Finite Group Databasebeta

Subgroups — #34014489

Order 18688.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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	#22626282	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989513	C₂	1	1	yes	maximal, characteristic
3	#22625093	C₂	1	2	yes	maximal, complemented
4	#22625092	C₂	1	2	yes	maximal, complemented
5	M₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989597	—	2	1	no	—
7	C₂×#14556660	—	2	1	no	—
8	#17989514	—	2	1	no	—
9	#17989513	—	2	1	no	—
10	#17989513	C₂²	1	1	yes	characteristic
11	#17989513	C₂²	1	2	yes	—
12	#17989513	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
14	#17988421	C₄	1	1	yes	characteristic, complemented
15	#17988420	C₄	1	2	yes	complemented
16	C₃₂×D₇₃	C₂²	1	2	yes	—
17	M₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
18	#14556670	—	4	1	no	—
19	C₂×C₄⋊#64408	—	4	1	no	—
20	#14556660	—	2	2	no	—
21	#14556660	—	2	1	no	—
22	#14556660	—	2	1	no	—
23	#14556660	—	4	1	no	—
24	#14556659	—	4	1	no	—
25	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
26	C₂×#11924003	Q₈	1	1	yes	characteristic
27	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×D₇₃	D₄	1	1	yes	characteristic
29	C₁₆×D₇₃	Q₈	1	1	yes	characteristic
30	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
31	#14556497	C₂×C₄	1	2	yes	complemented
32	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₃₂×C₇₃	C₂×C₄	1	2	yes	complemented
34	C₂²×#64408	—	8	1	no	—
35	C₄⋊#64408	—	4	1	no	—
36	C₄⋊#64408	—	4	1	no	—
37	C₄⋊#64408	—	4	2	no	—
38	C₄⋊#64408	—	8	1	no	—
39	C₂²⋊#64408	—	8	1	no	—
40	C₄×#64408	—	8	1	no	—
41	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
42	C₂×#10596831	Q₁₆	1	1	yes	characteristic
43	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₈×D₇₃	D₈	1	1	yes	characteristic
45	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
46	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
47	#11924003	C₄⋊C₄	1	1	yes	characteristic
48	#11924003	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₁₆×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₇₃	D₁₆	1	1	yes	characteristic
53	C₂×#64408	—	8	1	no	—
54	C₂×#64408	—	8	1	no	—
55	C₂×#64408	—	8	2	no	—
56	C₂×#64408	—	16	2	no	—
57	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
58	C₂×Dic₇₃	Q₃₂	1	1	yes	characteristic
59	C₄×D₇₃	#107	1	1	yes	characteristic
60	C₄×D₇₃	D₁₆	1	1	yes	characteristic
61	C₄×D₇₃	Q₃₂	1	1	yes	characteristic
62	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
63	C₈×C₇₃	#107	1	1	yes	characteristic
64	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	#10596831	#107	1	1	yes	characteristic
66	#10596831	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₇₃	#366	1	1	yes	characteristic
68	C₂×D₇₃	—	2	1	no	—
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	#366	1	1	yes	characteristic
71	C₂×D₇₃	C₂×D₁₆	1	1	yes	characteristic
72	#64408	—	16	1	no	—
73	#64408	—	16	2	no	—
74	#64408	—	16	1	no	—
75	C₄×C₇₃	#366	1	1	yes	characteristic
