋊C₄	—	73	48	no	—
147	C₄²	—	73	36	no	—
148	C₄²	—	73	12	no	—
149	C₂³	—	73	4	no	—
150	C₂³	—	73	4	no	—
151	C₂³	—	73	3	no	—
152	C₂³	—	73	3	no	—
153	C₂³	C₂²×#10596842	1	1	yes	characteristic
154	Q₈	—	73	48	no	—
155	Q₈	C₂²×#10596842	1	16	yes	direct product factor
156	C₂×C₄	—	73	1	no	—
157	C₂×C₄	—	73	3	no	—
158	C₂×C₄	—	73	6	no	—
159	C₂×C₄	—	73	6	no	—
160	C₂×C₄	—	73	12	no	—
161	C₂×C₄	—	73	18	no	—
162	C₂×C₄	—	73	12	no	—
163	C₂×C₄	—	73	36	no	—
164	C₂×C₄	—	73	12	no	—
165	C₂×C₄	—	146	36	no	—
166	C₂×C₄	C₂²×#10596842	1	18	yes	—
167	C₈	—	73	8	no	—
168	C₈	—	146	12	no	—
169	C₂²	—	73	6	no	—
170	C₂²	—	73	6	no	—
171	C₂²	—	73	12	no	—
172	C₂²	—	73	1	no	—
173	C₂²	—	73	3	no	—
174	C₂²	Q₈×#10596842	1	4	yes	direct product factor
175	C₂²	C₂³×#10596842	1	3	yes	—
176	C₄	—	73	1	no	—
177	C₄	—	73	1	no	—
178	C₄	—	73	6	no	—
179	C₄	—	73	12	no	—
180	C₄	—	146	12	no	—
181	C₄	C₂³×#10596842	1	12	yes	—
182	C₂	—	73	1	no	—
183	C₂	—	73	1	no	—
184	C₂	—	73	6	no	—
185	C₂	C₂×Q₈×#10596842	1	6	yes	direct product factor
186	C₂	C₂⁴×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×#10596842
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₂	C₂⁴×#10596842
Fitting	C₂²×Q₈×C₇₃	C₈
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂²×C₄
cosocle	C₂×#64408	C₂⁵
hypercenter	C₂²×Q₈	#10596842
hypocenter	C₇₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#10596842	C₂	direct (N × H)
7. Q₈×#10596842	C₂²	direct (N × H)
16. C₂²×#10596842	Q₈	direct (N × H)
28. C₂²×Dic₁₄₆	C₈	semidirect
29. C₂×Q₈×D₇₃	C₈	semidirect
32. C₂²×Q₈×C₇₃	C₈	semidirect
34. C₂×#10596842	C₂×Q₈	direct (N × H)
47. C₂×Dic₁₄₆	C₂×C₈	semidirect
48. Q₈×D₇₃	C₂×C₈	semidirect
53. C₂×Q₈×C₇₃	C₂×C₈	semidirect
68. #10596842	C₂²×Q₈	direct (N × H)
69. Q₈×C₇₃	C₂²×C₈	semidirect
74. Dic₁₄₆	C₂²×C₈	semidirect
75. C₂²×C₇₃	C₈×Q₈	semidirect
77. C₂×D₇₃	C₈×Q₈	semidirect
89. C₂×C₇₃	C₂×C₈×Q₈	semidirect
91. D₇₃	C₂×C₈×Q₈	semidirect
98. C₇₃	C₂²×C₈×Q₈	semidirect
112. C₂²×Q₈	#10596842	direct (N × H)
131. C₂×Q₈	C₂×#10596842	direct (N × H)
155. Q₈	C₂²×#10596842	direct (N × H)
174. C₂²	Q₈×#10596842	direct (N × H)
185. C₂	C₂×Q₈×#10596842	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×Q₈×#10596842	2
C₂²×C₄⋊#10596842	2
C₂²×C₄×#10596842	2
C₂²×Q₈×#64408	2
C₂²×C₈×Q₈	73

Subgroups — C22×Q8×#10596842 #34035041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×#10596842 #34035041