Subgroups — #34032704

Order 18688.

10930 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×#64408	C₂	1	32	yes	maximal, complemented
2	#22626911	C₂	1	20	yes	maximal, complemented
3	C₂×#17989655	C₂	1	10	yes	maximal, complemented
4	C₂×2₋¹⁺⁴×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989656	C₂²	1	160	yes	complemented
6	#17989655	C₂²	1	40	yes	—
7	#17989655	C₂²	1	80	yes	complemented
8	#17989654	C₂²	1	40	yes	—
9	#41×#64408	C₂²	1	160	yes	complemented
10	C₂×Q₈×#64408	C₂²	1	80	yes	complemented
11	C₂×#14556670	C₂²	1	60	yes	—
12	C₂×#14556639	C₄	1	5	yes	complemented
13	C₂×#14556638	C₄	1	10	yes	complemented
14	2₋¹⁺⁴×D₇₃	C₄	1	16	yes	complemented
15	2₋¹⁺⁴×D₇₃	C₂²	1	16	yes	—
16	C₂×#41×D₇₃	C₂²	1	10	yes	—
17	C₂²×Q₈×D₇₃	C₂²	1	5	yes	—
18	C₂×2₋¹⁺⁴×C₇₃	C₄	1	1	yes	characteristic, complemented
19	Q₈×#64408	C₂³	1	320	yes	—
20	D₄×#64408	C₂³	1	320	yes	—
21	#14556670	C₂³	1	80	yes	—
22	#14556670	C₂³	1	240	yes	—
23	C₂×C₄⋊#64408	C₂³	1	120	yes	—
24	C₂×C₂²⋊#64408	C₂³	1	40	yes	—
25	C₂×C₄×#64408	C₂³	1	120	yes	—
26	#14556639	C₂×C₄	1	80	yes	complemented
27	#14556638	C₂×C₄	1	160	yes	complemented
28	C₂×#11924029	C₂×C₄	1	10	yes	complemented
29	C₂×#11924028	C₂×C₄	1	30	yes	complemented
30	C₂×#11924027	C₂×C₄	1	30	yes	complemented
31	#41×D₇₃	C₂×C₄	1	80	yes	complemented
32	#41×D₇₃	C₂³	1	80	yes	—
33	C₂²×Dic₁₄₆	C₂×C₄	1	15	yes	complemented
34	C₂×Q₈×D₇₃	C₂×C₄	1	40	yes	complemented
35	C₂×Q₈×D₇₃	C₂×C₄	1	20	yes	complemented
36	C₂×Q₈×D₇₃	C₂³	1	40	yes	—
37	C₂×Q₈×D₇₃	C₂³	1	10	yes	—
38	C₂×D₄×D₇₃	C₂³	1	10	yes	—
39	C₂²×C₄×D₇₃	C₂³	1	15	yes	—
40	2₋¹⁺⁴×C₇₃	C₂×C₄	1	16	yes	complemented
41	C₂×#41×C₇₃	C₂×C₄	1	10	yes	complemented
42	C₂²×Q₈×C₇₃	C₂×C₄	1	5	yes	complemented
43	C₂²×#64408	C₂⁴	1	80	yes	—
44	C₄⋊#64408	C₂⁴	1	160	yes	—
45	C₂²⋊#64408	C₂⁴	1	16	yes	—
46	C₂²⋊#64408	C₂⁴	1	80	yes	—
47	C₄×#64408	C₂⁴	1	160	yes	—
48	C₂³×D₇₃	C₂⁴	1	5	yes	—
49	#11924029	C₂²×C₄	1	80	yes	—
50	#11924028	C₂²×C₄	1	240	yes	—
51	#11924027	C₂²×C₄	1	240	yes	—
