Subgroups — #33001198

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988388	C₂	1	2	yes	maximal
2	C₂×#17988363	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608022	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	2	yes	—
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₇₃	—	2	1	no	—
9	#17988388	C₄	1	8	yes	complemented
10	#17988363	—	2	2	no	—
11	#17988350	—	2	1	no	—
12	C₂²×D₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₇₃	—	2	1	no	—
14	C₄×D₈×C₇₃	—	2	2	no	—
15	C₄×D₈×C₇₃	C₄	1	4	yes	complemented
16	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#102×C₇₃	—	2	1	no	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄×Dic₇₃	—	4	1	no	—
21	C₂×#11924003	—	2	2	no	—
22	C₂×#11924003	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	4	yes	complemented
24	#14556525	—	2	2	no	—
25	#14556525	C₂×C₄	1	4	yes	complemented
26	#14556498	—	4	1	no	—
27	C₂²×D₄×C₇₃	—	2	1	no	—
28	C₂³×C₄×C₇₃	—	4	1	no	—
29	C₂×D₈×C₇₃	—	2	1	no	—
30	C₂×D₈×C₇₃	—	2	2	no	—
31	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
32	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₇₃	—	2	4	no	—
36	C₄×D₄×C₇₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₇₃	—	4	1	no	—
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	#107×C₇₃	—	2	1	no	—
41	#107×C₇₃	C₂×C₄	1	2	yes	—
42	#102×C₇₃	—	2	4	no	—
43	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₇₃	D₄	1	2	yes	—
45	C₂²×Dic₇₃	—	4	2	no	—
46	C₄⋊Dic₇₃	—	2	4	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	C₄×Dic₇₃	—	8	2	no	—
49	#11924003	—	2	4	no	—
50	C₂⁴×C₇₃	—	4	1	no	—
51	C₂×D₄×C₇₃	—	2	2	no	—
52	C₂×D₄×C₇₃	—	2	4	no	—
53	C₂×D₄×C₇₃	—	4	1	no	—
54	C₂×D₄×C₇₃	—	4	2	no	—
55	C₂²×C₄×C₇₃	—	4	1	no	—
56	C₂²×C₄×C₇₃	—	4	4	no	—
57	C₂²×C₄×C₇₃	—	8	4	no	—
58	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	D₈×C₇₃	—	2	2	no	—
62	D₈×C₇₃	—	2	4	no	—
63	D₈×C₇₃	C₄²	1	4	yes	complemented
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	#41	1	2	yes	—
70	C₄⋊C₄×C₇₃	—	2	2	no	—
71	C₄⋊C₄×C₇₃	—	4	1	no	—
72	C₂²⋊C₄×C₇₃	—	4	4	no	—
73	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₇₃	SD₁₆	1	2	yes	—
75	C₂³×C₇₃	—	4	2	no	—
76	C₂³×C₇₃	—	4	4	no	—
77	C₂³×C₇₃	—	8	2	no	—
78	C₂³×C₇₃	—	8	1	no	—
79	C₂³×C₇₃	—	8	1	no	—
80	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
81	D₄×C₇₃	—	2	4	no	—
82	D₄×C₇₃	—	4	2	no	—
83	D₄×C₇₃	—	4	4	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	4	2	no	—
88	C₂×C₄×C₇₃	—	8	8	no	—
89	C₂×C₄×C₇₃	—	8	2	no	—
90	C₂×C₄×C₇₃	#102	1	2	yes	—
91	C₂×C₄×C₇₃	#102	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#135	1	2	yes	—
96	C₂×Dic₇₃	—	4	4	no	—
97	C₂×Dic₇₃	—	8	2	no	—
98	C₂×Dic₇₃	—	8	2	no	—
99	C₈×C₇₃	—	2	2	no	—
100	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
101	C₈×C₇₃	#117	1	1	yes	characteristic
102	C₈×C₇₃	C₄×D₄	1	2	yes	—
103	C₂²×C₇₃	—	4	4	no	—
104	C₂²×C₇₃	—	8	2	no	—
105	C₂²×C₇₃	—	8	2	no	—
106	C₂²×C₇₃	—	8	4	no	—
107	C₂²×C₇₃	—	8	4	no	—
108	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
109	C₂²×C₇₃	C₄×D₈	1	2	yes	—
110	C₂²×C₇₃	#509	1	1	yes	characteristic
111	C₂²×C₇₃	#509	1	2	yes	—
112	C₂²×C₇₃	#510	1	1	yes	characteristic
113	C₄×C₇₃	—	2	2	no	—
114	C₄×C₇₃	—	2	2	no	—
115	C₄×C₇₃	—	8	2	no	—
116	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₇₃	#416	1	1	yes	characteristic
118	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
119	Dic₇₃	—	8	4	no	—
120	#12637	—	73	1	no	maximal, Sylow
121	C₂×C₇₃	—	8	4	no	—
122	C₂×C₇₃	C₄×#102	1	1	yes	characteristic
123	C₂×C₇₃	#2139	1	1	yes	characteristic
124	C₂×C₇₃	#2140	1	1	yes	characteristic
125	C₂×C₇₃	#2175	1	2	yes	—
126	C₂×C₇₃	#2177	1	2	yes	—
127	C₂×C₄×D₈	D₇₃	1	1	yes	characteristic
128	C₂×#357	—	73	2	no	—
129	C₂×#346	—	73	1	no	—
130	C₄×#107	—	73	1	no	—
131	#1380	—	73	2	no	—
132	C₇₃	#12637	1	1	yes	characteristic, complemented, Sylow
133	C₂²×D₈	C₂×D₇₃	1	1	yes	characteristic
