Subgroups — C₂×#26011455 #36361999

Order 22144.

1008 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14583810	C₂	1	2	yes	maximal, complemented
2	#26011455	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊#13063436	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14583810	—	2	8	no	—
6	C₂×#14583810	C₂²	1	8	yes	complemented
7	C₂²×#13063436	C₂²	1	4	yes	complemented
8	C₂²⋊#13063436	C₂²	1	16	yes	complemented
9	C₂⁵×C₁₇₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₇₃	—	2	8	no	—
11	C₂³×C₄×C₁₇₃	C₄	1	1	yes	characteristic
12	C₂³×C₄×C₁₇₃	C₄	1	4	yes	—
13	C₂³×C₄×C₁₇₃	C₄	1	2	yes	—
14	C₂³×C₄×C₁₇₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₇₃	C₂²	1	4	yes	—
16	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
17	#14583810	—	2	16	no	—
18	C₂×#13063436	—	2	8	no	—
19	C₂×#13063436	C₂³	1	8	yes	complemented
20	C₂⁴×C₁₇₃	—	2	4	no	—
21	C₂⁴×C₁₇₃	—	2	8	no	—
22	C₂⁴×C₁₇₃	C₂×C₄	1	4	yes	—
23	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂⁴×C₁₇₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₇₃	—	2	4	no	—
26	C₂²×C₄×C₁₇₃	—	2	16	no	—
27	C₂²×C₄×C₁₇₃	—	2	8	no	—
28	C₂²×C₄×C₁₇₃	—	2	8	no	—
29	C₂²×C₄×C₁₇₃	—	2	16	no	—
30	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₇₃	C₂×C₄	1	8	yes	—
34	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇₃	D₄	1	8	yes	—
37	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
40	#13063436	—	2	8	no	—
41	C₂³×C₁₇₃	—	2	32	no	—
42	C₂³×C₁₇₃	—	2	8	no	—
43	C₂³×C₁₇₃	—	2	4	no	—
44	C₂³×C₁₇₃	—	2	4	no	—
45	C₂³×C₁₇₃	—	2	4	no	—
46	C₂³×C₁₇₃	—	2	16	no	—
47	C₂³×C₁₇₃	C₂²⋊C₄	1	4	yes	—
48	C₂³×C₁₇₃	M₁₆	1	8	yes	complemented
49	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
50	C₂³×C₁₇₃	C₂²×C₄	1	4	yes	—
51	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇₃	—	2	8	no	—
53	C₂×C₄×C₁₇₃	—	2	4	no	—
54	C₂×C₄×C₁₇₃	—	2	8	no	—
55	C₂×C₄×C₁₇₃	—	2	16	no	—
56	C₂×C₄×C₁₇₃	—	2	8	no	—
57	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	8	yes	—
58	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	4	yes	—
59	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₁₇₃	C₂×D₄	1	8	yes	—
64	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
65	C₂²×C₁₇₃	—	2	32	no	—
66	C₂²×C₁₇₃	—	2	4	no	—
67	C₂²×C₁₇₃	—	2	4	no	—
68	C₂²×C₁₇₃	—	2	4	no	—
69	C₂²×C₁₇₃	—	2	8	no	—
70	C₂²×C₁₇₃	—	2	16	no	—
71	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	4	yes	—
72	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂²×C₁₇₃	C₂×M₁₆	1	4	yes	—
74	C₂²×C₁₇₃	C₂×M₁₆	1	8	yes	complemented
75	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
76	C₄×C₁₇₃	—	2	4	no	—
77	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
78	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	4	yes	—
79	C₄×C₁₇₃	C₂²×D₄	1	2	yes	—
80	C₂×C₁₇₃	—	2	4	no	—
81	C₂×C₁₇₃	—	2	8	no	—
82	C₂×C₁₇₃	#407	1	4	yes	complemented
83	C₂×C₁₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₁₇₃	C₂²×M₁₆	1	2	yes	—
85	C₁₇₃	C₂×#407	1	1	yes	characteristic, complemented, Sylow
86	C₂×#407	—	173	1	no	maximal, Sylow
87	C₂⁴×C₄	D₁₇₃	1	1	yes	characteristic
88	C₂²×M₁₆	—	173	2	no	—
89	#407	—	173	8	no	—
90	C₂×C₂²⋊C₈	—	173	4	no	—
91	C₂⁵	Dic₁₇₃	1	1	yes	characteristic
92	C₂³×C₄	—	2	8	no	—
93	C₂³×C₄	Dic₁₇₃	1	1	yes	characteristic
