Subgroups — C₂²×D₈×Dic₇₃ #33949986

Order 18688.

3694 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×Dic₇₃	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×Dic₇₃	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14556512	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₇₃	—	2	2	no	—
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂×D₄×Dic₇₃	—	2	12	no	—
10	C₂×D₄×Dic₇₃	C₂²	1	24	yes	complemented
11	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊Dic₇₃	—	2	2	no	—
13	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	D₈×Dic₇₃	C₂²	1	64	yes	direct product factor
15	C₂×#14556525	C₂²	1	12	yes	complemented
16	C₂×#14556512	C₂²	1	24	yes	complemented
17	C₂×C₈×Dic₇₃	C₂²	1	12	yes	complemented
18	C₂³×D₄×C₇₃	C₂²	1	2	yes	—
19	C₂²×D₈×C₇₃	C₄	1	16	yes	complemented
20	C₂²×D₈×C₇₃	C₂²	1	12	yes	—
21	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₇₃	—	2	24	no	—
23	C₂³×Dic₇₃	—	2	2	no	—
24	C₂³×Dic₇₃	—	4	16	no	—
25	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
27	C₂²×#10596831	—	2	4	no	—
28	C₂²×#10596831	C₂³	1	6	yes	—
29	D₄×Dic₇₃	—	2	48	no	—
30	D₄×Dic₇₃	C₂³	1	32	yes	complemented
31	C₂×C₄⋊Dic₇₃	—	2	6	no	—
32	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
33	C₂×C₂²⋊Dic₇₃	—	2	24	no	—
34	C₂×C₄×Dic₇₃	D₄	1	6	yes	—
35	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
36	#14556525	C₂³	1	16	yes	complemented
37	#14556512	C₂³	1	32	yes	complemented
38	C₈×Dic₇₃	C₂³	1	16	yes	complemented
39	C₂⁵×C₇₃	—	2	2	no	—
40	C₂²×D₄×C₇₃	—	2	6	no	—
41	C₂²×D₄×C₇₃	—	2	8	no	—
42	C₂²×D₄×C₇₃	C₂×C₄	1	16	yes	complemented
43	C₂²×D₄×C₇₃	C₂³	1	12	yes	—
44	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₇₃	C₂×C₄	1	96	yes	complemented
46	C₂×D₈×C₇₃	C₂³	1	16	yes	—
47	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
49	C₂²×Dic₇₃	—	2	4	no	—
50	C₂²×Dic₇₃	—	2	12	no	—
51	C₂²×Dic₇₃	—	2	32	no	—
52	C₂²×Dic₇₃	—	4	8	no	—
53	C₂²×Dic₇₃	—	4	96	no	—
54	C₂²×Dic₇₃	D₈	1	8	yes	direct product factor
55	C₂²×Dic₇₃	C₂×D₄	1	6	yes	—
56	C₂²×Dic₇₃	C₂×D₄	1	6	yes	—
57	C₂×#10596831	—	2	12	no	—
58	C₂×#10596831	C₂⁴	1	4	yes	—
59	C₄⋊Dic₇₃	—	2	12	no	—
60	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
61	C₂²⋊Dic₇₃	—	2	32	no	—
62	C₄×Dic₇₃	C₂×D₄	1	12	yes	—
63	C₄×Dic₇₃	C₂⁴	1	4	yes	—
64	C₂⁴×C₇₃	—	2	16	no	—
65	C₂⁴×C₇₃	—	2	12	no	—
66	C₂⁴×C₇₃	—	4	8	no	—
67	C₂⁴×C₇₃	—	4	8	no	—
68	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×D₄×C₇₃	—	2	48	no	—
70	C₂×D₄×C₇₃	—	2	12	no	—
71	C₂×D₄×C₇₃	—	2	24	no	—
72	C₂×D₄×C₇₃	C₂²×C₄	1	48	yes	complemented