52	C₂×Dic₁₄₆	C₂²×C₄	1	30	yes	—
53	C₂×Dic₁₄₆	C₂²×C₄	1	120	yes	—
54	Q₈×D₇₃	C₂²×C₄	1	40	yes	—
55	Q₈×D₇₃	C₂²×C₄	1	160	yes	—
56	Q₈×D₇₃	C₂⁴	1	40	yes	—
57	C₂×D₂₉₂	C₂²×C₄	1	10	yes	—
58	C₂×C₂²⋊D₇₃	C₂²×C₄	1	20	yes	—
59	D₄×D₇₃	C₂²×C₄	1	40	yes	—
60	D₄×D₇₃	C₂⁴	1	40	yes	—
61	C₂²×Dic₇₃	C₂²×C₄	1	15	yes	—
62	C₂×C₄×D₇₃	C₂²×C₄	1	60	yes	—
63	C₂×C₄×D₇₃	C₂²×C₄	1	30	yes	—
64	C₂×C₄×D₇₃	C₂⁴	1	10	yes	—
65	C₂×C₄×D₇₃	C₂⁴	1	60	yes	—
66	#41×C₇₃	C₂²×C₄	1	80	yes	—
67	C₂×Q₈×C₇₃	C₂²×C₄	1	10	yes	—
68	C₂×Q₈×C₇₃	C₂²×C₄	1	40	yes	—
69	C₂×D₄×C₇₃	C₂²×C₄	1	10	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	15	yes	—
71	C₂³×C₇₃	C₂³×C₄	1	5	yes	—
72	C₂²×D₇₃	—	4	10	no	—
73	C₂²×D₇₃	C₂³×C₄	1	10	yes	—
74	C₂²×D₇₃	C₂³×C₄	1	5	yes	—
75	C₂²×D₇₃	C₂⁵	1	1	yes	characteristic
76	C₂²×D₇₃	C₂⁵	1	10	yes	—
77	C₂×#64408	—	4	80	no	—
78	C₂×#64408	C₂⁵	1	32	yes	—
79	Q₈×C₇₃	C₂³×C₄	1	40	yes	—
80	D₄×C₇₃	C₂³×C₄	1	40	yes	—
81	C₂×C₄×C₇₃	C₂³×C₄	1	10	yes	—
82	C₂×C₄×C₇₃	C₂³×C₄	1	60	yes	—
83	C₂²⋊D₇₃	C₂³×C₄	1	80	yes	—
84	C₂×Dic₇₃	C₂³×C₄	1	10	yes	—
85	C₂×Dic₇₃	C₂³×C₄	1	60	yes	—
86	D₂₉₂	C₂³×C₄	1	40	yes	—
87	C₄×D₇₃	C₂³×C₄	1	20	yes	—
88	C₄×D₇₃	C₂³×C₄	1	120	yes	—
89	C₄×D₇₃	C₂⁵	1	20	yes	—
90	Dic₁₄₆	C₂³×C₄	1	120	yes	—
91	C₂²×C₇₃	—	4	5	no	—
92	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
93	C₂²×C₇₃	C₂⁴×C₄	1	10	yes	—
94	C₂×D₇₃	—	2	20	no	—
95	C₂×D₇₃	—	2	2	no	—
96	C₂×D₇₃	—	4	10	no	—
97	C₂×D₇₃	—	4	5	no	—
98	C₂×D₇₃	C₂⁴×C₄	1	1	yes	characteristic
99	C₂×D₇₃	C₂⁴×C₄	1	10	yes	—
100	C₂×D₇₃	C₂⁶	1	1	yes	characteristic
101	#64408	—	2	32	no	—
102	C₄×C₇₃	C₂⁴×C₄	1	20	yes	—
103	Dic₇₃	C₂⁴×C₄	1	20	yes	—
104	#62653	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	—	2	10	no	—
107	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
108	D₇₃	—	2	1	no	—
109	D₇₃	—	2	10	no	—
110	D₇₃	2₋¹⁺⁶	1	2	yes	—
111	C₂²×2₋¹⁺⁴	—	73	1	no	—
112	C₄×2₋¹⁺⁴	—	73	32	no	—
113	#3428	—	73	20	no	—
114	C₂×#519	—	73	10	no	—
115	C₇₃	#62653	1	1	yes	characteristic, complemented, Sylow
