Subgroups — #33241493

Order 18688.

474 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	#22620470	C₂	1	2	yes	maximal
2	#22612217	C₂	1	4	yes	maximal, complemented
3	#22612210	C₂	1	2	yes	maximal, complemented
4	#22612206	C₂	1	2	yes	maximal, complemented
5	#22611719	C₂	1	2	yes	maximal
6	#22611715	C₂	1	1	yes	maximal, characteristic
7	C₂×#17988413	C₂	1	1	yes	maximal, characteristic
8	C₇₃×#2108	C₂	1	1	yes	maximal, characteristic
9	#17989215	—	2	2	no	—
10	#17988996	—	2	4	no	—
11	#17988992	C₂²	1	4	yes	—
12	#17988989	C₂²	1	4	yes	—
13	#17988629	—	2	2	no	—
14	#17988625	C₂²	1	2	yes	—
15	C₂×#14556525	C₂²	1	2	yes	—
16	#17988415	C₂²	1	2	yes	—
17	#17988415	C₂²	1	4	yes	—
18	#17988413	C₂²	1	1	yes	characteristic
19	#17988413	C₂²	1	1	yes	characteristic
20	#17988413	C₂²	1	4	yes	—
21	#17988413	C₂²	1	2	yes	—
22	#17988413	C₂²	1	2	yes	—
23	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	#405×C₇₃	C₂²	1	1	yes	characteristic
26	#346×C₇₃	C₂²	1	2	yes	—
27	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
28	#14556591	—	2	4	no	—
29	#14556590	—	4	2	no	—
30	#14556583	—	2	4	no	—
31	#14556582	—	4	2	no	—
32	#14556568	—	4	4	no	—
33	#14556550	—	2	2	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	#14556525	—	2	2	no	—
36	#14556525	—	2	4	no	—
37	#14556525	C₂³	1	4	yes	—
38	#14556525	C₂³	1	2	yes	—
39	#14556525	C₂³	1	2	yes	—
40	#14556523	—	2	2	no	—
41	#14556523	—	2	4	no	—
42	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
44	#117×C₇₃	D₄	1	1	yes	characteristic
45	M₃₂×C₇₃	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₇₃	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
50	#97×C₇₃	D₄	1	2	yes	—
51	C₄×C₈×C₇₃	C₂³	1	2	yes	—
52	C₂×Dic₁₄₆	—	4	4	no	—
53	C₂²×Dic₇₃	—	4	2	no	—
54	C₄⋊Dic₇₃	—	2	2	no	—
55	C₄⋊Dic₇₃	—	2	4	no	—
56	C₄⋊Dic₇₃	—	2	2	no	—
57	C₄⋊Dic₇₃	—	4	2	no	—
58	C₄⋊Dic₇₃	—	4	4	no	—
59	#11924007	—	4	4	no	—
60	#11924007	—	4	4	no	—
61	C₂²⋊Dic₇₃	—	4	2	no	—
62	C₄×Dic₇₃	—	4	2	no	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₇₃	Q₁₆	1	4	yes	—
65	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
71	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₇₃	D₈	1	2	yes	—
73	C₄²×C₇₃	C₂×D₄	1	2	yes	—
74	C₁₆×C₇₃	C₂×Q₈	1	2	yes	—
75	C₁₆×C₇₃	C₂×Q₈	1	2	yes	—
76	C₁₆×C₇₃	C₂×Q₈	1	4	yes	—
77	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	—	4	2	no	—
86	C₂×Dic₇₃	—	4	4	no	—
87	C₂×Dic₇₃	—	16	2	no	—
88	Dic₁₄₆	—	16	4	no	—
89	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
90	C₈×C₇₃	C₂×Q₁₆	1	4	yes	—
91	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₇₃	—	2	1	no	—
94	C₂²×C₇₃	—	2	1	no	—
95	C₂²×C₇₃	#500	1	2	yes	—
96	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₇₃	—	2	2	no	—
98	C₄×C₇₃	—	2	2	no	—
99	C₄×C₇₃	#500	1	2	yes	—
100	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₇₃	—	8	2	no	—
103	Dic₇₃	—	8	4	no	—
104	Dic₇₃	—	8	2	no	—
105	#33729	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	—	2	2	no	—
107	C₂×C₇₃	C₂×#500	1	1	yes	characteristic
108	C₂×C₇₃	#3415	1	1	yes	characteristic
109	C₂×C₇₃	#3417	1	1	yes	characteristic
110	#3164	—	73	2	no	—
111	#2255	—	73	4	no	—
112	#2253	—	73	2	no	—
113	#2252	—	73	2	no	—
114	#2159	—	73	2	no	—
115	#2158	—	73	1	no	—
116	C₂×#366	—	73	1	no	—
117	#2108	D₇₃	1	1	yes	characteristic
118	C₇₃	#33729	1	1	yes	characteristic, complemented, Sylow
119	#544	—	146	2	no	—