73	C₂×D₄×C₇₃	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₇₃	#41	1	4	yes	—
77	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
78	D₈×C₇₃	C₂²×C₄	1	64	yes	complemented
79	C₂×C₈×C₇₃	C₂²×C₄	1	24	yes	complemented
80	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
81	C₂³×C₇₃	—	2	48	no	—
82	C₂³×C₇₃	—	2	8	no	—
83	C₂³×C₇₃	—	4	24	no	—
84	C₂³×C₇₃	—	4	24	no	—
85	C₂³×C₇₃	—	4	64	no	—
86	C₂³×C₇₃	C₄×D₄	1	4	yes	—
87	C₂³×C₇₃	C₂×D₈	1	4	yes	—
88	C₂³×C₇₃	#135	1	4	yes	—
89	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
90	D₄×C₇₃	—	2	48	no	—
91	D₄×C₇₃	—	2	8	no	—
92	D₄×C₇₃	C₂³×C₄	1	16	yes	complemented
93	C₂×C₄×C₇₃	C₄×D₄	1	12	yes	—
94	C₂×C₄×C₇₃	C₂³×C₄	1	6	yes	—
95	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₇₃	C₂×#41	1	6	yes	—
97	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
98	C₂×Dic₇₃	—	2	12	no	—
99	C₂×Dic₇₃	—	2	8	no	—
100	C₂×Dic₇₃	—	4	24	no	—
101	C₂×Dic₇₃	—	4	64	no	—
102	C₂×Dic₇₃	C₂×D₈	1	24	yes	direct product factor
103	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
104	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
105	C₈×C₇₃	C₂³×C₄	1	8	yes	complemented
106	#10596831	—	2	4	no	—
107	C₂²×C₇₃	—	2	16	no	—
108	C₂²×C₇₃	—	4	96	no	—
109	C₂²×C₇₃	—	4	8	no	—
110	C₂²×C₇₃	—	4	8	no	—
111	C₂²×C₇₃	C₄×D₈	1	16	yes	complemented
112	C₂²×C₇₃	C₂×C₄×D₄	1	6	yes	—
113	C₂²×C₇₃	C₂²×D₈	1	6	yes	—
114	C₂²×C₇₃	C₂×#135	1	6	yes	—
115	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
116	C₄×C₇₃	C₂×C₄×D₄	1	6	yes	—
117	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
118	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
119	Dic₇₃	—	2	4	no	—
120	Dic₇₃	—	4	8	no	—
121	Dic₇₃	C₂²×D₈	1	8	yes	direct product factor
122	C₂²×C₄×D₈	—	73	1	no	maximal, Sylow
123	C₂×C₇₃	—	4	16	no	—
124	C₂×C₇₃	C₂×C₄×D₈	1	12	yes	complemented
125	C₂×C₇₃	C₂²×C₄×D₄	1	1	yes	characteristic
126	C₂×C₇₃	C₂³×D₈	1	1	yes	characteristic
127	C₂×C₇₃	C₂²×#135	1	1	yes	characteristic
128	C₂³×D₈	D₇₃	1	1	yes	characteristic
129	C₂²×C₄×D₄	—	73	2	no	—
130	C₂×C₄×D₈	—	73	24	no	—
131	C₂²×#107	—	73	1	no	—
132	C₂²×#102	—	73	2	no	—
133	C₂²×C₄×C₈	—	73	1	no	—
134	C₇₃	C₂²×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
135	C₂³×D₄	C₂×D₇₃	1	2	yes	—
136	C₂⁴×C₄	—	146	2	no	—
137	C₂²×D₈	Dic₇₃	1	16	yes	direct product factor
138	C₂²×D₈	C₂×D₇₃	1	12	yes	—
139	C₂³×C₈	—	73	1	no	—
140	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
141	C₂×C₄×D₄	—	73	24	no	—
142	C₂×C₄×D₄	—	146	12	no	—
143	C₂²×C₄⋊C₄	—	73	2	no	—
144	C₂²×C₂²⋊C₄	—	146	2	no	—
145	C₂²×C₄²	—	73	1	no	—
146	C₄×D₈	—	73	64	no	—
