Subgroups — #33241342

Order 18688.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620445	C₂	1	1	yes	maximal, characteristic
2	#22612176	C₂	1	4	yes	maximal, complemented
3	#22612158	C₂	1	4	yes	maximal, complemented
4	#22611651	C₂	1	4	yes	maximal
5	C₄×#14556531	C₂	1	1	yes	maximal, characteristic
6	C₂×#494×C₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989144	—	2	1	no	—
8	C₂×#14556531	C₂²	1	2	yes	—
9	#17988992	C₂²	1	2	yes	—
10	#17988989	—	2	1	no	—
11	#17988989	C₂²	1	2	yes	—
12	#17988985	—	2	1	no	—
13	#17988625	—	2	1	no	—
14	#17988625	C₂²	1	2	yes	—
15	#17988391	C₂²	1	16	yes	—
16	#17988363	C₂²	1	2	yes	—
17	C₄×#11924003	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₇₃	—	2	1	no	—
19	C₂²×Q₁₆×C₇₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₇₃	—	2	1	no	—
21	#494×C₇₃	—	2	1	no	—
22	#494×C₇₃	—	2	1	no	—
23	#494×C₇₃	C₂²	1	2	yes	—
24	#494×C₇₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#14556591	—	2	2	no	—
27	#14556583	—	2	2	no	—
28	#14556570	—	4	2	no	—
29	C₄×Dic₁₄₆	—	4	2	no	—
30	#14556550	—	2	2	no	—
31	#14556531	—	2	4	no	—
32	C₂×#11924003	C₂³	1	4	yes	—
33	#14556525	—	2	2	no	—
34	#14556525	C₂³	1	4	yes	—
35	#14556523	—	2	2	no	—
36	#14556523	—	2	2	no	—
37	C₂²×Q₈×C₇₃	—	2	2	no	—
38	C₂×Q₁₆×C₇₃	—	2	2	no	—
39	C₂×Q₁₆×C₇₃	—	2	2	no	—
40	C₂×Q₁₆×C₇₃	—	2	2	no	—
41	C₂×Q₁₆×C₇₃	—	2	8	no	—
42	C₂×Q₁₆×C₇₃	—	4	4	no	—
43	C₂×Q₁₆×C₇₃	C₂³	1	4	yes	—
44	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₇₃	—	2	2	no	—
47	C₄⋊Q₈×C₇₃	—	2	2	no	—
48	C₄⋊Q₈×C₇₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₇₃	—	4	2	no	—
50	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₄₆	—	4	2	no	—
56	C₄⋊Dic₇₃	—	2	4	no	—
57	C₄⋊Dic₇₃	—	4	2	no	—
58	#11924007	—	4	4	no	—
59	C₂²⋊Dic₇₃	—	4	2	no	—
60	C₄×Dic₇₃	—	4	2	no	—
61	#11924003	—	2	4	no	—
62	C₂×Q₈×C₇₃	—	2	4	no	—
63	C₂×Q₈×C₇₃	—	4	2	no	—
64	C₂×Q₈×C₇₃	—	4	4	no	—
65	C₂×Q₈×C₇₃	—	4	4	no	—
66	C₂²×C₄×C₇₃	—	4	2	no	—
67	C₂²×C₄×C₇₃	D₈	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	Q₁₆×C₇₃	—	2	8	no	—
70	Q₁₆×C₇₃	—	4	4	no	—
71	Q₁₆×C₇₃	—	4	4	no	—
72	Q₁₆×C₇₃	—	4	4	no	—
73	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
74	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	#41	1	4	yes	—
78	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₇₃	—	4	4	no	—
80	C₄⋊C₄×C₇₃	—	4	4	no	—
81	C₄²×C₇₃	D₈	1	2	yes	—
82	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₇₃	—	4	4	no	—
86	Q₈×C₇₃	—	4	4	no	—
87	Q₈×C₇₃	—	8	4	no	—
88	C₂×C₄×C₇₃	—	2	4	no	—
89	C₂×C₄×C₇₃	—	4	4	no	—
90	C₂×C₄×C₇₃	—	8	4	no	—
91	C₂×C₄×C₇₃	#124	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₇₃	—	4	4	no	—
98	Dic₁₄₆	—	8	4	no	—
99	C₈×C₇₃	#124	1	2	yes	—
100	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
101	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
102	C₈×C₇₃	C₂×#41	1	2	yes	—
103	C₂²×C₇₃	—	2	2	no	—
104	C₂²×C₇₃	#486	1	2	yes	—
105	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₇₃	—	2	2	no	—
107	C₄×C₇₃	—	8	4	no	—
108	C₄×C₇₃	#486	1	2	yes	—
109	C₄×C₇₃	#510	1	4	yes	—
110	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
111	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
