Subgroups — C₂×#22611808 #33239288

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₈₄	C₂	1	2	yes	maximal, complemented
3	#22611808	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988415	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988389	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₈₄	C₂²	1	2	yes	—
8	C₂×C₄×D₂₉₂	—	2	2	no	—
9	C₂×D₁₁₆₈	C₄	1	8	yes	complemented
10	C₂×D₁₁₆₈	C₂²	1	4	yes	—
11	C₄×D₅₈₄	—	2	4	no	—
12	C₄×D₅₈₄	C₂²	1	8	yes	complemented
13	C₂×#14556525	C₂²	1	2	yes	—
14	C₂×#14556513	—	2	2	no	—
15	#17988415	C₂²	1	4	yes	complemented
16	#17988389	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	#346×C₇₃	C₂²	1	4	yes	complemented
21	C₂²×D₂₉₂	—	2	2	no	—
22	C₂²×C₄×D₇₃	—	4	2	no	—
23	C₂×D₅₈₄	—	2	2	no	—
24	C₂×D₅₈₄	—	2	4	no	—
25	C₂×D₅₈₄	C₂×C₄	1	8	yes	complemented
26	C₂×D₅₈₄	C₂³	1	4	yes	—
27	C₄×D₂₉₂	—	2	8	no	—
28	C₄×D₂₉₂	—	4	4	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×#11924005	—	4	2	no	—
31	D₁₁₆₈	C₂×C₄	1	16	yes	complemented
32	#14556525	—	2	2	no	—
33	#14556525	C₂³	1	4	yes	—
34	#14556513	—	2	8	no	—
35	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₇₃	—	2	2	no	—
39	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
42	C₄×C₈×C₇₃	D₄	1	2	yes	—
43	C₄×C₈×C₇₃	C₂³	1	2	yes	—
44	C₂³×D₇₃	—	4	2	no	—
45	C₂×D₂₉₂	—	2	8	no	—
46	C₂×D₂₉₂	—	2	4	no	—
47	C₂×D₂₉₂	—	4	2	no	—
48	C₂×D₂₉₂	—	4	4	no	—
49	C₂²×Dic₇₃	—	4	2	no	—
50	C₂×C₄×D₇₃	—	4	8	no	—
51	C₂×C₄×D₇₃	—	8	8	no	—
52	D₅₈₄	—	2	8	no	—
53	D₅₈₄	—	2	4	no	—
54	D₅₈₄	C₂²×C₄	1	8	yes	complemented
55	C₄⋊Dic₇₃	—	2	4	no	—
56	C₄⋊Dic₇₃	—	4	2	no	—
57	#11924005	—	4	8	no	—
58	C₂²×C₄×C₇₃	D₈	1	2	yes	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	—	2	2	no	—
61	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	2	yes	—
65	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
66	C₄²×C₇₃	D₈	1	2	yes	—
67	C₄²×C₇₃	C₂×D₄	1	2	yes	—
68	C₁₆×C₇₃	—	2	2	no	—
69	C₁₆×C₇₃	C₂²×C₄	1	4	yes	complemented
70	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₇₃	—	4	4	no	—
72	C₂²×D₇₃	—	4	8	no	—
73	C₂²×D₇₃	—	8	4	no	—
74	C₂²×D₇₃	—	8	4	no	—
75	C₂×C₄×C₇₃	—	2	4	no	—
76	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#135	1	2	yes	—
80	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₇₃	—	4	4	no	—
82	C₂×Dic₇₃	—	8	4	no	—
83	D₂₉₂	—	2	8	no	—
84	D₂₉₂	—	4	4	no	—
85	D₂₉₂	—	4	8	no	—
86	C₄×D₇₃	—	8	16	no	—
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	C₄×D₄	1	2	yes	—
89	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
91	C₂²×C₇₃	C₄×D₈	1	2	yes	—
92	C₂²×C₇₃	#509	1	4	yes	—
93	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
94	C₂×D₇₃	—	4	8	no	—
95	C₂×D₇₃	—	8	4	no	—
96	C₂×D₇₃	—	8	4	no	—
97	C₂×D₇₃	—	8	16	no	—
98	C₄×C₇₃	—	2	4	no	—
99	C₄×C₇₃	C₄×D₈	1	2	yes	—
100	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
101	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
102	Dic₇₃	—	8	4	no	—
103	C₂×#2177	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	#2177	1	4	yes	complemented
105	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
106	C₂×C₇₃	C₂×#509	1	2	yes	—
107	D₇₃	—	8	8	no	—
108	C₂²×D₁₆	—	73	1	no	—
109	C₂×C₄×D₈	—	73	2	no	—
110	#2177	—	73	8	no	—
111	C₂×#367	—	73	1	no	—
112	C₂×#357	—	73	2	no	—
113	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
114	C₇₃	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
115	C₂²×D₈	—	73	2	no	—
116	C₂×C₄×D₄	—	146	2	no	—
117	C₂×D₁₆	—	73	8	no	—
118	C₂×D₁₆	—	73	4	no	—
119	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
121	C₄×D₈	—	73	8	no	—
122	C₄×D₈	—	146	4	no	—
123	C₂×#107	—	73	2	no	—
124	C₂×#102	—	146	2	no	—
125	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
126	#367	—	73	4	no	—
127	#357	—	73	8	no	—
