Finite Group Databasebeta

Subgroups — #34014549

Order 18688.

182 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 50 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	#22626334	C₂	1	1	yes	maximal, characteristic
2	#22626328	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625096	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625087	C₂	1	1	yes	maximal, characteristic, complemented
5	#22625085	C₂	1	1	yes	maximal, characteristic
6	#22625040	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612250	C₂	1	1	yes	maximal, characteristic
8	#17989631	—	2	1	no	—
9	Q₁₆×#64408	—	2	1	no	—
10	#17989520	C₂²	1	1	yes	characteristic
11	#17989516	C₂²	1	1	yes	characteristic
12	#17989514	C₂²	1	1	yes	characteristic
13	#17989502	—	2	1	no	—
14	#17989500	—	2	1	no	—
15	#17989474	C₂²	1	1	yes	characteristic
16	#17989020	C₂²	1	1	yes	characteristic
17	Q₃₂×D₇₃	C₂²	1	1	yes	characteristic
18	#17988427	C₄	1	1	yes	characteristic, complemented
19	#17988426	C₄	1	1	yes	characteristic, complemented
20	#17988424	C₄	1	1	yes	characteristic, complemented
21	#17988420	C₂²	1	1	yes	characteristic
22	Q₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
23	#14556679	—	4	1	no	—
24	Q₈×#64408	—	4	1	no	—
25	#14556664	D₄	1	1	yes	characteristic
26	#14556662	—	2	1	no	—
27	#14556660	—	2	1	no	—
28	#14556656	—	4	1	no	—
29	#14556654	—	4	1	no	—
30	C₈×#64408	D₄	1	1	yes	characteristic
31	#14556641	—	2	1	no	—
32	#14556614	—	2	1	no	—
33	Q₁₆×D₇₃	—	2	1	no	—
34	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic
35	#14556537	—	2	1	no	—
36	#14556534	—	2	1	no	—
37	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic
38	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
39	#14556497	C₂×C₄	1	1	yes	characteristic
40	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
41	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924038	—	4	1	no	—
44	#11924038	—	8	1	no	—
45	#11924036	—	8	1	no	—
46	#11924036	D₈	1	1	yes	characteristic
47	C₄⋊#64408	—	4	1	no	—
48	C₄⋊#64408	—	8	1	no	—
49	C₄×#64408	—	8	1	no	—
50	C₄×#64408	D₈	1	1	yes	characteristic
51	#11924031	—	2	1	no	—
52	#11924029	—	4	1	no	—
53	Q₈×D₇₃	—	4	1	no	—
54	Dic₂₉₂	—	2	1	no	—
55	#11924018	—	4	1	no	—
56	#11924015	—	4	1	no	—
57	D₅₈₄	—	2	1	no	—
58	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
59	#11924003	#41	1	1	yes	characteristic
60	Q₁₆×C₇₃	—	2	1	no	—
61	Q₁₆×C₇₃	—	2	1	no	—
62	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×#64408	—	2	1	no	—
64	C₂×#64408	—	8	1	no	—
65	Q₈×C₇₃	—	4	1	no	—
66	Q₈×C₇₃	—	4	1	no	—
67	D₂₉₂	—	4	1	no	—
68	D₂₉₂	—	8	1	no	—
69	C₄×D₇₃	—	8	1	no	—
70	C₄×D₇₃	—	8	1	no	—
71	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
72	Dic₁₄₆	—	4	1	no	—
73	Dic₁₄₆	—	8	1	no	—
74	#10596833	—	2	1	no	—
75	#10596833	—	8	1	no	—
76	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
77	#10596831	#135	1	1	yes	characteristic
78	C₂×D₇₃	—	8	1	no	—
79	C₂×D₇₃	#509	1	1	yes	characteristic
80	#64408	—	4	1	no	—
81	#64408	—	16	1	no	—
82	C₄×C₇₃	—	8	1	no	—
83	C₄×C₇₃	—	8	1	no	—
84	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
