Subgroups — C₂×#22610090 #33086385

Order 18688.

1358 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556524	C₂	1	4	yes	maximal
2	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988457	C₂	1	1	yes	maximal, characteristic
4	#22610090	C₂	1	8	yes	maximal, direct product factor
5	C₂×#384×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10596831	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₇₃	—	2	2	no	—
8	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
9	C₂×#14556524	—	2	16	no	—
10	C₂×#14556524	C₂²	1	16	yes	—
11	C₂×C₄×#10596831	C₄	1	8	yes	complemented
12	C₂×C₄×#10596831	C₂²	1	4	yes	—
13	#17988457	—	2	2	no	—
14	#17988457	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₇₃	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#384×C₇₃	—	2	2	no	—
19	#384×C₇₃	C₂²	1	4	yes	—
20	C₂³×Dic₇₃	—	2	2	no	—
21	C₂²×#10596831	C₂×C₄	1	8	yes	—
22	C₂²×#10596831	D₄	1	8	yes	—
23	C₂²×#10596831	Q₈	1	8	yes	—
24	C₂²×#10596831	C₂³	1	4	yes	—
25	C₂×C₄⋊Dic₇₃	—	2	8	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	8	no	—
28	C₂×C₄⋊Dic₇₃	—	4	8	no	—
29	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
30	#14556524	—	2	32	no	—
31	C₄×#10596831	C₂×C₄	1	16	yes	complemented
32	C₂³×C₄×C₇₃	—	2	2	no	—
33	C₂³×C₄×C₇₃	D₄	1	2	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
38	C₂×C₄⋊C₄×C₇₃	—	4	8	no	—
39	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₃	D₄	1	2	yes	—
42	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
43	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
44	C₂²×Dic₇₃	—	2	4	no	—
45	C₂²×Dic₇₃	—	4	4	no	—
46	C₂²×Dic₇₃	—	4	8	no	—
47	C₂×#10596831	C₄⋊C₄	1	32	yes	complemented
48	C₂×#10596831	C₂²×C₄	1	8	yes	—
49	C₂×#10596831	C₂×D₄	1	8	yes	—
50	C₂×#10596831	C₂×Q₈	1	8	yes	—
51	C₄⋊Dic₇₃	—	2	8	no	—
52	C₄⋊Dic₇₃	—	4	4	no	—
53	C₄⋊Dic₇₃	—	4	16	no	—
54	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	—	2	4	no	—
56	C₂²×C₄×C₇₃	—	4	4	no	—
57	C₂²×C₄×C₇₃	—	4	8	no	—
58	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₇₃	D₈	1	8	yes	—
60	C₂²×C₄×C₇₃	Q₁₆	1	8	yes	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₇₃	—	2	8	no	—
69	C₄⋊C₄×C₇₃	—	4	4	no	—
70	C₄⋊C₄×C₇₃	—	4	16	no	—
71	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
72	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
73	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
74	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₇₃	#127	1	2	yes	—
76	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
77	C₂³×C₇₃	C₂×D₈	1	4	yes	—
78	C₂³×C₇₃	C₂×Q₁₆	1	4	yes	—
79	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	—	4	4	no	—
81	C₂×C₄×C₇₃	—	4	8	no	—
82	C₂×C₄×C₇₃	—	8	8	no	—
83	C₂×C₄×C₇₃	#107	1	32	yes	complemented
84	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₇₃	#127	1	2	yes	—
