Subgroups — C₂×#22625853 #34021234

Order 18688.

1006 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989531	C₂	1	2	yes	maximal
2	#22625853	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989461	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#64408	C₂²	1	4	yes	—
6	#17989531	—	2	4	no	—
7	#17989531	C₂²	1	8	yes	—
8	C₂×(C₂×#64408)⋊C₄	—	2	4	no	—
9	#17989461	C₂²	1	16	yes	—
10	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
11	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₄	1	1	yes	characteristic, complemented
13	C₂×C₈×Dic₇₃	C₄	1	2	yes	complemented
14	C₄×C₈×D₇₃	C₄	1	4	yes	complemented
15	C₄×C₈×D₇₃	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂³×#64408	—	2	4	no	—
18	C₂×C₄⋊#64408	—	2	4	no	—
19	C₂×C₄⋊#64408	—	2	16	no	—
20	C₂×C₄⋊#64408	C₂³	1	8	yes	—
21	(C₂×#64408)⋊C₄	—	2	16	no	—
22	C₂²×C₄×D₇₃	D₄	1	2	yes	—
23	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
24	C₂²×#10596831	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₇₃	—	2	8	no	—
26	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₇₃	C₂³	1	4	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
30	C₄²×D₇₃	C₂×C₄	1	2	yes	—
31	C₄²×D₇₃	D₄	1	2	yes	—
32	C₄²×D₇₃	Q₈	1	2	yes	—
33	C₄²×D₇₃	C₂³	1	2	yes	—
34	C₄×#10596831	C₂×C₄	1	4	yes	complemented
35	C₈×Dic₇₃	C₂×C₄	1	8	yes	complemented
36	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
39	C₂²×#64408	—	2	8	no	—
40	C₂²×#64408	—	4	8	no	—
41	C₂²×#64408	—	4	16	no	—
42	C₄⋊#64408	—	2	16	no	—
43	C₄⋊#64408	—	4	8	no	—
44	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
46	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×D₇₃	—	2	8	no	—
48	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₂×D₄	1	4	yes	—
51	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×D₇₃	#41	1	8	yes	—
54	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
55	C₂×#10596831	—	2	4	no	—
56	C₂×#10596831	C₂²×C₄	1	4	yes	—
57	C₈×D₇₃	—	2	8	no	—
58	C₈×D₇₃	—	2	8	no	—
59	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
60	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
61	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
62	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	—	2	4	no	—
66	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×D₇₃	#124	1	4	yes	—
72	C₂²×D₇₃	#135	1	8	yes	—
73	C₂²×D₇₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×#64408	—	4	8	no	—
75	C₂×#64408	—	4	16	no	—
76	C₂×#64408	—	8	16	no	—
77	C₂×C₄×C₇₃	—	2	4	no	—
78	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#117	1	4	yes	—
81	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×Dic₇₃	—	2	4	no	—
83	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×Dic₇₃	#125	1	4	yes	—
86	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
87	C₄×D₇₃	—	2	8	no	—
88	C₄×D₇₃	—	2	8	no	—
89	C₄×D₇₃	#117	1	4	yes	—
90	C₄×D₇₃	#124	1	4	yes	—
91	C₄×D₇₃	#125	1	4	yes	—
92	C₄×D₇₃	C₂×#41	1	4	yes	—
93	C₈×C₇₃	—	2	4	no	—
94	#10596831	—	2	4	no	—
95	C₂²×C₇₃	#382	1	2	yes	—
96	C₂²×C₇₃	#427	1	4	yes	—
97	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂×D₇₃	#382	1	2	yes	—
99	C₂×D₇₃	#427	1	4	yes	—
100	C₂×D₇₃	#488	1	16	yes	—
101	C₂×D₇₃	C₂×#124	1	2	yes	—
102	C₂×D₇₃	C₂×#135	1	4	yes	—
103	#64408	—	8	8	no	—
104	C₄×C₇₃	—	2	4	no	—
105	C₄×C₇₃	#382	1	2	yes	—
106	C₄×C₇₃	C₂×#117	1	2	yes	—
107	Dic₇₃	—	2	4	no	—
108	Dic₇₃	#382	1	2	yes	—
109	Dic₇₃	C₂×#125	1	2	yes	—
110	C₂×#1835	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	#1835	1	4	yes	complemented
112	C₂×C₇₃	C₂×#382	1	1	yes	characteristic
113	C₂×C₇₃	C₂×#427	1	2	yes	—
114	D₇₃	#1835	1	4	yes	complemented
115	D₇₃	C₂×#488	1	4	yes	—
116	C₂²×C₄×C₈	—	73	1	no	—
117	C₂×#382	—	73	2	no	—
118	#1835	—	73	8	no	—
