Subgroups — C₂×#26012570 #36370556

Order 22144.

1460 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012570	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19149767	C₂	1	2	yes	maximal, complemented
3	C₂×#19149766	C₂	1	6	yes	maximal, complemented
4	C₂²×#14583836	C₂	1	3	yes	maximal, complemented
5	C₂×D₁₇₃⋊M₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₁₇₃	C₂	1	1	yes	maximal, characteristic
7	#19149767	C₂²	1	16	yes	complemented
8	#19149766	C₂²	1	48	yes	complemented
9	C₂×#14583836	C₂²	1	1	yes	characteristic
10	C₂×#14583836	C₂²	1	3	yes	—
11	C₂×#14583836	C₂²	1	6	yes	complemented
12	C₂×#14583836	C₂²	1	24	yes	complemented
13	D₁₇₃⋊M₁₆	C₂²	1	24	yes	complemented
14	C₂×#14583835	C₂²	1	6	yes	complemented
15	C₂×D₁₇₃⋊C₈	C₂²	1	6	yes	complemented
16	C₂²×#13063443	C₂²	1	6	yes	complemented
17	C₂×#14583827	C₂²	1	1	yes	characteristic
18	C₂×#14583826	C₄	1	3	yes	—
19	C₂×#14583825	C₂²	1	3	yes	—
20	#41×D₁₇₃	C₄	1	8	yes	—
21	#41×D₁₇₃	C₂²	1	8	yes	—
22	C₂×Q₈×D₁₇₃	C₄	1	1	yes	characteristic
23	C₂×D₄×D₁₇₃	C₄	1	3	yes	—
24	C₂²×C₄×D₁₇₃	C₂²	1	3	yes	—
25	C₂×#41×C₁₇₃	C₄	1	1	yes	characteristic
26	#14583836	C₂³	1	4	yes	—
27	#14583836	C₂³	1	12	yes	—
28	#14583836	C₂³	1	24	yes	complemented
29	#14583835	C₂³	1	24	yes	complemented
30	D₁₇₃⋊C₈	C₂³	1	24	yes	complemented
31	C₂×#13063443	C₂³	1	6	yes	—
32	C₂×#13063443	C₂³	1	2	yes	—
33	C₂×#13063443	C₂³	1	24	yes	complemented
34	C₂³×D₁₇₃	C₂×C₄	1	3	yes	—
35	#14583827	C₂×C₄	1	4	yes	—
36	#14583827	C₂³	1	4	yes	—
37	#14583826	C₂×C₄	1	24	yes	—
38	#14583825	C₂×C₄	1	12	yes	—
39	#14583825	C₂³	1	12	yes	—
40	C₂×Dic₃₄₆	C₂×C₄	1	3	yes	—
41	Q₈×D₁₇₃	C₂×C₄	1	8	yes	—
42	C₂×D₆₉₂	C₂×C₄	1	3	yes	—
43	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	6	yes	—
44	D₄×D₁₇₃	C₂×C₄	1	24	yes	—
45	C₂²×Dic₁₇₃	C₂³	1	3	yes	—
46	C₂×C₄×D₁₇₃	C₂×C₄	1	3	yes	—
47	C₂×C₄×D₁₇₃	C₂×C₄	1	12	yes	—
48	C₂×C₄×D₁₇₃	C₂×C₄	1	3	yes	—
49	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₇₃	C₂³	1	12	yes	—
51	C₂×C₄×D₁₇₃	C₂³	1	3	yes	—
52	#41×C₁₇₃	C₂×C₄	1	8	yes	—
53	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×D₄×C₁₇₃	C₂×C₄	1	3	yes	—
55	C₂²×C₄×C₁₇₃	C₂×C₄	1	3	yes	—
56	#13063443	C₂⁴	1	12	yes	—
57	#13063443	C₂⁴	1	4	yes	—
58	C₂²×D₁₇₃	—	4	6	no	—
59	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₁₇₃	C₂²×C₄	1	3	yes	—
61	C₂²×D₁₇₃	C₂²×C₄	1	12	yes	—
62	Dic₃₄₆	C₂²×C₄	1	12	yes	—
63	D₆₉₂	C₂²×C₄	1	12	yes	—
64	C₂²⋊D₁₇₃	C₂²×C₄	1	24	yes	—
65	C₂×Dic₁₇₃	C₂²×C₄	1	3	yes	—
