Finite Group Databasebeta

← #34010016

Subgroups — #34010016

Order 18688.

364 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626289C212yesmaximal, complemented
2#22624854C212yesmaximal, complemented
3#22624853C212yesmaximal, complemented
4#22618641C211yesmaximal, characteristic
5#17989508—22no—
6C22⋊#11924031C2214yes—
7#17989473—22no—
8C2×#14556597C411yescharacteristic
9#17989133C411yescharacteristic
10#17988663—22no—
11C8×C22⋊D73—22no—
12#17988610C411yescharacteristic
13#17988555C2211yescharacteristic
14#17988554C2211yescharacteristic
15C2×C8×Dic73C2211yescharacteristic
16C2×#131×C73C411yescharacteristic
17C2×#11924031—24no—
18C2×#11924028C2×C411yescharacteristic
19#14556597—42no—
20C2×#11924012C2×C411yescharacteristic
21C2×#11924011—21no—
22C22×#10596831C2311yescharacteristic
23C2×C8×D73—21no—
24#14556589C811yescharacteristic
25#14556587C811yescharacteristic
26#14556580C811yescharacteristic
27#14556578C811yescharacteristic
28#14556563—22no—
29C4×C22⋊D73—22no—
30C2×C4×Dic73C2×C411yescharacteristic
31#14556520—22no—
32C22⋊#10596831—22no—
33#14556503—22no—
34C8×Dic73D412yes—
35C8×Dic73D412yes—
36C2×#41×C73C2×C411yescharacteristic
37#131×C73—42no—
38C2×M16×C73—21no—
39C2×M16×C73C2×C411yescharacteristic
40C22×C8×C73—21no—
41C22×C8×C73C2×C411yescharacteristic
42#11924031—44no—
43#11924028—41no—
44#11924028—41no—
45C2×Dic146C2×C811yescharacteristic
46C2×D292C2×C811yescharacteristic
47C2×C22⋊D73—21no—
48C2×C22⋊D73—22no—
49C22×Dic73M1612yes—
50C2×C4×D73—21no—
51#11924012—41no—
52#11924011—42no—
53C2×#10596831—21no—
54C2×#10596831—41no—
55C2×#10596831C2×D412yes—
56C8×D73—42no—
57#11924007—22no—
58C22⋊Dic73—22no—
59#11924005—22no—
60C4×Dic73C22⋊C412yes—
61C4×Dic73C2×C811yescharacteristic
62C4×Dic73C2×C811yescharacteristic
63#41×C73—41no—
64#41×C73—41no—
65C2×Q8×C73C2×C811yescharacteristic
66C2×D4×C73—21no—
67C2×D4×C73C2×C811yescharacteristic
68C22×C4×C73—21no—
69C22×C4×C73C22×C411yescharacteristic
70M16×C73—42no—
71M16×C73—41no—
72C2×C8×C73—22no—
73C2×C8×C73—21no—
74C2×C8×C73—42no—
75C2×C8×C73C22⋊C411yescharacteristic
76C2×C8×C73C22⋊C411yescharacteristic
77C23×C73—21no—
78C23×C73C2×M1611yescharacteristic
79C22×D73—21no—
80Q8×C73—41no—
81D4×C73—41no—
82D4×C73—42no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73C2×C22⋊C411yescharacteristic
89C2×C4×C73C22×C811yescharacteristic
90C22⋊D73—42no—
91C22⋊D73—44no—
92C2×Dic73—24no—
93C2×Dic73—21no—
94C2×Dic73C22⋊C812yes—
95C2×Dic73C22⋊C812yes—
96D292—41no—
97C4×D73—41no—
98C4×D73—41no—
99Dic146—41no—
100C8×C73—21no—
101C8×C73—41no—
102C8×C73#10012yes—
103#10596831—22no—
104#10596831—41no—
105C22×C73—22no—
106C22×C73—21no—
107C22×C73—42no—
108C22×C73—41no—
109C22×C73C2×C22⋊C811yescharacteristic
110C2×D73—42no—
111C2×D73—41no—
112C4×C73—21no—
113C4×C73—41no—
114C4×C73#41011yescharacteristic