112	Dic₇₃	—	8	4	no	—
113	#33705	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#510	1	2	yes	—
117	#3160	—	73	1	no	—
118	C₂×#494	D₇₃	1	1	yes	characteristic
119	#2245	—	73	4	no	—
120	#2241	—	73	4	no	—
121	#2141	—	73	4	no	—
122	C₄×M₃₂	—	73	1	no	—
123	C₇₃	#33705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	—	2	1	no	—
125	C₂²×Q₁₆	C₂×D₇₃	1	2	yes	—
126	#533	—	146	1	no	—
127	C₂×C₄⋊Q₈	—	2	1	no	—
128	C₂×M₃₂	—	73	2	no	—
129	#500	—	73	2	no	—
130	#499	—	73	2	no	—
131	#499	—	146	1	no	—
132	#498	—	146	1	no	—
133	#494	—	2	1	no	—
134	#494	—	2	1	no	—
135	#494	C₂×D₇₃	1	2	yes	—
136	#494	C₂×D₇₃	1	2	yes	—
137	#427	—	73	2	no	—
138	#427	—	146	1	no	—
139	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
140	#358	—	73	16	no	—
141	#346	—	73	2	no	—
142	C₄×C₁₆	—	73	2	no	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	4	4	no	—
149	C₂×Q₁₆	C₂²×D₇₃	1	4	yes	—
150	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	146	2	no	—
156	#125	—	146	2	no	—
157	C₂²⋊Q₈	—	292	2	no	—
158	C₄×Q₈	—	292	2	no	—
159	#117	—	146	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
162	M₃₂	—	146	4	no	—
163	C₂×C₁₆	—	73	4	no	—
164	#107	—	73	4	no	—
165	#107	—	146	2	no	—
166	#106	—	146	2	no	—
167	#106	—	146	2	no	—
168	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	292	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#11924013	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	Q₁₆	—	4	4	no	—
184	C₂×C₈	#11924016	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
188	C₂×C₈	#11924027	1	4	yes	—
189	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	292	4	no	—
193	C₄⋊C₄	—	292	2	no	—
194	C₄⋊C₄	—	146	4	no	—
195	C₂²⋊C₄	—	292	2	no	—
196	C₄²	—	292	2	no	—
197	C₄²	#11924013	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
200	C₁₆	—	146	4	no	—
201	C₂³	C₂×#11924013	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	584	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	292	4	no	—
210	C₂×C₄	#14556578	1	2	yes	—
211	C₂×C₄	C₂×#11924013	1	2	yes	—
212	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
213	C₂×C₄	C₂×#11924013	1	2	yes	—
214	C₂×C₄	C₂×#11924016	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
216	C₈	#14556578	1	2	yes	—
217	C₈	C₂×#11924016	1	2	yes	—
218	C₈	C₂×#11924016	1	2	yes	—
219	C₈	C₂×#11924027	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#17988933	1	2	yes	—
222	C₂²	C₂²×#11924013	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	584	4	no	—
226	C₄	#17988933	1	2	yes	—
227	C₄	#17989028	1	4	yes	—
228	C₄	C₂×#14556578	1	1	yes	characteristic
229	C₄	C₂²×#11924016	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#17988933	1	1	yes	characteristic
232	C₂	C₂×#17989028	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#494×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₂×#494	D₇₃
hypocenter	C₇₃	#33705
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #22612176	C₂	semidirect
3. #22612158	C₂	semidirect
123. C₇₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#22620445	2
#22612176	2
#22612158	2
#22611651	2
C₄×#14556531	2
C₂×#494×C₇₃	2
#33705	73