66	C₂×Dic₁₇₃	C₂²×C₄	1	6	yes	—
67	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₇₃	C₂⁴	1	6	yes	—
69	C₄×D₁₇₃	C₂²×C₄	1	2	yes	—
70	C₄×D₁₇₃	C₂²×C₄	1	12	yes	—
71	C₄×D₁₇₃	C₂²×C₄	1	6	yes	—
72	C₄×D₁₇₃	C₂²×C₄	1	12	yes	—
73	C₄×D₁₇₃	C₂⁴	1	2	yes	—
74	C₄×D₁₇₃	C₂⁴	1	6	yes	—
75	C₂³×C₁₇₃	C₂²×C₄	1	3	yes	—
76	Q₈×C₁₇₃	C₂²×C₄	1	4	yes	—
77	D₄×C₁₇₃	C₂²×C₄	1	12	yes	—
78	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇₃	C₂²×C₄	1	3	yes	—
80	C₂×C₄×C₁₇₃	C₂²×C₄	1	12	yes	—
81	C₂²×C₁₇₃	—	2	6	no	—
82	C₂²×C₁₇₃	C₂³×C₄	1	6	yes	—
83	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×D₁₇₃	—	2	2	no	—
85	C₂×D₁₇₃	—	2	6	no	—
86	C₂×D₁₇₃	—	4	12	no	—
87	C₂×D₁₇₃	C₂³×C₄	1	2	yes	—
88	C₂×D₁₇₃	C₂³×C₄	1	6	yes	—
89	C₄×C₁₇₃	C₂³×C₄	1	2	yes	—
90	C₄×C₁₇₃	C₂³×C₄	1	6	yes	—
91	Dic₁₇₃	C₂³×C₄	1	1	yes	characteristic
92	Dic₁₇₃	C₂³×C₄	1	6	yes	—
93	Dic₁₇₃	C₂⁵	1	1	yes	characteristic
94	C₂×C₁₇₃	—	2	6	no	—
95	C₂×C₁₇₃	#568	1	2	yes	complemented
96	C₂×C₁₇₃	C₂⁴×C₄	1	1	yes	characteristic
97	D₁₇₃	—	2	2	no	—
98	D₁₇₃	—	2	6	no	—
99	C₁₇₃	C₂×#568	1	1	yes	characteristic, complemented, Sylow
100	C₂×#568	—	173	1	no	maximal, Sylow
101	C₂²×#41	—	173	1	no	—
102	#568	—	173	16	no	—
103	C₂×#131	—	173	6	no	—
104	C₂×#131	—	173	2	no	—
105	C₂²×M₁₆	—	173	3	no	—
106	C₂²×M₁₆	—	173	3	no	—
107	C₂×#41	—	173	8	no	—
108	C₂×#41	—	173	8	no	—
109	C₂×#41	—	173	1	no	—
110	C₂×#41	—	173	3	no	—
111	C₂×#41	—	173	3	no	—
112	C₂×#41	#10621924	1	1	yes	characteristic
113	C₂²×Q₈	—	173	1	no	—
114	C₂²×D₄	—	173	3	no	—
115	C₂³×C₄	—	173	3	no	—
116	#131	—	173	48	no	—
117	#131	—	173	16	no	—
118	C₂×M₁₆	—	173	1	no	—
119	C₂×M₁₆	—	173	3	no	—
120	C₂×M₁₆	—	173	6	no	—
121	C₂×M₁₆	—	173	6	no	—
122	C₂×M₁₆	—	173	24	no	—
123	C₂×M₁₆	—	173	24	no	—
124	C₂²×C₈	—	173	6	no	—
125	C₂²×C₈	—	173	6	no	—
126	C₂⁴	—	173	3	no	—
127	#41	—	173	4	no	—
128	#41	—	173	4	no	—
129	#41	—	173	12	no	—
130	#41	—	173	12	no	—
131	#41	—	173	24	no	—
132	#41	C₂×#10621924	1	8	yes	—
133	C₂×Q₈	—	173	8	no	—
134	C₂×Q₈	—	173	3	no	—
135	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic
136	C₂×D₄	—	173	6	no	—
137	C₂×D₄	—	173	24	no	—
138	C₂×D₄	—	173	3	no	—
139	C₂×D₄	C₂×#10621924	1	3	yes	—
140	C₂²×C₄	—	173	3	no	—
141	C₂²×C₄	—	173	3	no	—
142	C₂²×C₄	—	173	12	no	—
143	C₂²×C₄	—	173	3	no	—
144	C₂²×C₄	—	173	3	no	—
145	C₂²×C₄	—	173	1	no	—
146	C₂²×C₄	—	173	12	no	—
147	C₂²×C₄	C₂×#10621924	1	3	yes	—
148	M₁₆	—	173	24	no	—
149	M₁₆	—	173	4	no	—
150	M₁₆	—	173	12	no	—
151	M₁₆	—	173	24	no	—
152	C₂×C₈	—	173	6	no	—
