Subgroups — C₂×#17989656 #22626910

Order 9344.

1640 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989656	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#64408	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#64408	C₂	1	6	yes	maximal, complemented
4	C₂×#14556670	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄⋊#64408	C₂	1	3	yes	maximal, complemented
6	C₂×#41×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#64408	C₂²	1	6	yes	—
8	Q₈×#64408	C₂²	1	16	yes	complemented
9	D₄×#64408	C₂²	1	48	yes	complemented
10	#14556670	C₂²	1	24	yes	complemented
11	C₂×C₄⋊#64408	C₂²	1	3	yes	—
12	C₂×C₄⋊#64408	C₂²	1	6	yes	—
13	C₂×C₄⋊#64408	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#64408	C₂²	1	24	yes	complemented
15	C₂×C₂²⋊#64408	C₂²	1	6	yes	—
16	C₂×C₄×#64408	C₂²	1	6	yes	—
17	C₂×#11924029	C₄	1	1	yes	characteristic, complemented
18	C₂×#11924028	C₄	1	3	yes	complemented
19	C₂×#11924027	C₄	1	3	yes	complemented
20	#41×D₇₃	C₄	1	8	yes	complemented
21	#41×D₇₃	C₂²	1	8	yes	—
22	C₂×Q₈×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×D₇₃	C₂²	1	3	yes	—
24	C₂²×C₄×D₇₃	C₂²	1	3	yes	—
25	C₂×#41×C₇₃	C₄	1	1	yes	characteristic, complemented
26	C₂²×#64408	—	4	12	no	—
27	C₂²×#64408	C₂³	1	6	yes	—
28	C₂²×#64408	C₂³	1	2	yes	—
29	C₂²×#64408	C₂³	1	24	yes	—
30	C₄⋊#64408	C₂³	1	24	yes	—
31	C₄⋊#64408	C₂³	1	12	yes	—
32	C₄⋊#64408	C₂³	1	4	yes	—
33	C₂²⋊#64408	C₂³	1	24	yes	—
34	C₄×#64408	C₂³	1	24	yes	—
35	C₂³×D₇₃	C₂³	1	3	yes	—
36	#11924029	C₂×C₄	1	8	yes	complemented
37	#11924028	C₂×C₄	1	24	yes	complemented
38	#11924027	C₂×C₄	1	24	yes	complemented
39	C₂×Dic₁₄₆	C₂×C₄	1	3	yes	complemented
40	Q₈×D₇₃	C₂×C₄	1	4	yes	complemented
41	Q₈×D₇₃	C₂³	1	4	yes	—
42	C₂×D₂₉₂	C₂×C₄	1	3	yes	complemented
43	C₂×C₂²⋊D₇₃	C₂×C₄	1	6	yes	complemented
44	D₄×D₇₃	C₂×C₄	1	12	yes	complemented
45	D₄×D₇₃	C₂³	1	12	yes	—
46	C₂²×Dic₇₃	C₂×C₄	1	3	yes	complemented
47	C₂×C₄×D₇₃	C₂×C₄	1	12	yes	complemented
48	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	complemented
49	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	complemented
50	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₇₃	C₂³	1	3	yes	—
52	C₂×C₄×D₇₃	C₂³	1	12	yes	—
53	#41×C₇₃	C₂×C₄	1	8	yes	complemented