85	Dic₇₃	—	8	1	no	—
86	Dic₇₃	#509	1	1	yes	characteristic
87	#13657	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	#2177	1	1	yes	characteristic
89	D₇₃	—	2	1	no	—
90	D₇₃	—	16	1	no	—
91	#2264	—	73	1	no	—
92	#2179	—	73	1	no	—
93	#2176	—	73	1	no	—
94	#1426	—	73	1	no	—
95	#1420	—	73	1	no	—
96	#1418	—	73	1	no	—
97	#1398	—	73	1	no	—
98	C₇₃	#13657	1	1	yes	characteristic, complemented, Sylow
99	#508	—	73	1	no	—
100	C₂×Q₃₂	—	73	1	no	—
101	#443	—	146	1	no	—
102	C₄×Q₁₆	—	146	1	no	—
103	Q₆₄	—	73	1	no	—
104	Q₆₄	#64408	1	1	yes	characteristic, complemented
105	SD₆₄	—	73	1	no	—
106	SD₆₄	—	73	1	no	—
107	M₆₄	—	73	1	no	—
108	M₆₄	—	73	1	no	—
109	#368	—	73	1	no	—
110	#367	—	73	1	no	—
111	#359	—	146	1	no	—
112	#358	—	146	1	no	—
113	#346	—	73	1	no	—
114	#135	—	146	1	no	—
115	C₂×Q₁₆	—	146	1	no	—
116	#131	—	292	1	no	—
117	C₄×Q₈	—	292	1	no	—
118	Q₃₂	—	73	1	no	—
119	Q₃₂	—	146	1	no	—
120	Q₃₂	C₂×#64408	1	1	yes	characteristic
121	Q₃₂	C₂×#64408	1	1	yes	characteristic
122	SD₃₂	—	146	1	no	—
123	D₁₆	—	73	1	no	—
124	C₂×C₁₆	—	73	1	no	—
125	C₂×C₁₆	—	73	1	no	—
126	#108	—	146	1	no	—
127	#107	—	146	1	no	—
128	#104	—	292	1	no	—
129	#103	—	292	1	no	—
130	C₄×C₈	—	73	1	no	—
131	C₃₂	—	73	1	no	—
132	C₃₂	—	146	1	no	—
133	C₃₂	C₂×#64408	1	1	yes	characteristic
134	#41	—	292	1	no	—
135	C₂×Q₈	—	292	1	no	—
136	Q₁₆	—	2	1	no	—
137	Q₁₆	—	2	1	no	—
138	Q₁₆	—	292	1	no	—
139	Q₁₆	—	146	1	no	—
140	SD₁₆	—	292	1	no	—
141	D₈	—	146	1	no	—
142	M₁₆	—	292	1	no	—
143	M₁₆	—	584	1	no	—
144	C₂×C₈	—	584	1	no	—
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₄⋊C₄	—	292	1	no	—
148	C₄⋊C₄	—	584	1	no	—
149	C₄²	—	584	1	no	—
150	C₄²	—	73	1	no	—
151	C₁₆	—	73	1	no	—
152	C₁₆	—	146	1	no	—
153	C₁₆	C₂²×#64408	1	1	yes	characteristic
154	Q₈	—	4	1	no	—
155	Q₈	—	4	1	no	—
156	Q₈	—	292	1	no	—
157	Q₈	—	584	1	no	—
158	D₄	—	292	1	no	—
159	D₄	—	584	1	no	—
160	C₂×C₄	—	584	1	no	—
161	C₂×C₄	—	584	1	no	—
162	C₂×C₄	—	584	1	no	—
163	C₂×C₄	—	73	1	no	—
164	C₂×C₄	—	146	1	no	—
165	C₈	—	584	1	no	—
166	C₈	—	73	1	no	—
167	C₈	—	146	1	no	—
168	C₈	D₄×#64408	1	1	yes	characteristic
169	C₂²	—	584	1	no	—
170	C₂²	—	73	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	8	1	no	—
173	C₄	—	292	1	no	—
174	C₄	—	584	1	no	—
175	C₄	—	73	1	no	—
176	C₄	—	1168	1	no	—
177	C₄	D₈×#64408	1	1	yes	characteristic
178	C₂	—	1168	1	no	—
179	C₂	—	146	1	no	—
180	C₂	#22626329	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22626329
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₁₆	C₂²×#64408
Fitting	Q₆₄×C₇₃	C₄
socle	C₂×C₇₃	#2177
prime radical	#17989020	C₂²
cosocle	C₁₆×D₇₃	C₂³
hypercenter	Q₆₄	#64408
hypocenter	C₇₃	#13657
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
2. #22626328	C₂	semidirect
3. #22625096	C₂	semidirect
4. #22625087	C₂	semidirect
6. #22625040	C₂	semidirect
18. #17988427	C₄	semidirect
19. #17988426	C₄	semidirect
20. #17988424	C₄	semidirect
22. Q₆₄×C₇₃	C₄	semidirect
98. C₇₃	#13657	semidirect
104. Q₆₄	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626334	2
#22626328	2
#22625096	2
#22625087	2
#22625085	2
#22625040	2
#22612250	2
#13657	73