153	C₂×C₈	—	173	2	no	—
154	C₂×C₈	—	173	24	no	—
155	C₂×C₈	—	173	24	no	—
156	C₂³	—	173	1	no	—
157	C₂³	—	173	3	no	—
158	C₂³	—	173	12	no	—
159	C₂³	—	692	6	no	—
160	C₂³	C₂²×#10621924	1	3	yes	—
161	Q₈	—	173	12	no	—
162	Q₈	C₂²×#10621924	1	4	yes	—
163	D₄	—	173	24	no	—
164	D₄	—	173	12	no	—
165	D₄	C₂²×#10621924	1	12	yes	—
166	C₂×C₄	—	173	6	no	—
167	C₂×C₄	—	173	6	no	—
168	C₂×C₄	—	173	3	no	—
169	C₂×C₄	—	173	1	no	—
170	C₂×C₄	—	173	2	no	—
171	C₂×C₄	—	173	12	no	—
172	C₂×C₄	—	173	6	no	—
173	C₂×C₄	—	173	12	no	—
174	C₂×C₄	—	173	6	no	—
175	C₂×C₄	—	173	2	no	—
176	C₂×C₄	C₂²×#10621924	1	12	yes	—
177	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
178	C₂×C₄	C₂²×#10621924	1	3	yes	—
179	C₈	—	173	12	no	—
180	C₈	—	173	4	no	—
181	C₂²	—	2	6	no	—
182	C₂²	—	346	2	no	—
183	C₂²	—	346	6	no	—
184	C₂²	—	173	2	no	—
185	C₂²	—	173	6	no	—
186	C₂²	—	692	12	no	—
187	C₂²	C₂³×#10621924	1	6	yes	—
188	C₂²	C₂³×#10621924	1	1	yes	characteristic
189	C₄	—	173	1	no	—
190	C₄	—	173	1	no	—
191	C₄	—	173	6	no	—
192	C₄	C₂³×#10621924	1	2	yes	—
193	C₄	C₂³×#10621924	1	6	yes	—
194	C₂	—	2	6	no	—
195	C₂	—	346	6	no	—
196	C₂	—	346	2	no	—
197	C₂	#26012570	1	2	yes	direct product factor
198	C₂	C₂⁴×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#10621924
commutator	C₂×C₁₇₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#10621924
Fitting	C₂×#41×C₁₇₃	C₄
socle	C₂²×C₁₇₃	C₂³×C₄
prime radical	C₂×#41×D₁₇₃	C₂
cosocle	Dic₁₇₃	C₂⁵
hypercenter	C₂×#41	#10621924
hypocenter	C₁₇₃	C₂×#568
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #26012570	C₂	direct (N × H)
2. C₂×#19149767	C₂	semidirect
3. C₂×#19149766	C₂	semidirect
4. C₂²×#14583836	C₂	semidirect
5. C₂×D₁₇₃⋊M₁₆	C₂	semidirect
7. #19149767	C₂²	semidirect
8. #19149766	C₂²	semidirect
11. C₂×#14583836	C₂²	semidirect
12. C₂×#14583836	C₂²	semidirect
13. D₁₇₃⋊M₁₆	C₂²	semidirect
14. C₂×#14583835	C₂²	semidirect
15. C₂×D₁₇₃⋊C₈	C₂²	semidirect
16. C₂²×#13063443	C₂²	semidirect
28. #14583836	C₂³	semidirect
29. #14583835	C₂³	semidirect
30. D₁₇₃⋊C₈	C₂³	semidirect
33. C₂×#13063443	C₂³	semidirect
95. C₂×C₁₇₃	#568	semidirect
99. C₁₇₃	C₂×#568	semidirect
197. C₂	#26012570	direct (N × H)

Maximal subgroups

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

subgroup	index
#26012570	2
C₂×#19149767	2
C₂×#19149766	2
C₂²×#14583836	2
C₂×D₁₇₃⋊M₁₆	2
C₂×#41×D₁₇₃	2
C₂×#568	173

Subgroups — C2×#26012570 #36370556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26012570 #36370556