Subgroups — C₂²×#17988649 #33949702

Order 18688.

2310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556548	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988649	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556520	C₂	1	2	yes	maximal, complemented
4	C₂²×#14556503	C₂	1	2	yes	maximal
5	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
8	C₂³×#10596831	C₂²	1	2	yes	—
9	C₂×#14556548	C₄	1	12	yes	—
10	C₂×#14556548	C₂²	1	12	yes	—
11	C₂²×#11924007	C₄	1	2	yes	—
12	C₂²×#11924005	C₄	1	2	yes	—
13	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	#17988649	C₂²	1	64	yes	direct product factor
15	C₂×#14556520	C₂²	1	24	yes	complemented
16	C₂×#14556503	C₂²	1	24	yes	—
17	C₂×C₄×#10596831	C₂²	1	12	yes	complemented
18	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×#97×C₇₃	C₂²	1	12	yes	complemented
21	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×D₇₃	—	2	8	no	—
25	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
26	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
27	C₂²×#10596831	—	2	8	no	—
28	C₂²×#10596831	C₂³	1	12	yes	—
29	#14556548	C₂×C₄	1	48	yes	—
30	#14556548	C₂³	1	16	yes	—
31	C₂×#11924007	C₂×C₄	1	24	yes	—
32	C₂×#11924005	C₂×C₄	1	24	yes	—
33	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
34	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
35	#14556520	C₂³	1	32	yes	complemented
36	#14556503	C₂³	1	32	yes	—
37	C₄×#10596831	C₂³	1	16	yes	complemented
38	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	—	2	8	no	—
41	C₂²×C₈×C₇₃	C₂³	1	12	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
44	#97×C₇₃	C₂³	1	16	yes	complemented
45	C₂³×D₇₃	—	2	4	no	—
46	C₂³×D₇₃	—	2	8	no	—
47	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
48	C₂²×Dic₇₃	—	2	4	no	—
49	C₂²×Dic₇₃	—	2	8	no	—
50	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
51	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
52	C₂×C₄×D₇₃	—	2	24	no	—
53	C₂×C₄×D₇₃	—	2	24	no	—
54	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
55	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
56	C₂×#10596831	—	2	24	no	—
57	C₂×#10596831	C₂⁴	1	8	yes	—
58	#11924007	C₂²×C₄	1	32	yes	—
59	#11924005	C₂²×C₄	1	32	yes	—
60	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
61	C₄×Dic₇₃	C₂⁴	1	4	yes	—
62	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	8	no	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₇₃	#41	1	8	yes	—
67	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
68	C₂×C₈×C₇₃	—	2	24	no	—
69	C₂×C₈×C₇₃	C₂⁴	1	8	yes	—
70	C₄²×C₇₃	C₂²×C₄	1	12	yes	—
71	C₄²×C₇₃	C₂⁴	1	4	yes	—
72	C₂³×C₇₃	#117	1	4	yes	—
73	C₂³×C₇₃	#131	1	8	yes	—
74	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
75	C₂²×D₇₃	—	2	12	no	—
76	C₂²×D₇₃	—	2	24	no	—
77	C₂²×D₇₃	—	4	16	no	—
78	C₂²×D₇₃	C₂³×C₄	1	4	yes	—
79	C₂×C₄×C₇₃	—	2	24	no	—
80	C₂×C₄×C₇₃	#117	1	12	yes	—
81	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
82	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₇₃	C₂×#41	1	12	yes	—
84	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₇₃	—	2	12	no	—
86	C₂×Dic₇₃	—	2	24	no	—
87	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
88	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
89	C₄×D₇₃	—	2	24	no	—
90	C₄×D₇₃	—	2	8	no	—
91	C₈×C₇₃	—	2	8	no	—
92	#10596831	—	2	8	no	—
93	C₂²×C₇₃	#433	1	16	yes	complemented
94	C₂²×C₇₃	C₂×#117	1	6	yes	—
95	C₂²×C₇₃	C₂×#131	1	12	yes	—
96	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
97	C₂×D₇₃	—	2	4	no	—
98	C₂×D₇₃	—	2	8	no	—
99	C₂×D₇₃	—	4	24	no	—
100	C₄×C₇₃	—	2	8	no	—
101	C₄×C₇₃	C₂×#117	1	6	yes	—
102	C₄×C₇₃	C₂²×#41	1	2	yes	—
103	Dic₇₃	—	2	4	no	—
104	Dic₇₃	—	2	8	no	—
105	C₂²×#433	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	C₂×#433	1	12	yes	complemented
107	C₂×C₇₃	C₂²×#117	1	1	yes	characteristic
108	C₂×C₇₃	C₂²×#131	1	2	yes	—
109	D₇₃	—	4	4	no	—
110	C₂²×#117	—	73	1	no	—
111	C₂×#433	—	73	24	no	—
112	C₂²×C₄⋊C₈	—	73	2	no	—
113	C₂²×C₂²⋊C₈	—	73	2	no	—