119	C₂×#340	—	73	4	no	—
120	C₇₃	C₂×#1835	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₈	—	73	2	no	—
122	C₂²×C₄⋊C₄	—	73	4	no	—
123	C₂²×C₄²	—	73	1	no	—
124	C₂×C₄×C₈	—	73	2	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	4	no	—
127	C₂×C₄×C₈	—	73	4	no	—
128	C₂×C₄×C₈	#64408	1	1	yes	characteristic, complemented
129	#382	—	73	8	no	—
130	#382	—	146	4	no	—
131	C₂×#95	—	146	4	no	—
132	#340	—	73	16	no	—
133	C₂³×C₄	—	73	1	no	—
134	C₂³×C₄	—	73	2	no	—
135	C₂³×C₄	—	146	4	no	—
136	C₂²×C₈	—	73	2	no	—
137	C₂²×C₈	—	73	4	no	—
138	C₂²×C₈	—	73	4	no	—
139	C₂²×C₈	—	146	8	no	—
140	C₂²×C₈	C₂×#64408	1	2	yes	—
141	C₂×C₄⋊C₄	—	73	8	no	—
142	C₂×C₄⋊C₄	—	146	4	no	—
143	C₂×C₄⋊C₄	—	146	16	no	—
144	C₂×C₄²	—	73	2	no	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	73	2	no	—
147	C₂×C₄²	—	73	2	no	—
148	C₂×C₄²	—	73	2	no	—
149	C₂×C₄²	—	73	2	no	—
150	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
151	C₄×C₈	—	73	8	no	—
152	C₄×C₈	—	73	4	no	—
153	C₄×C₈	C₂×#64408	1	4	yes	complemented
154	#95	—	146	16	no	—
155	C₂⁴	—	73	1	no	—
156	C₂²×C₄	—	292	8	no	—
157	C₂²×C₄	—	292	16	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	—	73	8	no	—
164	C₂²×C₄	—	73	2	no	—
165	C₂²×C₄	—	73	4	no	—
166	C₂²×C₄	—	73	4	no	—
167	C₂²×C₄	—	146	8	no	—
168	C₂²×C₄	—	146	8	no	—
169	C₂²×C₄	C₄⋊#64408	1	2	yes	—
170	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	73	4	no	—
173	C₂×C₈	—	146	8	no	—
174	C₂×C₈	—	146	8	no	—
175	C₂×C₈	—	146	4	no	—
176	C₂×C₈	C₂²×#64408	1	4	yes	—
177	C₄⋊C₄	—	292	8	no	—
178	C₄⋊C₄	—	146	16	no	—
179	C₄²	—	73	2	no	—
180	C₄²	—	73	4	no	—
181	C₄²	—	73	2	no	—
182	C₄²	—	73	4	no	—
183	C₄²	C₄⋊#64408	1	2	yes	—
184	C₄²	C₂²×#64408	1	2	yes	—
185	C₂³	—	73	1	no	—
186	C₂³	—	73	1	no	—
187	C₂³	—	73	4	no	—
188	C₂³	—	73	8	no	—
189	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	292	8	no	—
192	C₂×C₄	—	292	16	no	—
193	C₂×C₄	—	584	16	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	4	no	—
197	C₂×C₄	—	73	4	no	—
198	C₂×C₄	—	73	4	no	—
199	C₂×C₄	—	73	4	no	—
200	C₂×C₄	—	73	4	no	—
201	C₂×C₄	—	73	4	no	—
202	C₂×C₄	—	146	8	no	—
203	C₂×C₄	—	146	8	no	—
204	C₂×C₄	—	146	4	no	—
205	C₂×C₄	C₂×C₄⋊#64408	1	4	yes	—
206	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
207	C₂×C₄	#14556670	1	4	yes	—
208	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
209	C₈	—	2	4	no	—
210	C₈	—	146	4	no	—
211	C₂²	—	73	4	no	—
212	C₂²	—	73	4	no	—
213	C₂²	—	73	16	no	—
214	C₂²	—	73	2	no	—
215	C₂²	—	73	2	no	—
216	C₂²	#17989531	1	2	yes	—
217	C₂²	#17989597	1	4	yes	—
218	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	584	8	no	—
221	C₄	—	73	2	no	—
222	C₄	—	73	2	no	—
223	C₄	—	146	4	no	—
224	C₄	#17989531	1	2	yes	—
225	C₄	C₂×#14556670	1	2	yes	—
226	C₂	—	73	4	no	—
227	C₂	—	73	4	no	—
228	C₂	#22625853	1	4	yes	direct product factor
229	C₂	C₂×#17989531	1	1	yes	characteristic
230	C₂	C₂×#17989597	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×D₄
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1835
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22625853	C₂	direct (N × H)
12. C₂×C₄×#10596831	C₄	semidirect
13. C₂×C₈×Dic₇₃	C₄	semidirect
14. C₄×C₈×D₇₃	C₄	semidirect
16. C₂×C₄×C₈×C₇₃	C₄	semidirect
34. C₄×#10596831	C₂×C₄	semidirect
35. C₈×Dic₇₃	C₂×C₄	semidirect
38. C₄×C₈×C₇₃	C₂×C₄	semidirect
111. C₂×C₇₃	#1835	semidirect
114. D₇₃	#1835	semidirect
120. C₇₃	C₂×#1835	semidirect
128. C₂×C₄×C₈	#64408	semidirect
153. C₄×C₈	C₂×#64408	semidirect
228. C₂	#22625853	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989531	2
#22625853	2
C₂×#17989461	2
C₂×C₄×C₈×D₇₃	2
C₂×#1835	73

Subgroups — C2×#22625853 #34021234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625853 #34021234