116	C₂×2₋¹⁺⁴	—	73	16	no	—
117	C₂×2₋¹⁺⁴	—	73	16	no	—
118	C₂×2₋¹⁺⁴	—	73	10	no	—
119	C₂×2₋¹⁺⁴	—	73	5	no	—
120	C₂×2₋¹⁺⁴	#64408	1	1	yes	characteristic, complemented
121	C₂²×#41	—	73	10	no	—
122	C₂³×Q₈	—	73	5	no	—
123	#520	—	73	160	no	—
124	#519	—	73	40	no	—
125	#519	—	73	80	no	—
126	#518	—	73	40	no	—
127	C₄×#41	—	73	160	no	—
128	C₂×C₄×Q₈	—	73	80	no	—
129	C₂×#117	—	73	60	no	—
130	2₋¹⁺⁴	—	73	160	no	—
131	2₋¹⁺⁴	—	73	80	no	—
132	2₋¹⁺⁴	C₂×#64408	1	16	yes	complemented
133	C₂×#41	—	73	80	no	—
134	C₂×#41	—	73	80	no	—
135	C₂×#41	—	73	10	no	—
136	C₂×#41	—	73	30	no	—
137	C₂×#41	—	73	30	no	—
138	C₂×#41	C₂×#64408	1	10	yes	complemented
139	C₂²×Q₈	—	73	40	no	—
140	C₂²×Q₈	—	73	10	no	—
141	C₂²×Q₈	—	73	40	no	—
142	C₂²×Q₈	—	73	20	no	—
143	C₂²×Q₈	—	73	15	no	—
144	C₂²×Q₈	C₂×#64408	1	5	yes	complemented
145	C₂²×D₄	—	73	10	no	—
146	C₂³×C₄	—	73	15	no	—
147	C₄×Q₈	—	73	320	no	—
148	C₄×D₄	—	73	320	no	—
149	#117	—	73	80	no	—
150	#117	—	73	240	no	—
151	C₂×C₄⋊C₄	—	73	120	no	—
152	C₂×C₂²⋊C₄	—	73	40	no	—
153	C₂×C₄²	—	73	120	no	—
154	C₂⁴	—	73	5	no	—
155	#41	—	73	80	no	—
156	#41	—	73	240	no	—
157	#41	—	73	240	no	—
158	#41	C₂²×#64408	1	80	yes	—
159	C₂×Q₈	—	73	40	no	—
160	C₂×Q₈	—	73	160	no	—
161	C₂×Q₈	—	73	40	no	—
162	C₂×Q₈	—	73	30	no	—
163	C₂×Q₈	—	73	120	no	—
164	C₂×Q₈	C₂²×#64408	1	10	yes	—
165	C₂×Q₈	C₂²×#64408	1	40	yes	—
166	C₂×D₄	—	73	20	no	—
167	C₂×D₄	—	73	40	no	—
168	C₂×D₄	—	73	40	no	—
169	C₂×D₄	—	73	10	no	—
170	C₂×D₄	C₂²×#64408	1	10	yes	—
171	C₂²×C₄	—	73	80	no	—
172	C₂²×C₄	—	73	15	no	—
173	C₂²×C₄	—	73	30	no	—
174	C₂²×C₄	—	73	60	no	—
175	C₂²×C₄	—	73	60	no	—
176	C₂²×C₄	—	73	10	no	—
177	C₂²×C₄	C₂²×#64408	1	15	yes	—
178	C₄⋊C₄	—	73	160	no	—
179	C₂²⋊C₄	—	73	16	no	—
180	C₂²⋊C₄	—	73	80	no	—
181	C₄²	—	73	160	no	—
182	C₂³	—	292	10	no	—
183	C₂³	—	73	10	no	—
184	C₂³	—	73	5	no	—
185	C₂³	—	73	1	no	—
186	C₂³	—	73	10	no	—
187	C₂³	C₂³×#64408	1	5	yes	—
188	Q₈	—	73	120	no	—
189	Q₈	C₂³×#64408	1	40	yes	—
190	D₄	—	73	80	no	—
191	D₄	—	73	40	no	—