134	C₂×C₄×D₄	—	2	1	no	—
135	C₂²×C₁₆	—	73	2	no	—
136	C₄×D₈	—	2	2	no	—
137	C₄×D₈	Dic₇₃	1	4	yes	complemented
138	C₂×#107	—	73	2	no	—
139	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
140	C₂×#102	—	2	1	no	—
141	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
142	C₄×C₄⋊C₄	—	146	1	no	—
143	#357	—	73	8	no	—
144	#346	—	146	2	no	—
145	#340	—	146	1	no	—
146	C₂²×D₄	—	2	1	no	—
147	C₂³×C₄	—	4	1	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	C₄×D₇₃	1	4	yes	—
151	C₂×D₈	C₂×Dic₇₃	1	2	yes	—
152	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
154	C₄×D₄	—	2	4	no	—
155	C₄×D₄	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	146	2	no	—
158	C₂×C₂²⋊C₄	—	4	1	no	—
159	C₂×C₄²	—	292	1	no	—
160	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
161	C₂×C₁₆	—	73	4	no	—
162	C₂×C₁₆	—	146	2	no	—
163	C₂×C₁₆	—	146	2	no	—
164	#107	—	2	1	no	—
165	#107	—	73	4	no	—
166	#107	—	146	2	no	—
167	#107	C₂×Dic₇₃	1	2	yes	—
168	#102	—	2	4	no	—
169	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
170	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
171	#95	—	292	1	no	—
172	C₂⁴	—	4	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	8	4	no	—
180	C₂²×C₄	—	292	2	no	—
181	C₂²×C₄	#11924013	1	1	yes	characteristic
182	C₂²×C₄	#11924013	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
184	D₈	—	2	2	no	—
185	D₈	—	2	4	no	—
186	D₈	C₄×Dic₇₃	1	4	yes	complemented
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
190	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
192	C₂×C₈	#11924028	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	292	2	no	—
196	C₄⋊C₄	—	146	4	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₄²	—	584	2	no	—
199	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
200	C₄²	#11924016	1	2	yes	—
201	C₁₆	—	146	4	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	1	no	—
207	C₂³	C₂×#11924013	1	1	yes	characteristic
208	D₄	—	2	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	8	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	292	4	no	—
218	C₂×C₄	—	584	2	no	—
219	C₂×C₄	—	584	2	no	—
220	C₂×C₄	#14556512	1	2	yes	—
221	C₂×C₄	#14556512	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
223	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
224	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
225	C₂×C₄	#14556611	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	C₂×C₄×Dic₇₃	1	1	yes	characteristic
228	C₈	#14556550	1	1	yes	characteristic
229	C₈	C₄×C₂²⋊D₇₃	1	2	yes	—
230	C₂²	—	4	4	no	—
231	C₂²	—	8	4	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₂×#14556512	1	1	yes	characteristic
236	C₂²	C₄×#11924013	1	2	yes	—
237	C₂²	#17989022	1	1	yes	characteristic
238	C₂²	#17989023	1	2	yes	—
239	C₂²	#17989027	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	584	4	no	—
244	C₄	C₄×C₂²⋊Dic₇₃	1	1	yes	characteristic
245	C₄	#17988583	1	1	yes	characteristic
246	C₄	C₄×#11924016	1	2	yes	—
247	C₂	—	8	4	no	—
248	C₂	C₄×#14556512	1	1	yes	characteristic
249	C₂	#22611643	1	1	yes	characteristic
250	C₂	#22611647	1	1	yes	characteristic
251	C₂	#22611795	1	2	yes	—
252	C₂	#22611805	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×D₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂²×D₈×C₇₃	C₂²
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×D₈	D₇₃
hypocenter	C₇₃	#12637
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17988363	C₂	semidirect
3. C₄×#14556525	C₂	semidirect
4. #22608022	C₂	semidirect
9. #17988388	C₄	semidirect
15. C₄×D₈×C₇₃	C₄	semidirect
23. C₂×#11924003	C₂×C₄	semidirect
25. #14556525	C₂×C₄	semidirect
63. D₈×C₇₃	C₄²	semidirect
132. C₇₃	#12637	semidirect
137. C₄×D₈	Dic₇₃	semidirect
186. D₈	C₄×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988388	2
C₂×#17988363	2
C₄×#14556525	2
#22608022	2
C₂×C₄×D₈×C₇₃	2
#12637	73