Subgroups — #33239484

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618865	C₂	1	1	yes	maximal, characteristic
2	#22611837	C₂	1	8	yes	maximal, complemented
3	#22611689	C₂	1	2	yes	maximal
4	#22611687	C₂	1	1	yes	maximal, characteristic
5	#22611527	C₂	1	2	yes	maximal
6	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989144	C₄	1	1	yes	characteristic
8	C₂×#14556531	C₂²	1	2	yes	—
9	#17988717	C₄	1	4	yes	—
10	C₈×Dic₁₄₆	C₂²	1	4	yes	—
11	#17988613	C₄	1	2	yes	—
12	#17988611	C₄	1	1	yes	characteristic
13	#17988542	C₂²	1	2	yes	—
14	C₄⋊#11924003	C₂²	1	4	yes	—
15	#17988405	C₂²	1	8	yes	—
16	#17988362	C₂²	1	4	yes	—
17	C₁₆×Dic₇₃	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	#346×C₇₃	C₂²	1	4	yes	—
22	C₂×#11924012	C₂×C₄	1	2	yes	—
23	#14556591	C₈	1	2	yes	—
24	#14556583	C₈	1	2	yes	—
25	#14556570	C₈	1	4	yes	—
26	C₄×Dic₁₄₆	C₂×C₄	1	4	yes	—
27	#14556550	C₂×C₄	1	2	yes	—
28	#14556531	—	2	4	no	—
29	C₂×#11924003	C₂³	1	4	yes	—
30	C₄⋊#10596831	C₂×C₄	1	4	yes	—
31	#14556520	C₂×C₄	1	8	yes	—
32	#14556501	C₂×C₄	1	4	yes	—
33	C₈×Dic₇₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₇₃	—	2	2	no	—
38	C₂×C₁₆×C₇₃	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
40	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
41	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂³	1	2	yes	—
43	C₂×Dic₁₄₆	C₂×C₈	1	4	yes	—
44	#11924012	—	2	4	no	—
45	C₂×#10596831	C₂²×C₄	1	4	yes	—
46	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
48	#11924007	C₂×C₈	1	8	yes	—
49	C₂²⋊Dic₇₃	C₂×C₈	1	4	yes	—
50	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
51	#11924003	—	2	4	no	—
52	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	—	2	2	no	—
55	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₇₃	#41	1	2	yes	—
59	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
63	C₁₆×C₇₃	—	2	2	no	—
64	C₁₆×C₇₃	C₂×Q₈	1	4	yes	—
65	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	4	2	no	—
67	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#131	1	2	yes	—
72	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₇₃	C₂²×C₈	1	8	yes	—
74	Dic₁₄₆	—	4	4	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	C₄×Q₈	1	2	yes	—
77	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
79	#10596831	—	2	4	no	—
80	C₂²×C₇₃	—	2	2	no	—
81	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	2	2	no	—
85	C₄×C₇₃	C₈×Q₈	1	2	yes	—
86	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
87	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
88	Dic₇₃	—	2	8	no	—
89	#33475	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	2	no	—
91	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₇₃	#3407	1	2	yes	—
93	#2960	—	73	1	no	—
94	#2183	—	73	8	no	—
95	#2151	—	73	2	no	—
96	#2150	—	73	1	no	—
97	#2108	—	73	2	no	—
98	C₂×#346	D₇₃	1	1	yes	characteristic
99	C₇₃	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	—	73	1	no	—
101	C₂×M₃₂	—	73	2	no	—
102	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
104	#446	—	73	4	no	—
105	C₈×Q₈	—	73	4	no	—
106	#424	—	73	2	no	—
107	#423	—	73	1	no	—
108	#405	—	73	2	no	—
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₄⋊C₁₆	—	73	8	no	—
111	C₄⋊C₁₆	—	73	4	no	—
112	#346	—	73	4	no	—
113	#346	C₂×D₇₃	1	4	yes	—
114	C₄×C₁₆	—	73	4	no	—
115	C₂×M₁₆	—	73	2	no	—
116	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
118	C₄⋊Q₈	—	73	2	no	—
119	#125	—	73	2	no	—
120	C₂²⋊Q₈	—	73	4	no	—
121	C₄×Q₈	—	73	4	no	—
122	#117	—	73	2	no	—
123	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
124	M₃₂	—	146	4	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	73	4	no	—
127	C₂×C₁₆	Dic₁₄₆	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
129	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
130	C₄⋊C₈	—	73	8	no	—
131	C₄⋊C₈	—	73	4	no	—
132	#97	—	73	4	no	—
133	C₄×C₈	—	73	4	no	—
134	C₄×C₈	C₄×D₇₃	1	2	yes	—
135	C₄×C₈	C₂²×D₇₃	1	2	yes	—
136	C₂×Q₈	—	73	4	no	—
137	C₂²×C₄	C₈×D₇₃	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
139	M₁₆	—	146	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	73	4	no	—
142	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
145	C₂×C₈	#11924028	1	2	yes	—
146	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
147	C₄⋊C₄	—	73	8	no	—
148	C₄⋊C₄	—	73	4	no	—
149	C₄⋊C₄	—	73	4	no	—
150	C₂²⋊C₄	—	73	4	no	—
151	C₄²	—	73	4	no	—
152	C₄²	C₈×D₇₃	1	1	yes	characteristic
153	C₄²	C₈×D₇₃	1	1	yes	characteristic
154	C₄²	C₂×C₄×D₇₃	1	2	yes	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	146	4	no	—
157	C₁₆	C₂×Dic₁₄₆	1	4	yes	—
158	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
159	Q₈	—	292	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	73	8	no	—
162	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
163	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
165	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
166	C₂×C₄	#14556597	1	2	yes	—
167	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	146	4	no	—
170	C₈	C₄×Dic₁₄₆	1	2	yes	—
171	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
172	C₈	C₂×#11924028	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₈×Dic₁₄₆	1	2	yes	—
175	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	146	8	no	—
179	C₄	C₈×Dic₁₄₆	1	2	yes	—
180	C₄	C₂×C₄×Dic₁₄₆	1	1	yes	characteristic
181	C₄	C₂×#14556597	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₈×Dic₁₄₆	1	1	yes	characteristic
184	C₂	#22623033	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂×#346×C₇₃	C₂
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#33475
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #22611837	C₂	semidirect
38. C₂×C₁₆×C₇₃	Q₈	semidirect
99. C₇₃	#33475	semidirect
127. 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
#22618865	2
#22611837	2
#22611689	2
#22611687	2
#22611527	2
C₂×#346×C₇₃	2
#33475	73