54	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂×D₄×C₇₃	C₂×C₄	1	3	yes	complemented
56	C₂²×C₄×C₇₃	C₂×C₄	1	3	yes	complemented
57	C₂³×C₇₃	C₂²×C₄	1	3	yes	—
58	C₂²×D₇₃	—	2	6	no	—
59	C₂²×D₇₃	—	2	6	no	—
60	C₂²×D₇₃	C₂²×C₄	1	6	yes	—
61	C₂²×D₇₃	C₂²×C₄	1	3	yes	—
62	C₂²×D₇₃	C₂⁴	1	6	yes	—
63	C₂²×D₇₃	C₂⁴	1	1	yes	characteristic
64	C₂×#64408	—	2	12	no	—
65	C₂×#64408	—	2	4	no	—
66	C₂×#64408	—	4	24	no	—
67	C₂×#64408	C₂⁴	1	4	yes	—
68	C₂×#64408	C₂⁴	1	12	yes	—
69	Q₈×C₇₃	C₂²×C₄	1	4	yes	complemented
70	D₄×C₇₃	C₂²×C₄	1	12	yes	complemented
71	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
73	C₂×C₄×C₇₃	C₂²×C₄	1	12	yes	complemented
74	C₂²⋊D₇₃	C₂²×C₄	1	24	yes	complemented
75	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
77	C₂×Dic₇₃	C₂²×C₄	1	12	yes	complemented
78	D₂₉₂	C₂²×C₄	1	12	yes	complemented
79	C₄×D₇₃	C₂²×C₄	1	2	yes	—
80	C₄×D₇₃	C₂²×C₄	1	6	yes	—
81	C₄×D₇₃	C₂²×C₄	1	12	yes	complemented
82	C₄×D₇₃	C₂²×C₄	1	12	yes	complemented
83	C₄×D₇₃	C₂⁴	1	2	yes	—
84	C₄×D₇₃	C₂⁴	1	6	yes	—
85	Dic₁₄₆	C₂²×C₄	1	12	yes	complemented
86	C₂²×C₇₃	—	2	6	no	—
87	C₂²×C₇₃	C₂³×C₄	1	6	yes	—
88	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₇₃	—	2	6	no	—
90	C₂×D₇₃	—	2	12	no	—
91	C₂×D₇₃	—	2	6	no	—
92	C₂×D₇₃	—	2	6	no	—
93	C₂×D₇₃	C₂³×C₄	1	6	yes	—
94	C₂×D₇₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₇₃	H(5,𝔽₂)	1	2	yes	—
96	C₂×D₇₃	2₋¹⁺⁴	1	2	yes	—
97	C₂×D₇₃	C₂⁵	1	1	yes	characteristic
98	#64408	—	2	12	no	—
99	#64408	—	2	4	no	—
100	C₄×C₇₃	C₂³×C₄	1	6	yes	—
101	C₄×C₇₃	C₂³×C₄	1	2	yes	—
102	Dic₇₃	C₂³×C₄	1	6	yes	—
103	Dic₇₃	C₂³×C₄	1	2	yes	—
104	C₂×C₇₃	—	2	6	no	—
105	C₂×C₇₃	#520	1	2	yes	complemented
106	C₂×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
107	D₇₃	—	2	6	no	—
108	D₇₃	#520	1	2	yes	complemented
109	D₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
110	D₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×#520	—	73	1	no	maximal, Sylow
112	C₇₃	C₂×#520	1	1	yes	characteristic, complemented, Sylow
113	C₂²×#41	—	73	1	no	—
114	#520	—	73	16	no	—
115	C₂×C₄×Q₈	—	73	2	no	—
116	C₂×C₄×D₄	—	73	6	no	—
117	C₂×#117	—	73	3	no	—
118	C₂²×C₄⋊C₄	—	73	3	no	—
119	C₂×#41	—	73	8	no	—
120	C₂×#41	—	73	8	no	—
121	C₂×#41	—	73	1	no	—
122	C₂×#41	—	73	3	no	—
123	C₂×#41	—	73	3	no	—
124	C₂×#41	#64408	1	1	yes	characteristic, complemented