147	C₂×#107	—	73	12	no	—
148	C₂×#102	—	73	24	no	—
149	C₂×C₄×C₈	—	73	12	no	—
150	C₂⁵	—	2	2	no	—
151	C₂²×D₄	—	2	6	no	—
152	C₂²×D₄	—	2	8	no	—
153	C₂²×D₄	C₂×Dic₇₃	1	16	yes	complemented
154	C₂²×D₄	C₂²×D₇₃	1	12	yes	—
155	C₂³×C₄	—	292	16	no	—
156	C₂³×C₄	—	73	1	no	—
157	C₂³×C₄	—	73	1	no	—
158	C₂³×C₄	—	146	24	no	—
159	C₂³×C₄	—	146	2	no	—
160	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
161	C₂×D₈	C₂×Dic₇₃	1	96	yes	direct product factor
162	C₂×D₈	C₂²×D₇₃	1	16	yes	—
163	C₂²×C₈	—	73	6	no	—
164	C₂²×C₈	—	146	4	no	—
165	C₂²×C₈	C₂×Dic₇₃	1	8	yes	complemented
166	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
167	C₄×D₄	—	73	32	no	—
168	C₄×D₄	—	146	48	no	—
169	C₂×C₄⋊C₄	—	73	12	no	—
170	C₂×C₄⋊C₄	—	146	6	no	—
171	C₂×C₂²⋊C₄	—	146	24	no	—
172	C₂×C₄²	—	73	6	no	—
173	C₂×C₄²	—	73	6	no	—
174	#107	—	73	16	no	—
175	#102	—	73	32	no	—
176	C₄×C₈	—	73	16	no	—
177	C₂⁴	—	2	16	no	—
178	C₂⁴	—	2	12	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	—	4	8	no	—
181	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
182	C₂×D₄	—	2	48	no	—
183	C₂×D₄	—	2	12	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	C₂²×Dic₇₃	1	48	yes	complemented
186	C₂×D₄	C₂³×D₇₃	1	8	yes	—
187	C₂²×C₄	—	292	8	no	—
188	C₂²×C₄	—	292	96	no	—
189	C₂²×C₄	—	73	8	no	—
190	C₂²×C₄	—	73	6	no	—
191	C₂²×C₄	—	73	6	no	—
192	C₂²×C₄	—	146	32	no	—
193	C₂²×C₄	—	146	12	no	—
194	C₂²×C₄	—	146	4	no	—
195	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
196	C₂²×C₄	D₄×D₇₃	1	3	yes	—
197	C₂²×C₄	#11924027	1	4	yes	—
198	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
199	D₈	C₂²×Dic₇₃	1	64	yes	direct product factor
200	C₂×C₈	—	73	4	no	—
201	C₂×C₈	—	146	12	no	—
202	C₂×C₈	C₂²×Dic₇₃	1	24	yes	complemented
203	C₂×C₈	C₂³×D₇₃	1	4	yes	—
204	C₄⋊C₄	—	73	8	no	—
205	C₄⋊C₄	—	146	12	no	—
206	C₂²⋊C₄	—	146	32	no	—
207	C₄²	—	73	12	no	—
208	C₄²	—	73	4	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	48	no	—
211	C₂³	—	4	64	no	—
212	C₂³	—	4	24	no	—
213	C₂³	—	4	24	no	—
214	C₂³	D₄×Dic₇₃	1	4	yes	—
215	C₂³	D₈×D₇₃	1	4	yes	—
216	C₂³	#14556610	1	4	yes	—
217	C₂³	C₂×D₄×D₇₃	1	3	yes	—
218	D₄	—	2	8	no	—
219	D₄	—	2	48	no	—
220	D₄	C₂³×Dic₇₃	1	16	yes	complemented
221	C₂×C₄	—	292	24	no	—
222	C₂×C₄	—	292	64	no	—
223	C₂×C₄	—	73	4	no	—
224	C₂×C₄	—	73	24	no	—
225	C₂×C₄	—	73	4	no	—
226	C₂×C₄	—	146	8	no	—
227	C₂×C₄	—	146	12	no	—
228	C₂×C₄	D₄×Dic₇₃	1	12	yes	—
229	C₂×C₄	C₂³×Dic₇₃	1	6	yes	—
230	C₂×C₄	C₂×D₄×D₇₃	1	3	yes	—
231	C₂×C₄	C₂×#11924027	1	6	yes	—
232	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