88	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₇₃	C₄⋊Q₈	1	8	yes	—
90	C₂×C₄×C₇₃	C₂×D₈	1	8	yes	—
91	C₂×C₄×C₇₃	C₂×Q₁₆	1	8	yes	—
92	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×Dic₇₃	—	4	4	no	—
97	C₂×Dic₇₃	—	4	8	no	—
98	C₂×Dic₇₃	—	8	8	no	—
99	#10596831	C₂×C₄⋊C₄	1	16	yes	complemented
100	C₂²×C₇₃	#384	1	4	yes	—
101	C₂²×C₇₃	C₂×#107	1	8	yes	—
102	C₂²×C₇₃	#493	1	4	yes	—
103	C₂²×C₇₃	#494	1	4	yes	—
104	C₂²×C₇₃	#500	1	8	yes	—
105	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
107	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₇₃	C₂²×D₈	1	2	yes	—
109	C₂²×C₇₃	C₂²×Q₁₆	1	2	yes	—
110	C₄×C₇₃	—	8	4	no	—
111	C₄×C₇₃	#384	1	4	yes	—
112	C₄×C₇₃	C₂×#107	1	16	yes	complemented
113	C₄×C₇₃	C₂²×C₄⋊C₄	1	2	yes	—
114	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
115	Dic₇₃	—	8	4	no	—
116	C₂×#1845	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	#1845	1	8	yes	complemented
118	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
119	C₂×C₇₃	C₂²×#107	1	2	yes	—
120	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
121	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
122	C₂×C₇₃	C₂×#500	1	2	yes	—
123	C₂²×#107	—	73	4	no	—
124	C₂²×C₄×C₈	—	73	1	no	—
125	C₂×#384	—	73	1	no	—
126	C₂×#384	D₇₃	1	1	yes	characteristic
127	#1845	—	73	8	no	—
128	C₇₃	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₈	—	73	2	no	—
130	C₂²×C₄⋊C₄	—	2	2	no	—
131	C₂²×C₄⋊C₄	—	73	2	no	—
132	C₂²×C₄⋊C₄	—	146	2	no	—
133	C₂²×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
134	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
135	C₂×#107	—	73	16	no	—
136	C₂×#107	—	146	16	no	—
137	C₂×C₄×C₈	—	73	8	no	—
138	C₂×C₄×C₈	—	73	4	no	—
139	#384	—	2	2	no	—
140	#384	—	73	4	no	—
141	#384	—	146	2	no	—
142	#384	C₂×D₇₃	1	4	yes	—
143	C₂³×C₄	—	2	2	no	—
144	C₂³×C₄	—	146	2	no	—
145	C₂³×C₄	C₂²⋊D₇₃	1	2	yes	—
146	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
147	C₂²×C₈	—	73	8	no	—
148	C₂²×C₈	—	73	8	no	—
149	C₂²×C₈	—	73	8	no	—
150	C₂²×C₈	—	73	4	no	—
151	C₂×C₄⋊C₄	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	8	no	—
155	C₂×C₄⋊C₄	—	292	8	no	—
156	C₂×C₄⋊C₄	—	73	4	no	—
157	C₂×C₄⋊C₄	—	146	2	no	—
158	C₂×C₄⋊C₄	—	146	8	no	—
159	C₂×C₄⋊C₄	—	146	8	no	—
160	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
161	C₂×C₄²	Dic₁₄₆	1	4	yes	—
162	C₂×C₄²	C₄×D₇₃	1	4	yes	—
163	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
164	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
165	#107	—	146	32	no	—
166	C₄×C₈	—	73	16	no	—
167	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	4	8	no	—
171	C₂²×C₄	—	292	4	no	—
172	C₂²×C₄	—	292	8	no	—
173	C₂²×C₄	—	146	4	no	—
174	C₂²×C₄	#11924007	1	8	yes	—
175	C₂²×C₄	#11924013	1	8	yes	—
176	C₂²×C₄	#11924018	1	8	yes	—
177	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
178	C₂²×C₄	D₄×D₇₃	1	4	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