120	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
121	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
122	#501	—	146	4	no	—
123	#500	—	73	4	no	—
124	#499	—	73	4	no	—
125	#428	—	146	2	no	—
126	#427	—	73	2	no	—
127	C₂×#107	—	73	2	no	—
128	#405	C₂×D₇₃	1	1	yes	characteristic
129	#367	—	73	2	no	—
130	#367	—	73	4	no	—
131	#366	—	73	2	no	—
132	#366	—	73	2	no	—
133	#366	—	73	1	no	—
134	#366	—	73	1	no	—
135	#366	—	73	4	no	—
136	#346	C₂×D₇₃	1	2	yes	—
137	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
138	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
140	C₄⋊Q₈	—	292	2	no	—
141	C₄⋊Q₈	—	146	4	no	—
142	#125	—	292	2	no	—
143	#125	—	146	4	no	—
144	C₂²⋊Q₈	—	292	4	no	—
145	#117	—	146	2	no	—
146	#117	D₂₉₂	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	146	2	no	—
148	M₃₂	Dic₁₄₆	1	4	yes	complemented
149	C₂×C₁₆	Dic₁₄₆	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
152	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
153	#107	—	73	4	no	—
154	#107	—	73	2	no	—
155	#107	—	73	2	no	—
156	#107	—	146	2	no	—
157	#107	—	146	4	no	—
158	#106	—	146	2	no	—
159	#106	—	146	4	no	—
160	#97	D₂₉₂	1	2	yes	—
161	C₄×C₈	C₂²×D₇₃	1	2	yes	—
162	C₂×Q₈	—	292	4	no	—
163	C₂²×C₄	—	292	2	no	—
164	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
165	M₁₆	Dic₂₉₂	1	4	yes	—
166	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
170	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
171	C₄⋊C₄	—	292	4	no	—
172	C₄⋊C₄	—	292	4	no	—
173	C₄⋊C₄	—	292	2	no	—
174	C₄⋊C₄	—	292	4	no	—
175	C₄⋊C₄	—	146	2	no	—
176	C₄⋊C₄	—	146	4	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
179	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
180	C₂²⋊C₄	—	292	2	no	—
181	C₂²⋊C₄	D₅₈₄	1	2	yes	—
182	C₄²	—	292	2	no	—
183	C₄²	C₂×D₂₉₂	1	2	yes	—
184	C₁₆	C₂×Dic₁₄₆	1	2	yes	—
185	C₁₆	C₂×Dic₁₄₆	1	2	yes	—
186	C₁₆	C₂×Dic₁₄₆	1	4	yes	—
187	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
188	Q₈	—	1168	4	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	4	no	—
192	C₂×C₄	—	1168	2	no	—
193	C₂×C₄	#14556591	1	2	yes	—
194	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
195	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
196	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
198	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
199	C₈	#14556591	1	2	yes	—
200	C₈	C₂×Dic₂₉₂	1	4	yes	—
201	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#17988992	1	2	yes	—
206	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	584	4	no	—
210	C₄	—	584	2	no	—
211	C₄	—	584	2	no	—
212	C₄	#17988992	1	2	yes	—
213	C₄	C₂×#14556591	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#17988992	1	1	yes	characteristic
217	C₂	#22623072	1	1	yes	characteristic
218	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2108	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#2108	D₇₃
hypocenter	C₇₃	#33729
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22612217	C₂	semidirect
3. #22612210	C₂	semidirect
4. #22612206	C₂	semidirect
45. M₃₂×C₇₃	Q₈	semidirect
46. C₂×C₁₆×C₇₃	Q₈	semidirect
118. C₇₃	#33729	semidirect
148. M₃₂	Dic₁₄₆	semidirect
149. C₂×C₁₆	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
#22620470	2
#22612217	2
#22612210	2
#22612206	2
#22611719	2
#22611715	2
C₂×#17988413	2
C₇₃×#2108	2
#33729	73