Subgroups — C₂×#22627155 #34034848

Order 18688.

582 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627155	C₂	1	8	yes	maximal, direct product factor
2	C₂×#17989690	C₂	1	2	yes	maximal
3	C₂×C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#11924040	C₂	1	2	yes	maximal
5	C₂×C₄×#11924040	C₂	1	1	yes	maximal, characteristic
6	C₂×#17989636	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924040	C₂²	1	2	yes	—
8	C₂²×#11924040	C₂²	1	1	yes	characteristic
9	C₂²×#11924040	C₂²	1	1	yes	characteristic
10	#17989690	C₂²	1	8	yes	—
11	C₄⋊#11924040	C₂²	1	4	yes	—
12	C₄×#11924040	C₂²	1	8	yes	—
13	C₄×#11924040	C₂²	1	4	yes	—
14	#17989636	C₄	1	4	yes	—
15	#17989636	C₂²	1	4	yes	—
16	C₂×#14556658	C₄	1	2	yes	—
17	C₂×C₄⋊#10596833	C₄	1	1	yes	characteristic
18	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10596833	C₂²	1	2	yes	—
20	C₂×#14556556	C₄	1	1	yes	characteristic
21	C₂×#11924040	—	2	4	no	—
22	C₂×#11924040	—	2	2	no	—
23	C₂×#11924040	Q₈	1	4	yes	complemented
24	C₂×#11924040	C₂³	1	4	yes	—
25	C₂×#11924040	C₂³	1	2	yes	—
26	C₂×#11924040	C₂³	1	2	yes	—
27	C₂²×#10596833	C₂×C₄	1	2	yes	—
28	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596833	C₂³	1	1	yes	characteristic
30	#14556658	C₂×C₄	1	8	yes	—
31	C₄⋊#10596833	C₂×C₄	1	4	yes	—
32	C₄×#10596833	C₂×C₄	1	2	yes	—
33	C₄×#10596833	C₂×C₄	1	4	yes	—
34	C₄×#10596833	C₂³	1	2	yes	—
35	C₄×#10596833	C₂³	1	4	yes	—
36	C₂²×Dic₁₄₆	C₈	1	1	yes	characteristic
37	#14556556	C₈	1	4	yes	—
38	#14556556	C₂×C₄	1	4	yes	—
39	C₂×#11924007	C₈	1	2	yes	—
40	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
43	#11924040	—	2	4	no	—
44	#11924040	—	2	2	no	—
45	#11924040	C₂×Q₈	1	4	yes	complemented
46	C₂×#10596833	—	2	4	no	—
47	C₂×#10596833	—	2	2	no	—
48	C₂×#10596833	C₂²×C₄	1	4	yes	—
49	C₂×#10596833	C₂²×C₄	1	2	yes	—
50	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
51	C₂×#10596833	C₂×Q₈	1	2	yes	—
52	C₂×#10596833	#41	1	2	yes	—
53	C₂×#10596833	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₁₄₆	C₁₆	1	8	yes	complemented
55	C₂×Dic₁₄₆	C₂×C₈	1	4	yes	—
56	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
58	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
59	#11924007	C₂×C₈	1	8	yes	—
60	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
61	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
62	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
63	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
67	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	4	no	—
69	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
71	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
72	C₂×Dic₇₃	—	2	2	no	—
73	C₂×Dic₇₃	C₂×C₁₆	1	8	yes	—
74	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
75	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
76	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
78	C₂×Dic₇₃	#131	1	2	yes	—
79	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
80	Dic₁₄₆	C₂×C₁₆	1	16	yes	complemented
81	#10596833	—	2	4	no	—
82	#10596833	—	2	2	no	—
83	#10596833	C₄×Q₈	1	2	yes	—
84	#10596833	C₂²×Q₈	1	1	yes	characteristic
85	#10596833	C₂×#41	1	1	yes	characteristic
86	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
87	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
88	C₂²×C₇₃	#504	1	2	yes	—
89	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₇₃	—	2	4	no	—
91	C₄×C₇₃	C₂²×C₁₆	1	4	yes	—
92	Dic₇₃	—	2	2	no	—
93	Dic₇₃	C₈×Q₈	1	2	yes	—
94	Dic₇₃	C₂²×C₁₆	1	8	yes	—
95	Dic₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
96	Dic₇₃	C₂×#131	1	1	yes	characteristic
97	C₂×Q₈×C₁₆	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	Q₈×C₁₆	1	4	yes	complemented