192	D₄	C₂³×#64408	1	40	yes	—
193	C₂×C₄	—	292	80	no	—
194	C₂×C₄	—	73	10	no	—
195	C₂×C₄	—	73	60	no	—
196	C₂×C₄	—	73	20	no	—
197	C₂×C₄	—	73	20	no	—
198	C₂×C₄	—	73	120	no	—
199	C₂×C₄	—	73	32	no	—
200	C₂×C₄	C₂³×#64408	1	10	yes	—
201	C₂×C₄	C₂³×#64408	1	60	yes	—
202	C₂²	—	4	5	no	—
203	C₂²	—	292	5	no	—
204	C₂²	—	292	10	no	—
205	C₂²	—	73	1	no	—
206	C₂²	—	73	1	no	—
207	C₂²	—	73	10	no	—
208	C₂²	—	146	20	no	—
209	C₂²	—	146	2	no	—
210	C₂²	C₂⁴×#64408	1	1	yes	characteristic
211	C₂²	C₂⁴×#64408	1	10	yes	—
212	C₄	—	73	20	no	—
213	C₄	—	146	32	no	—
214	C₄	C₂⁴×#64408	1	20	yes	—
215	C₂	—	2	1	no	—
216	C₂	—	2	10	no	—
217	C₂	—	73	2	no	—
218	C₂	—	146	1	no	—
219	C₂	—	146	10	no	—
220	C₂	C₂⁵×#64408	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#64408
commutator, socle	C₂×C₇₃	C₂⁵×C₄
Fitting	C₂×2₋¹⁺⁴×C₇₃	C₄
prime radical	C₂×2₋¹⁺⁴×D₇₃	C₂
cosocle	C₂×D₇₃	C₂⁶
hypercenter	C₂×2₋¹⁺⁴	#64408
hypocenter	C₇₃	#62653
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×#64408	C₂	semidirect
2. #22626911	C₂	semidirect
3. C₂×#17989655	C₂	semidirect
5. #17989656	C₂²	semidirect
7. #17989655	C₂²	semidirect
9. #41×#64408	C₂²	semidirect
10. C₂×Q₈×#64408	C₂²	semidirect
12. C₂×#14556639	C₄	semidirect
13. C₂×#14556638	C₄	semidirect
14. 2₋¹⁺⁴×D₇₃	C₄	semidirect
18. C₂×2₋¹⁺⁴×C₇₃	C₄	semidirect
26. #14556639	C₂×C₄	semidirect
27. #14556638	C₂×C₄	semidirect
28. C₂×#11924029	C₂×C₄	semidirect
29. C₂×#11924028	C₂×C₄	semidirect
30. C₂×#11924027	C₂×C₄	semidirect
31. #41×D₇₃	C₂×C₄	semidirect
33. C₂²×Dic₁₄₆	C₂×C₄	semidirect
34. C₂×Q₈×D₇₃	C₂×C₄	semidirect
35. C₂×Q₈×D₇₃	C₂×C₄	semidirect
40. 2₋¹⁺⁴×C₇₃	C₂×C₄	semidirect
41. C₂×#41×C₇₃	C₂×C₄	semidirect
42. C₂²×Q₈×C₇₃	C₂×C₄	semidirect
115. C₇₃	#62653	semidirect
120. C₂×2₋¹⁺⁴	#64408	semidirect
132. 2₋¹⁺⁴	C₂×#64408	semidirect
138. C₂×#41	C₂×#64408	semidirect
144. C₂²×Q₈	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×#64408	2
#22626911	2
C₂×#17989655	2
C₂×2₋¹⁺⁴×D₇₃	2
#62653	73