94	C₂³×C₄	Dic₁₇₃	1	4	yes	—
95	C₂³×C₄	Dic₁₇₃	1	2	yes	—
96	C₂³×C₄	C₂×D₁₇₃	1	4	yes	—
97	C₂³×C₄	C₂×D₁₇₃	1	2	yes	—
98	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
99	C₂×M₁₆	—	346	8	no	—
100	C₂×M₁₆	—	173	8	no	—
101	C₂²×C₈	—	173	4	no	—
102	C₂²⋊C₈	—	173	16	no	—
103	C₂⁴	—	2	8	no	—
104	C₂⁴	—	2	4	no	—
105	C₂⁴	C₂×Dic₁₇₃	1	4	yes	—
106	C₂⁴	C₂×Dic₁₇₃	1	2	yes	—
107	C₂⁴	C₂×Dic₁₇₃	1	1	yes	characteristic
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	8	no	—
110	C₂²×C₄	—	2	16	no	—
111	C₂²×C₄	—	2	16	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	C₂×Dic₁₇₃	1	4	yes	—
114	C₂²×C₄	C₂×Dic₁₇₃	1	2	yes	—
115	C₂²×C₄	C₂×Dic₁₇₃	1	8	yes	—
116	C₂²×C₄	C₂×Dic₁₇₃	1	4	yes	—
117	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
118	C₂²×C₄	C₂×Dic₁₇₃	1	2	yes	—
119	C₂²×C₄	C₂²⋊D₁₇₃	1	8	yes	—
120	C₂²×C₄	C₂²×D₁₇₃	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
123	M₁₆	—	346	16	no	—
124	C₂×C₈	—	346	8	no	—
125	C₂×C₈	—	173	8	no	—
126	C₂³	—	2	32	no	—
127	C₂³	—	2	8	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	16	no	—
132	C₂³	C₂²⋊Dic₁₇₃	1	4	yes	—
133	C₂³	#14583810	1	8	yes	complemented
134	C₂³	C₂²×Dic₁₇₃	1	2	yes	—
135	C₂³	C₂²×Dic₁₇₃	1	4	yes	—
136	C₂³	C₂²×Dic₁₇₃	1	1	yes	characteristic
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	16	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	C₂²⋊Dic₁₇₃	1	8	yes	—
143	C₂×C₄	C₂²⋊Dic₁₇₃	1	4	yes	—
144	C₂×C₄	C₂²×Dic₁₇₃	1	4	yes	—
145	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
146	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	8	yes	—
149	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
150	C₈	—	346	8	no	—
151	C₂²	—	2	32	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂×C₂²⋊Dic₁₇₃	1	4	yes	—
158	C₂²	C₂×C₂²⋊Dic₁₇₃	1	2	yes	—
159	C₂²	C₂×#14583810	1	4	yes	—
160	C₂²	C₂×#14583810	1	8	yes	complemented
161	C₂²	C₂³×Dic₁₇₃	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	C₂×C₂²⋊Dic₁₇₃	1	4	yes	—
164	C₄	C₂×C₂²⋊Dic₁₇₃	1	2	yes	—
165	C₄	C₂²×C₂²⋊D₁₇₃	1	2	yes	—
166	C₂	—	2	4	no	—
167	C₂	—	2	8	no	—
168	C₂	#26011455	1	4	yes	direct product factor
169	C₂	C₂²×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
170	C₂	C₂²×#14583810	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₇₃
commutator	C₂²×C₁₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	C₂⁴×C₄×C₁₇₃	C₂
socle	C₂³×C₁₇₃	C₂²×C₄
prime radical	C₂⁵×C₁₇₃	C₄
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	C₂⁴×C₄	D₁₇₃
hypocenter	C₁₇₃	C₂×#407
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14583810	C₂	semidirect
2. #26011455	C₂	direct (N × H)
3. C₂×C₂²⋊#13063436	C₂	semidirect
6. C₂×#14583810	C₂²	semidirect
7. C₂²×#13063436	C₂²	semidirect
8. C₂²⋊#13063436	C₂²	semidirect
19. C₂×#13063436	C₂³	semidirect
48. C₂³×C₁₇₃	M₁₆	semidirect
74. C₂²×C₁₇₃	C₂×M₁₆	semidirect
82. C₂×C₁₇₃	#407	semidirect
85. C₁₇₃	C₂×#407	semidirect
133. C₂³	#14583810	semidirect
160. C₂²	C₂×#14583810	semidirect
168. C₂	#26011455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14583810	2
#26011455	2
C₂×C₂²⋊#13063436	2
C₂⁴×C₄×C₁₇₃	2
C₂×#407	173

Subgroups — C2×#26011455 #36361999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26011455 #36361999