125	C₂²×Q₈	—	73	1	no	—
126	C₂²×D₄	—	73	3	no	—
127	C₂³×C₄	—	73	6	no	—
128	C₂³×C₄	—	73	3	no	—
129	C₄×Q₈	—	73	16	no	—
130	C₄×D₄	—	73	48	no	—
131	#117	—	73	24	no	—
132	C₂×C₄⋊C₄	—	73	3	no	—
133	C₂×C₄⋊C₄	—	73	6	no	—
134	C₂×C₄⋊C₄	—	73	24	no	—
135	C₂×C₄⋊C₄	—	73	1	no	—
136	C₂×C₂²⋊C₄	—	73	6	no	—
137	C₂×C₄²	—	73	6	no	—
138	C₂⁴	—	73	3	no	—
139	#41	—	73	8	no	—
140	#41	—	73	24	no	—
141	#41	—	73	24	no	—
142	#41	C₂×#64408	1	8	yes	complemented
143	C₂×Q₈	—	73	4	no	—
144	C₂×Q₈	—	73	4	no	—
145	C₂×Q₈	—	73	3	no	—
146	C₂×Q₈	C₂×#64408	1	1	yes	characteristic, complemented
147	C₂×D₄	—	73	6	no	—
148	C₂×D₄	—	73	12	no	—
149	C₂×D₄	—	73	12	no	—
150	C₂×D₄	—	73	3	no	—
151	C₂×D₄	C₂×#64408	1	3	yes	complemented
152	C₂²×C₄	—	292	12	no	—
153	C₂²×C₄	—	73	24	no	—
154	C₂²×C₄	—	73	3	no	—
155	C₂²×C₄	—	73	3	no	—
156	C₂²×C₄	—	73	3	no	—
157	C₂²×C₄	—	73	3	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	12	no	—
160	C₂²×C₄	—	73	12	no	—
161	C₂²×C₄	—	73	2	no	—
162	C₂²×C₄	—	73	6	no	—
163	C₂²×C₄	C₂×#64408	1	3	yes	complemented
164	C₄⋊C₄	—	73	12	no	—
165	C₄⋊C₄	—	73	24	no	—
166	C₄⋊C₄	—	73	4	no	—
167	C₂²⋊C₄	—	73	24	no	—
168	C₄²	—	73	24	no	—
169	C₂³	—	73	1	no	—
170	C₂³	—	73	6	no	—
171	C₂³	—	73	6	no	—
172	C₂³	—	73	3	no	—
173	C₂³	—	146	6	no	—
174	C₂³	—	146	6	no	—
175	C₂³	C₂²×#64408	1	3	yes	—
176	Q₈	—	73	12	no	—
177	Q₈	C₂²×#64408	1	4	yes	complemented
178	D₄	—	73	24	no	—
179	D₄	—	73	12	no	—
180	D₄	C₂²×#64408	1	12	yes	complemented
181	C₂×C₄	—	292	24	no	—
182	C₂×C₄	—	73	12	no	—
183	C₂×C₄	—	73	1	no	—
184	C₂×C₄	—	73	3	no	—
185	C₂×C₄	—	73	6	no	—
186	C₂×C₄	—	73	2	no	—
187	C₂×C₄	—	73	2	no	—
188	C₂×C₄	—	73	6	no	—
189	C₂×C₄	—	73	12	no	—
190	C₂×C₄	—	73	12	no	—
191	C₂×C₄	—	73	4	no	—
192	C₂×C₄	—	73	12	no	—
193	C₂×C₄	—	146	12	no	—
194	C₂×C₄	—	146	4	no	—
195	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
196	C₂×C₄	C₂²×#64408	1	3	yes	—
197	C₂×C₄	C₂²×#64408	1	12	yes	complemented
198	C₂²	—	2	6	no	—
199	C₂²	—	73	2	no	—
200	C₂²	—	73	2	no	—
201	C₂²	—	73	1	no	—
202	C₂²	—	73	1	no	—
203	C₂²	—	73	6	no	—
204	C₂²	—	146	6	no	—
205	C₂²	—	146	6	no	—
206	C₂²	—	146	12	no	—
207	C₂²	—	146	6	no	—
208	C₂²	C₂³×#64408	1	6	yes	—
209	C₂²	C₂³×#64408	1	1	yes	characteristic
210	C₄	—	73	2	no	—
211	C₄	—	73	6	no	—