181	C₂²×C₄	Q₈×D₇₃	1	4	yes	—
182	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
183	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
184	C₂×C₈	—	73	32	no	—
185	C₂×C₈	—	73	8	no	—
186	C₂×C₈	—	73	8	no	—
187	C₂×C₈	—	73	8	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	16	no	—
191	C₄⋊C₄	—	292	4	no	—
192	C₄⋊C₄	—	292	16	no	—
193	C₄⋊C₄	—	146	8	no	—
194	C₄²	#11924007	1	8	yes	—
195	C₄²	C₂×C₄×D₇₃	1	4	yes	—
196	C₄²	C₂×Dic₁₄₆	1	4	yes	—
197	C₂³	C₂×#11924007	1	2	yes	—
198	C₂³	#14556587	1	2	yes	—
199	C₂³	#14556589	1	2	yes	—
200	C₂³	C₂×#11924013	1	4	yes	—
201	C₂³	C₂×#11924018	1	4	yes	—
202	C₂³	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	8	8	no	—
206	C₂×C₄	—	292	4	no	—
207	C₂×C₄	—	292	8	no	—
208	C₂×C₄	—	584	8	no	—
209	C₂×C₄	#14556524	1	32	yes	complemented
210	C₂×C₄	C₄⋊C₄×D₇₃	1	8	yes	—
211	C₂×C₄	C₂×#11924007	1	8	yes	—
212	C₂×C₄	C₂×#11924007	1	2	yes	—
213	C₂×C₄	#14556587	1	2	yes	—
214	C₂×C₄	#14556589	1	2	yes	—
215	C₂×C₄	#14556590	1	8	yes	—
216	C₂×C₄	C₂×#11924013	1	8	yes	—
217	C₂×C₄	C₂×#11924018	1	8	yes	—
218	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
221	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
222	C₈	—	73	16	no	—
223	C₂²	#17988456	1	4	yes	—
224	C₂²	C₂×#14556524	1	8	yes	—
225	C₂²	#17988963	1	4	yes	—
226	C₂²	#17988966	1	4	yes	—
227	C₂²	#17988990	1	8	yes	—
228	C₂²	C₂²×#11924007	1	1	yes	characteristic
229	C₂²	C₂×#14556587	1	1	yes	characteristic
230	C₂²	C₂×#14556589	1	1	yes	characteristic
231	C₂²	C₂²×#11924013	1	2	yes	—
232	C₂²	C₂²×#11924018	1	2	yes	—
233	C₄	—	8	4	no	—
234	C₄	—	584	4	no	—
235	C₄	#17988456	1	4	yes	—
236	C₄	C₂×#14556524	1	16	yes	complemented
237	C₄	C₂×C₄⋊C₄×D₇₃	1	2	yes	—
238	C₄	C₂×#14556590	1	2	yes	—
239	C₂	#22610090	1	8	yes	direct product factor
240	C₂	C₂×#17988456	1	1	yes	characteristic
241	C₂	C₂²×#14556524	1	2	yes	—
242	C₂	C₂×#17988963	1	1	yes	characteristic
243	C₂	C₂×#17988966	1	1	yes	characteristic
244	C₂	C₂×#17988990	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#384×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#384	D₇₃
hypocenter	C₇₃	C₂×#1845
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
4. #22610090	C₂	direct (N × H)
11. C₂×C₄×#10596831	C₄	semidirect
31. C₄×#10596831	C₂×C₄	semidirect
47. C₂×#10596831	C₄⋊C₄	semidirect
83. C₂×C₄×C₇₃	#107	semidirect
99. #10596831	C₂×C₄⋊C₄	semidirect
112. C₄×C₇₃	C₂×#107	semidirect
117. C₂×C₇₃	#1845	semidirect
128. C₇₃	C₂×#1845	semidirect
209. C₂×C₄	#14556524	semidirect
236. C₄	C₂×#14556524	semidirect
239. C₂	#22610090	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×#14556524	2
C₂²×C₄×#10596831	2
C₂×#17988457	2
#22610090	2
C₂×#384×C₇₃	2
C₂×#1845	73

Subgroups — C2×#22610090 #33086385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22610090 #33086385