99	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
100	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
101	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
102	C₂×C₈×Q₈	—	73	1	no	—
103	Q₈×C₁₆	—	73	8	no	—
104	C₂×C₄⋊C₁₆	—	73	1	no	—
105	C₂×C₄⋊C₁₆	—	73	2	no	—
106	C₂×C₄×C₁₆	—	73	1	no	—
107	C₂×C₄×C₁₆	—	73	2	no	—
108	C₇₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄×Q₈	—	73	1	no	—
110	C₂²×C₁₆	—	73	1	no	—
111	C₂²×C₁₆	—	73	2	no	—
112	C₂²×C₁₆	—	73	1	no	—
113	C₈×Q₈	—	73	4	no	—
114	C₈×Q₈	—	73	4	no	—
115	C₂×C₄⋊C₈	—	73	1	no	—
116	C₂×C₄⋊C₈	—	73	2	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	2	no	—
119	C₄⋊C₁₆	—	73	4	no	—
120	C₄⋊C₁₆	—	73	8	no	—
121	C₄×C₁₆	—	73	4	no	—
122	C₄×C₁₆	—	73	8	no	—
123	C₂²×Q₈	—	73	1	no	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	2	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₄×Q₈	—	73	4	no	—
128	C₄×Q₈	—	73	4	no	—
129	C₂×C₄⋊C₄	—	73	2	no	—
130	C₂×C₄⋊C₄	#10596842	1	1	yes	characteristic
131	C₂×C₄²	—	73	2	no	—
132	C₂×C₄²	—	73	1	no	—
133	C₂×C₁₆	—	73	4	no	—
134	C₂×C₁₆	—	73	2	no	—
135	C₂×C₁₆	—	73	4	no	—
136	C₂×C₁₆	—	73	2	no	—
137	C₂×C₁₆	—	146	4	no	—
138	C₂×C₁₆	—	146	2	no	—
139	C₄⋊C₈	—	73	4	no	—
140	C₄⋊C₈	—	73	8	no	—
141	C₄×C₈	—	73	2	no	—
142	C₄×C₈	—	73	4	no	—
143	C₄×C₈	—	73	4	no	—
144	C₄×C₈	—	73	2	no	—
145	C₂×Q₈	—	73	8	no	—
146	C₂×Q₈	—	73	4	no	—
147	C₂²×C₄	—	73	2	no	—
148	C₂²×C₄	—	73	1	no	—
149	C₂²×C₄	—	73	1	no	—
150	C₂²×C₄	C₂×#10596842	1	2	yes	—
151	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	2	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	4	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	146	4	no	—
159	C₂×C₈	—	146	2	no	—
160	C₄⋊C₄	—	73	8	no	—
161	C₄⋊C₄	C₂×#10596842	1	4	yes	—
162	C₄²	—	73	4	no	—
163	C₄²	—	73	2	no	—
164	C₄²	—	73	4	no	—
165	C₄²	—	73	2	no	—
166	C₁₆	—	73	4	no	—
167	C₁₆	—	146	4	no	—
168	C₁₆	—	146	2	no	—
169	C₂³	C₂²×#10596842	1	1	yes	characteristic
170	Q₈	—	73	16	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	73	2	no	—
173	C₂×C₄	—	73	2	no	—
174	C₂×C₄	—	73	8	no	—
175	C₂×C₄	—	73	1	no	—
176	C₂×C₄	—	73	1	no	—
177	C₂×C₄	—	73	4	no	—
178	C₂×C₄	—	73	2	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	#14556686	1	4	yes	—
181	C₂×C₄	C₂²×#10596842	1	4	yes	—
182	C₂×C₄	C₂²×#10596842	1	2	yes	—
183	C₈	—	73	1	no	—
184	C₈	—	73	1	no	—
185	C₈	—	73	2	no	—
186	C₈	—	146	4	no	—
187	C₈	—	146	2	no	—
188	C₂²	Q₈×#10596842	1	2	yes	—
189	C₂²	C₂×#14556686	1	2	yes	—
190	C₂²	#17989708	1	2	yes	—
191	C₂²	C₂³×#10596842	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	73	1	no	—
194	C₄	—	73	1	no	—
195	C₄	—	73	2	no	—
196	C₄	—	73	8	no	—
197	C₄	—	146	2	no	—
198	C₄	C₂×#14556686	1	4	yes	—
199	C₂	#22627155	1	4	yes	direct product factor
200	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
201	C₂	C₂²×#14556686	1	1	yes	characteristic
202	C₂	C₂×#17989708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×#10596842
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₂²	C₂³×#10596842
Fitting	C₂×C₄⋊C₄×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₂×C₄⋊C₄	#10596842
hypocenter	C₇₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #22627155	C₂	direct (N × H)
23. C₂×#11924040	Q₈	semidirect
45. #11924040	C₂×Q₈	semidirect
54. C₂×Dic₁₄₆	C₁₆	semidirect
80. Dic₁₄₆	C₂×C₁₆	semidirect
98. C₂×C₇₃	Q₈×C₁₆	semidirect
108. C₇₃	C₂×Q₈×C₁₆	semidirect
199. C₂	#22627155	direct (N × H)

Maximal subgroups

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

subgroup	index
#22627155	2
C₂×#17989690	2
C₂×C₄⋊#11924040	2
C₂×C₄×#11924040	2
C₂×#17989636	2
C₂×Q₈×C₁₆	73

Subgroups — C2×#22627155 #34034848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22627155 #34034848