212	C₄	—	146	12	no	—
213	C₄	—	146	4	no	—
214	C₄	C₂³×#64408	1	2	yes	—
215	C₄	C₂³×#64408	1	6	yes	—
216	C₂	—	2	6	no	—
217	C₂	—	73	1	no	—
218	C₂	—	73	1	no	—
219	C₂	—	73	2	no	—
220	C₂	—	146	6	no	—
221	C₂	#17989656	1	2	yes	direct product factor
222	C₂	C₂⁴×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#64408
commutator	C₂×C₇₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#64408
Fitting	C₂×#41×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₄
hypercenter	C₂×#41	#64408
hypocenter	C₇₃	C₂×#520
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #17989656	C₂	direct (N × H)
2. C₂×Q₈×#64408	C₂	semidirect
3. C₂×D₄×#64408	C₂	semidirect
4. C₂×#14556670	C₂	semidirect
5. C₂²×C₄⋊#64408	C₂	semidirect
8. Q₈×#64408	C₂²	semidirect
9. D₄×#64408	C₂²	semidirect
10. #14556670	C₂²	semidirect
14. C₂×C₄⋊#64408	C₂²	semidirect
17. C₂×#11924029	C₄	semidirect
18. C₂×#11924028	C₄	semidirect
19. C₂×#11924027	C₄	semidirect
20. #41×D₇₃	C₄	semidirect
25. C₂×#41×C₇₃	C₄	semidirect
36. #11924029	C₂×C₄	semidirect
37. #11924028	C₂×C₄	semidirect
38. #11924027	C₂×C₄	semidirect
39. C₂×Dic₁₄₆	C₂×C₄	semidirect
40. Q₈×D₇₃	C₂×C₄	semidirect
42. C₂×D₂₉₂	C₂×C₄	semidirect
43. C₂×C₂²⋊D₇₃	C₂×C₄	semidirect
44. D₄×D₇₃	C₂×C₄	semidirect
46. C₂²×Dic₇₃	C₂×C₄	semidirect
47. C₂×C₄×D₇₃	C₂×C₄	semidirect
48. C₂×C₄×D₇₃	C₂×C₄	semidirect
49. C₂×C₄×D₇₃	C₂×C₄	semidirect
53. #41×C₇₃	C₂×C₄	semidirect
54. C₂×Q₈×C₇₃	C₂×C₄	semidirect
55. C₂×D₄×C₇₃	C₂×C₄	semidirect
56. C₂²×C₄×C₇₃	C₂×C₄	semidirect
69. Q₈×C₇₃	C₂²×C₄	semidirect
70. D₄×C₇₃	C₂²×C₄	semidirect
73. C₂×C₄×C₇₃	C₂²×C₄	semidirect
74. C₂²⋊D₇₃	C₂²×C₄	semidirect
77. C₂×Dic₇₃	C₂²×C₄	semidirect
78. D₂₉₂	C₂²×C₄	semidirect
81. C₄×D₇₃	C₂²×C₄	semidirect
82. C₄×D₇₃	C₂²×C₄	semidirect
85. Dic₁₄₆	C₂²×C₄	semidirect
105. C₂×C₇₃	#520	semidirect
108. D₇₃	#520	semidirect
112. C₇₃	C₂×#520	semidirect
124. C₂×#41	#64408	semidirect
142. #41	C₂×#64408	semidirect
146. C₂×Q₈	C₂×#64408	semidirect
151. C₂×D₄	C₂×#64408	semidirect
163. C₂²×C₄	C₂×#64408	semidirect
177. Q₈	C₂²×#64408	semidirect
180. D₄	C₂²×#64408	semidirect
197. C₂×C₄	C₂²×#64408	semidirect
221. C₂	#17989656	direct (N × H)

Subgroups — C2×#17989656 #22626910

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989656 #22626910