114	C₂²×#97	D₇₃	1	1	yes	characteristic, complemented
115	C₂²×C₄×C₈	—	73	1	no	—
116	C₇₃	C₂²×#433	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴×C₄	—	73	1	no	—
118	C₂³×C₈	—	73	2	no	—
119	C₂³×C₈	C₂×D₇₃	1	2	yes	—
120	C₂×#117	—	73	12	no	—
121	C₂×#117	—	73	12	no	—
122	C₂²×C₄⋊C₄	—	73	2	no	—
123	C₂²×C₂²⋊C₄	—	73	2	no	—
124	C₂²×C₄²	—	73	1	no	—
125	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
126	#433	—	73	64	no	—
127	C₂×C₄⋊C₈	—	73	24	no	—
128	C₂×C₂²⋊C₈	—	73	24	no	—
129	C₂×#97	C₂×D₇₃	1	12	yes	complemented
130	C₂×C₄×C₈	—	73	12	no	—
131	C₂⁵	—	73	1	no	—
132	C₂³×C₄	—	73	2	no	—
133	C₂³×C₄	—	73	1	no	—
134	C₂³×C₄	—	73	6	no	—
135	C₂³×C₄	—	73	6	no	—
136	C₂³×C₄	—	146	8	no	—
137	C₂³×C₄	C₄×D₇₃	1	2	yes	—
138	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	—	73	12	no	—
141	C₂²×C₈	—	146	8	no	—
142	C₂²×C₈	C₂²×D₇₃	1	12	yes	—
143	#117	—	73	48	no	—
144	#117	—	73	16	no	—
145	C₂×C₄⋊C₄	—	73	24	no	—
146	C₂×C₂²⋊C₄	—	73	24	no	—
147	C₂×C₄²	—	73	6	no	—
148	C₂×C₄²	—	73	6	no	—
149	C₂×C₄²	C₄×D₇₃	1	6	yes	—
150	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
151	C₄⋊C₈	—	73	32	no	—
152	C₂²⋊C₈	—	73	32	no	—
153	#97	C₂²×D₇₃	1	16	yes	complemented
154	C₄×C₈	—	73	16	no	—
155	C₂⁴	—	73	6	no	—
156	C₂⁴	—	146	4	no	—
157	C₂⁴	—	146	8	no	—
158	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	73	12	no	—
161	C₂²×C₄	—	73	6	no	—
162	C₂²×C₄	—	73	12	no	—
163	C₂²×C₄	—	73	4	no	—
164	C₂²×C₄	—	146	4	no	—
165	C₂²×C₄	—	146	8	no	—
166	C₂²×C₄	—	146	24	no	—
167	C₂²×C₄	—	146	24	no	—
168	C₂²×C₄	C₂×C₄×D₇₃	1	12	yes	—
169	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
170	C₂²×C₄	#11924028	1	8	yes	—
171	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
172	C₂×C₈	—	2	24	no	—
173	C₂×C₈	—	73	8	no	—
174	C₂×C₈	—	146	24	no	—
175	C₂×C₈	C₂³×D₇₃	1	8	yes	—
176	C₄⋊C₄	—	73	32	no	—
177	C₂²⋊C₄	—	73	32	no	—
178	C₄²	—	73	12	no	—
179	C₄²	—	73	4	no	—
180	C₄²	C₂×C₄×D₇₃	1	12	yes	—
181	C₄²	C₂³×D₇₃	1	4	yes	—
182	C₂³	—	292	16	no	—
183	C₂³	—	73	4	no	—
184	C₂³	—	146	12	no	—
185	C₂³	—	146	24	no	—
186	C₂³	#14556548	1	4	yes	—
187	C₂³	#14556599	1	8	yes	—
188	C₂³	C₂²×C₄×D₇₃	1	3	yes	—
189	C₂×C₄	—	2	24	no	—
190	C₂×C₄	—	73	8	no	—
191	C₂×C₄	—	73	4	no	—
192	C₂×C₄	—	146	12	no	—
193	C₂×C₄	—	146	24	no	—
194	C₂×C₄	—	146	8	no	—
195	C₂×C₄	—	146	24	no	—
196	C₂×C₄	#14556548	1	12	yes	—
197	C₂×C₄	C₂²×C₄×D₇₃	1	8	yes	—
198	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
199	C₂×C₄	C₂×#11924028	1	12	yes	—
200	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
201	C₈	—	2	8	no	—
202	C₈	—	146	8	no	—
203	C₂²	—	292	24	no	—
204	C₂²	—	146	4	no	—
205	C₂²	—	146	8	no	—
206	C₂²	#17988649	1	16	yes	direct product factor
207	C₂²	C₂×#14556548	1	6	yes	—
208	C₂²	C₂×#14556599	1	12	yes	—
209	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	—	146	4	no	—
212	C₄	—	146	8	no	—
213	C₄	C₂×#14556548	1	6	yes	—
214	C₄	C₂²×#11924028	1	2	yes	—
215	C₂	—	292	4	no	—
216	C₂	C₂×#17988649	1	12	yes	direct product factor
217	C₂	C₂²×#14556548	1	1	yes	characteristic
218	C₂	C₂²×#14556599	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	C₂²×#433
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#17988649	C₂	direct (N × H)
3. C₂²×#14556520	C₂	semidirect
5. C₂²×C₄×#10596831	C₂	semidirect
6. C₂²×#97×C₇₃	C₂	semidirect
14. #17988649	C₂²	direct (N × H)
15. C₂×#14556520	C₂²	semidirect
17. C₂×C₄×#10596831	C₂²	semidirect
20. C₂×#97×C₇₃	C₂²	semidirect
35. #14556520	C₂³	semidirect
37. C₄×#10596831	C₂³	semidirect
44. #97×C₇₃	C₂³	semidirect
93. C₂²×C₇₃	#433	semidirect
106. C₂×C₇₃	C₂×#433	semidirect
114. C₂²×#97	D₇₃	semidirect
116. C₇₃	C₂²×#433	semidirect
129. C₂×#97	C₂×D₇₃	semidirect
153. #97	C₂²×D₇₃	semidirect
206. C₂²	#17988649	direct (N × H)
216. C₂	C₂×#17988649	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×#14556548	2
C₂×#17988649	2
C₂²×#14556520	2
C₂²×#14556503	2
C₂²×C₄×#10596831	2
C₂²×#97×C₇₃	2
C₂²×#433	73

Subgroups — C22×#17988649 #33949702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988649 #33949702