76	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
77	Dic₇₃	#366	1	1	yes	characteristic
78	Dic₇₃	C₂×Q₃₂	1	1	yes	characteristic
79	#13634	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	C₂×#366	1	1	yes	characteristic
82	D₇₃	—	2	1	no	—
83	D₇₃	#2263	1	1	yes	characteristic
84	D₇₃	#2264	1	1	yes	characteristic
85	C₂×M₆₄	—	73	1	no	—
86	#2158	—	73	1	no	—
87	C₂×#366	—	73	1	no	—
88	#1424	—	73	2	no	—
89	#1423	—	73	2	no	—
90	C₇₃	#13634	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	73	1	no	—
92	#427	—	146	1	no	—
93	C₂×#107	—	146	1	no	—
94	M₆₄	—	73	1	no	—
95	M₆₄	—	73	2	no	—
96	M₆₄	#64408	1	1	yes	characteristic, complemented
97	C₂×C₃₂	—	73	2	no	—
98	#367	—	146	1	no	—
99	#366	—	73	2	no	—
100	#366	—	73	1	no	—
101	#366	—	73	1	no	—
102	#366	—	146	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	#117	—	292	1	no	—
105	C₂×C₄⋊C₄	—	292	1	no	—
106	C₂×C₁₆	—	73	1	no	—
107	C₂×C₁₆	—	73	1	no	—
108	C₂×C₁₆	—	73	1	no	—
109	C₂×C₁₆	—	73	1	no	—
110	C₂×C₁₆	—	73	1	no	—
111	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
112	#107	—	292	1	no	—
113	#107	—	146	2	no	—
114	#107	—	146	1	no	—
115	#107	—	146	1	no	—
116	#106	—	292	1	no	—
117	C₃₂	—	73	2	no	—
118	C₃₂	C₂×#64408	1	2	yes	complemented
119	C₂²×C₄	—	584	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
127	C₄⋊C₄	—	292	1	no	—
128	C₄⋊C₄	—	292	1	no	—
129	C₄⋊C₄	—	292	2	no	—
130	C₄⋊C₄	—	584	1	no	—
131	C₂²⋊C₄	—	584	1	no	—
132	C₄²	—	584	1	no	—
133	C₁₆	—	73	1	no	—
134	C₁₆	—	73	1	no	—
135	C₁₆	C₄⋊#64408	1	1	yes	characteristic
136	C₁₆	C₂²×#64408	1	1	yes	characteristic
137	C₂³	—	73	1	no	—
138	C₂×C₄	—	584	1	no	—
139	C₂×C₄	—	584	1	no	—
140	C₂×C₄	—	584	2	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	1168	2	no	—
147	C₂×C₄	#14556660	1	1	yes	characteristic
148	C₈	—	73	1	no	—
149	C₈	—	73	1	no	—
150	C₈	#14556660	1	1	yes	characteristic
151	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
152	C₂²	—	73	1	no	—
153	C₂²	—	73	1	no	—
154	C₂²	—	146	1	no	—
155	C₂²	—	146	1	no	—
156	C₂²	#17989513	1	1	yes	characteristic
157	C₄	—	73	1	no	—
158	C₄	—	73	1	no	—
159	C₄	—	1168	2	no	—
160	C₄	—	1168	1	no	—
161	C₄	—	1168	1	no	—
162	C₄	#17989513	1	1	yes	characteristic
163	C₄	C₂×#14556660	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	73	1	no	—
166	C₂	—	73	1	no	—
167	C₂	—	146	1	no	—
168	C₂	C₂×#17989513	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17989513
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₁₆	C₂²×#64408
Fitting	M₆₄×C₇₃	C₄
socle	C₂×C₇₃	C₂×#366
prime radical	C₂²×D₇₃	D₁₆
cosocle	C₁₆×D₇₃	C₂³
hypercenter	M₆₄	#64408
hypocenter	C₇₃	#13634
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #22625093	C₂	semidirect
4. #22625092	C₂	semidirect
14. #17988421	C₄	semidirect
15. #17988420	C₄	semidirect
17. M₆₄×C₇₃	C₄	semidirect
31. #14556497	C₂×C₄	semidirect
33. C₃₂×C₇₃	C₂×C₄	semidirect
90. C₇₃	#13634	semidirect
96. M₆₄	#64408	semidirect
118. C₃₂	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626282	2
C₂×#17989513	2
#22625093	2
#22625092	2
M₆₄×D₇₃	2
#13634	73