128	#346	C₂×D₇₃	1	4	yes	complemented
129	C₂²×D₄	—	146	2	no	—
130	C₂³×C₄	—	292	2	no	—
131	C₂×D₈	—	73	8	no	—
132	C₂×D₈	—	73	4	no	—
133	C₂×D₈	—	146	2	no	—
134	C₂×D₈	—	146	4	no	—
135	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
137	C₄×D₄	—	292	4	no	—
138	C₄×D₄	—	146	8	no	—
139	C₂×C₄⋊C₄	—	146	2	no	—
140	C₂×C₂²⋊C₄	—	292	2	no	—
141	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
142	D₁₆	—	73	16	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₄×D₇₃	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
147	#107	—	73	4	no	—
148	#107	—	146	2	no	—
149	#102	—	146	8	no	—
150	C₄×C₈	D₂₉₂	1	2	yes	—
151	C₄×C₈	C₂²×D₇₃	1	2	yes	—
152	C₂⁴	—	292	2	no	—
153	C₂×D₄	—	292	2	no	—
154	C₂×D₄	—	292	4	no	—
155	C₂×D₄	—	146	8	no	—
156	C₂×D₄	—	146	4	no	—
157	C₂²×C₄	—	292	2	no	—
158	C₂²×C₄	—	292	8	no	—
159	C₂²×C₄	—	584	8	no	—
160	C₂²×C₄	D₅₈₄	1	2	yes	—
161	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
162	D₈	—	73	8	no	—
163	D₈	—	146	8	no	—
164	D₈	—	146	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
167	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
168	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
169	C₂×C₈	#11924028	1	2	yes	—
170	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
171	C₄⋊C₄	—	292	2	no	—
172	C₄⋊C₄	—	146	4	no	—
173	C₂²⋊C₄	—	292	8	no	—
174	C₄²	D₅₈₄	1	2	yes	—
175	C₄²	C₂×D₂₉₂	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₇₃	1	4	yes	complemented
178	C₂³	—	292	8	no	—
179	C₂³	—	292	4	no	—
180	C₂³	—	584	4	no	—
181	C₂³	—	584	4	no	—
182	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
183	D₄	—	292	8	no	—
184	D₄	—	292	4	no	—
185	D₄	—	146	8	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	292	4	no	—
188	C₂×C₄	—	584	16	no	—
189	C₂×C₄	—	584	4	no	—
190	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
191	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
192	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
193	C₂×C₄	#14556613	1	2	yes	—
194	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	C₄×D₂₉₂	1	2	yes	—
197	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
198	C₈	C₂×#11924028	1	1	yes	characteristic
199	C₂²	—	292	8	no	—
200	C₂²	—	584	4	no	—
201	C₂²	—	584	4	no	—
202	C₂²	—	584	16	no	—
203	C₂²	C₄×D₅₈₄	1	2	yes	—
204	C₂²	#17989025	1	4	yes	—
205	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	584	4	no	—
208	C₄	C₄×D₅₈₄	1	2	yes	—
209	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
210	C₄	C₂×#14556613	1	1	yes	characteristic
211	C₂	—	584	8	no	—
212	C₂	#22611808	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic
214	C₂	C₂×#17989025	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂²×D₁₁₆₈	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	C₂×#2177
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₈₄	C₂	semidirect
3. #22611808	C₂	direct (N × H)
4. C₂×#17988415	C₂	semidirect
5. C₂×#17988389	C₂	semidirect
6. C₂×#346×C₇₃	C₂	semidirect
9. C₂×D₁₁₆₈	C₄	semidirect
12. C₄×D₅₈₄	C₂²	semidirect
15. #17988415	C₂²	semidirect
16. #17988389	C₂²	semidirect
20. #346×C₇₃	C₂²	semidirect
25. C₂×D₅₈₄	C₂×C₄	semidirect
31. D₁₁₆₈	C₂×C₄	semidirect
39. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
54. D₅₈₄	C₂²×C₄	semidirect
69. C₁₆×C₇₃	C₂²×C₄	semidirect
104. C₂×C₇₃	#2177	semidirect
113. C₂×#346	D₇₃	semidirect
114. C₇₃	C₂×#2177	semidirect
128. #346	C₂×D₇₃	semidirect
144. C₂×C₁₆	C₄×D₇₃	semidirect
177. C₁₆	C₂×C₄×D₇₃	semidirect
212. C₂	#22611808	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×D₁₁₆₈	2
C₂×C₄×D₅₈₄	2
#22611808	2
C₂×#17988415	2
C₂×#17988389	2
C₂×#346×C₇₃	2
C₂×#2177	73

Subgroups — C2×#22611808 #33239288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611808 #33239288