233	C₈	—	146	4	no	—
234	C₈	C₂³×Dic₇₃	1	8	yes	complemented
235	C₂²	—	2	16	no	—
236	C₂²	—	4	96	no	—
237	C₂²	—	4	8	no	—
238	C₂²	—	4	8	no	—
239	C₂²	D₈×Dic₇₃	1	16	yes	direct product factor
240	C₂²	C₂×D₄×Dic₇₃	1	6	yes	—
241	C₂²	C₂×D₈×D₇₃	1	6	yes	—
242	C₂²	C₂×#14556610	1	6	yes	—
243	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
244	C₄	—	292	8	no	—
245	C₄	—	73	8	no	—
246	C₄	—	146	4	no	—
247	C₄	C₂×D₄×Dic₇₃	1	6	yes	—
248	C₄	C₂⁴×Dic₇₃	1	1	yes	characteristic
249	C₄	C₂²×#11924027	1	1	yes	characteristic
250	C₂	—	4	16	no	—
251	C₂	C₂×D₈×Dic₇₃	1	12	yes	direct product factor
252	C₂	C₂²×D₄×Dic₇₃	1	1	yes	characteristic
253	C₂	C₂²×D₈×D₇₃	1	1	yes	characteristic
254	C₂	C₂²×#14556610	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₃
commutator	C₄×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting, prime radical	C₂³×D₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂³×D₈	D₇₃
hypocenter	C₇₃	C₂²×C₄×D₈
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×Dic₇₃	C₂	semidirect
2. C₂×D₈×Dic₇₃	C₂	direct (N × H)
3. C₂²×#14556525	C₂	semidirect
4. C₂²×#14556512	C₂	semidirect
5. C₂²×C₈×Dic₇₃	C₂	semidirect
10. C₂×D₄×Dic₇₃	C₂²	semidirect
14. D₈×Dic₇₃	C₂²	direct (N × H)
15. C₂×#14556525	C₂²	semidirect
16. C₂×#14556512	C₂²	semidirect
17. C₂×C₈×Dic₇₃	C₂²	semidirect
19. C₂²×D₈×C₇₃	C₄	semidirect
30. D₄×Dic₇₃	C₂³	semidirect
36. #14556525	C₂³	semidirect
37. #14556512	C₂³	semidirect
38. C₈×Dic₇₃	C₂³	semidirect
42. C₂²×D₄×C₇₃	C₂×C₄	semidirect
45. C₂×D₈×C₇₃	C₂×C₄	semidirect
47. C₂²×C₈×C₇₃	C₂×C₄	semidirect
54. C₂²×Dic₇₃	D₈	direct (N × H)
72. C₂×D₄×C₇₃	C₂²×C₄	semidirect
78. D₈×C₇₃	C₂²×C₄	semidirect
79. C₂×C₈×C₇₃	C₂²×C₄	semidirect
92. D₄×C₇₃	C₂³×C₄	semidirect
102. C₂×Dic₇₃	C₂×D₈	direct (N × H)
105. C₈×C₇₃	C₂³×C₄	semidirect
111. C₂²×C₇₃	C₄×D₈	semidirect
121. Dic₇₃	C₂²×D₈	direct (N × H)
124. C₂×C₇₃	C₂×C₄×D₈	semidirect
134. C₇₃	C₂²×C₄×D₈	semidirect
137. C₂²×D₈	Dic₇₃	direct (N × H)
153. C₂²×D₄	C₂×Dic₇₃	semidirect
161. C₂×D₈	C₂×Dic₇₃	direct (N × H)
165. C₂²×C₈	C₂×Dic₇₃	semidirect
185. C₂×D₄	C₂²×Dic₇₃	semidirect
199. D₈	C₂²×Dic₇₃	direct (N × H)
202. C₂×C₈	C₂²×Dic₇₃	semidirect
220. D₄	C₂³×Dic₇₃	semidirect
234. C₈	C₂³×Dic₇₃	semidirect
239. C₂²	D₈×Dic₇₃	direct (N × H)
251. C₂	C₂×D₈×Dic₇₃	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×D₄×Dic₇₃	2
C₂×D₈×Dic₇₃	2
C₂²×#14556525	2
C₂²×#14556512	2
C₂²×C₈×Dic₇₃	2
C₂³×D₈×C₇₃	2
C₂²×C₄×D₈	73

Subgroups — C22×D8×Dic73 #33949986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₈×Dic₇₃ #33949986