Finite Group Databasebeta

Subgroups — #33006606

Order 18688.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612243	C₂	1	1	yes	maximal, characteristic
2	#22611817	C₂	1	1	yes	maximal, characteristic, complemented
3	#22611806	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608136	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608122	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608066	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
9	#17988704	—	2	1	no	—
10	#17988669	—	2	1	no	—
11	#17988424	C₄	1	2	yes	complemented
12	#17988422	C₄	1	2	yes	complemented
13	#17988421	C₂²	1	1	yes	characteristic
14	#17988417	C₂²	1	1	yes	characteristic
15	#17988394	—	2	1	no	—
16	#17988387	—	2	1	no	—
17	#17988362	C₂²	1	1	yes	characteristic
18	#508×C₇₃	C₂²	1	1	yes	characteristic
19	M₆₄×C₇₃	C₂²	1	1	yes	characteristic
20	#367×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×D₅₈₄	—	2	1	no	—
22	#14556598	—	4	1	no	—
23	#14556553	—	4	1	no	—
24	D₁₁₆₈	—	2	1	no	—
25	D₁₁₆₈	C₂×C₄	1	2	yes	—
26	C₂×#11924003	D₄	1	1	yes	characteristic
27	#14556527	—	2	1	no	—
28	#14556518	—	4	1	no	—
29	#14556511	—	4	1	no	—
30	C₈×Dic₇₃	D₄	1	1	yes	characteristic
31	#14556497	C₂×C₄	1	2	yes	complemented
32	#135×C₇₃	—	2	1	no	—
33	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
34	SD₃₂×C₇₃	—	2	1	no	—
35	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
37	#107×C₇₃	—	2	1	no	—
38	C₃₂×C₇₃	—	2	1	no	—
39	C₂×D₂₉₂	—	4	1	no	—
40	C₂×C₄×D₇₃	—	8	1	no	—
41	D₅₈₄	—	2	2	no	—
42	D₅₈₄	—	4	1	no	—
43	#11924012	—	4	1	no	—
44	#11924012	—	8	1	no	—
45	C₂×#10596831	—	8	1	no	—
46	C₂×#10596831	D₈	1	1	yes	characteristic
47	#11924005	—	8	1	no	—
48	C₄×Dic₇₃	D₈	1	1	yes	characteristic
49	#11924003	—	2	1	no	—
50	#41×C₇₃	—	4	1	no	—
51	Q₁₆×C₇₃	—	2	1	no	—
52	SD₁₆×C₇₃	—	4	1	no	—
53	D₈×C₇₃	—	2	1	no	—
54	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₇₃	—	4	1	no	—
56	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₇₃	#41	1	1	yes	characteristic
58	C₂²×D₇₃	—	8	1	no	—
59	Q₈×C₇₃	—	4	1	no	—
60	D₄×C₇₃	—	4	1	no	—
61	D₄×C₇₃	—	8	1	no	—
62	C₂×C₄×C₇₃	—	8	1	no	—
63	C₂×C₄×C₇₃	—	8	1	no	—
64	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
65	C₂×Dic₇₃	—	2	1	no	—
66	D₂₉₂	—	4	2	no	—
67	D₂₉₂	—	8	1	no	—
68	C₄×D₇₃	—	16	2	no	—
69	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₈×C₇₃	#135	1	1	yes	characteristic
71	#10596831	—	2	1	no	—
72	#10596831	—	8	1	no	—
73	C₂²×C₇₃	—	8	1	no	—
74	C₂²×C₇₃	#509	1	1	yes	characteristic
75	C₂×D₇₃	—	8	2	no	—
76	C₂×D₇₃	—	16	1	no	—
77	C₂×D₇₃	—	16	1	no	—
78	C₄×C₇₃	—	8	1	no	—
79	C₄×C₇₃	—	16	1	no	—
80	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
81	C₄×C₇₃	#509	1	1	yes	characteristic
82	Dic₇₃	—	4	1	no	—
83	#13658	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	—	16	1	no	—
86	C₂×C₇₃	#2177	1	1	yes	characteristic
87	D₇₃	—	16	2	no	—
88	#2263	—	73	1	no	—
89	#2179	—	73	1	no	—
90	#2177	—	73	1	no	—
91	#1426	—	73	1	no	—
92	#1419	—	73	1	no	—
93	#1418	D₇₃	1	1	yes	characteristic, complemented
94	#1398	—	73	1	no	—
95	C₇₃	#13658	1	1	yes	characteristic, complemented, Sylow
96	#508	C₂×D₇₃	1	1	yes	characteristic
97	C₂×D₁₆	—	73	1	no	—
98	#443	—	146	1	no	—
99	C₄×D₈	—	146	1	no	—
100	SD₆₄	—	73	2	no	—
101	D₃₂	—	73	2	no	—
102	M₆₄	—	73	1	no	—
103	M₆₄	C₂×D₇₃	1	1	yes	characteristic
104	#368	—	73	1	no	—
105	#367	C₂×D₇₃	1	1	yes	characteristic
106	#359	—	146	1	no	—
107	#357	—	146	1	no	—
108	#346	—	73	1	no	—
109	#135	—	2	1	no	—
110	C₂×D₈	—	146	1	no	—
111	#131	—	292	1	no	—
112	C₄×D₄	—	292	1	no	—
113	Q₃₂	C₄×D₇₃	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	73	2	no	—
116	D₁₆	—	146	1	no	—
117	D₁₆	C₄×D₇₃	1	1	yes	characteristic, complemented
118	C₂×C₁₆	—	73	1	no	—
119	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
120	#108	—	146	1	no	—
121	#107	—	2	1	no	—
122	#104	—	292	1	no	—
123	#102	—	292	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₃₂	—	2	1	no	—
126	C₃₂	—	73	2	no	—
127	#41	—	4	1	no	—
128	C₂×D₄	—	292	1	no	—
129	C₂²×C₄	—	584	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	4	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	292	1	no	—
134	D₈	—	146	2	no	—
135	M₁₆	—	292	1	no	—
136	M₁₆	—	584	1	no	—
137	C₂×C₈	—	584	1	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₂²⋊C₄	—	584	1	no	—
142	C₄²	—	73	1	no	—
143	C₁₆	—	146	1	no	—
144	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
145	C₁₆	#11924027	1	1	yes	characteristic
146	C₂³	—	584	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	292	2	no	—
151	D₄	—	584	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	1168	2	no	—
155	C₂×C₄	—	146	1	no	—
156	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
157	C₈	—	584	1	no	—
158	C₈	—	146	1	no	—
159	C₈	#14556551	1	1	yes	characteristic
160	C₈	#14556612	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	584	2	no	—
163	C₂²	—	1168	1	no	—
164	C₂²	—	1168	1	no	—
165	C₂²	#17989021	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	16	1	no	—
168	C₄	—	292	1	no	—
169	C₄	#17988667	1	1	yes	characteristic
170	C₄	#17989024	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	1168	2	no	—
174	C₂	#22611804	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22611804
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1418	C₂
socle	C₂×C₇₃	#2177
prime radical	#22612243	C₂
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#1418	D₇₃
hypocenter	C₇₃	#13658
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
2. #22611817	C₂	semidirect
3. #22611806	C₂	semidirect
4. #22608136	C₂	semidirect
5. #22608122	C₂	semidirect
6. #22608066	C₂	semidirect
7. C₇₃×#1418	C₂	semidirect
11. #17988424	C₄	semidirect
12. #17988422	C₄	semidirect
31. #14556497	C₂×C₄	semidirect
33. Q₃₂×C₇₃	C₂×C₄	semidirect
35. D₁₆×C₇₃	C₂×C₄	semidirect
93. #1418	D₇₃	semidirect
95. C₇₃	#13658	semidirect
113. Q₃₂	C₄×D₇₃	semidirect
117. D₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612243	2
#22611817	2
#22611806	2
#22608136	2
#22608122	2
#22608066	2
C₇₃×#1418	2
#13658	73