115C4×C73C2×#10011yescharacteristic
116Dic73—22no—
117Dic73—41no—
118Dic73#35014yes—
119#11490—731nomaximal, Sylow
120C2×C73—41no—
121C2×C73—81no—
122C2×C73#146011yescharacteristic
123C2×C73C2×#35012yes—
124D73—81no—
125#2931—731no—
126#2162—732no—
127#1315—732no—
128#1315—732no—
129C73#1149011yescharacteristic, complemented, Sylow
130C2×#131—731no—
131C2×#131#6440811yescharacteristic
132#532—731no—
133#436—1462no—
134C8×D4—1462no—
135#423—731no—
136#408—731no—
137#408—731no—
138C2×C4×C8—731no—
139C4⋊C16—1462no—
140C22⋊C16—734no—
141C4×C16—1462no—
142C2×#41—731no—
143C2×#41C2×#6440811yescharacteristic
144#131—42no—
145#131—2922no—
146C2×M16—21no—
147C2×M16—731no—
148C2×M16—1461no—
149C2×M16C2×#6440811yescharacteristic
150C22×C8—21no—
151C22×C8—731no—
152C22×C8—1461no—
153C22×C8C2×#6440811yescharacteristic
154C4⋊Q8—731no—
155#127—731no—
156#124—731no—
157#124—731no—
158#121—1462no—
159C4×D4—1462no—
160C2×C42—731no—
161C2×C16—1464no—
162C4⋊C8—1462no—
163C22⋊C8—1462no—
164C22⋊C8—1462no—
165C4×C8—732no—
166C4×C8—732no—
167#41—41no—
168#41—41no—
169#41—2921no—
170#41—2921no—
171C2×Q8—731no—
172C2×Q8#1192403611yescharacteristic
173C2×D4—21no—
174C2×D4—731no—
175C2×D4—1462no—
176C2×D4—1461no—
177C2×D4#1192403611yescharacteristic
178C22×C4—21no—
179C22×C4—732no—
180C22×C4—1461no—
181C22×C4C22×#6440811yescharacteristic
182M16—42no—
183M16—41no—
184M16—2922no—
185M16—2921no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—42no—
189C2×C8—2922no—
190C2×C8—2921no—
191C2×C8—732no—
192C2×C8—1461no—
193C2×C8C22⋊#6440811yescharacteristic
194C2×C8C22⋊#6440811yescharacteristic
195C4⋊C4—1462no—
196C22⋊C4—1462no—
197C22⋊C4—1462no—
198C42—731no—
199C42—731no—
200C42—732no—
201C16—2924no—
202C23—21no—
203C23—1461no—
204C23D73⋊M1611yescharacteristic
205Q8—41no—
206Q8—2921no—
207D4—41no—
208D4—42no—
209D4—2921no—
210D4—2922no—
211D4—2924no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—2921no—
218C2×C4—2921no—
219C2×C4—732no—
220C2×C4—732no—
221C2×C4—1461no—
222C2×C4—1464no—
223C2×C4C2×C22⋊#6440811yescharacteristic
224C2×C4C2×#1192403611yescharacteristic
225C8—21no—
226C8—41no—
227C8—2921no—
228C8—1462no—
229C8#1455665112yes—
230C22—22no—
231C22—21no—
232C22—42no—
233C22—41no—
234C22—2922no—
235C22—2921no—
236C22#1798955511yescharacteristic
237C4—21no—
238C4—41no—
239C4—2921no—
240C4—734no—
241C4—1462no—
242C4#1798956611yescharacteristic
243C4C2×#1455665111yescharacteristic
244C2—41no—
245C2—81no—
246C2—5841no—
247C2#2262517011yescharacteristic
248C2#2262617112yes—
subgroupG/H
centerC2×C8C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#131×C73C4
socleC22×C73C2×C22⋊C8
prime radical#17989133C4
cosocleC22×#10596831C23
hypercenterC2×#131#64408
hypocenterC73#11490
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. #22626289C2semidirect
2. #22624854C2semidirect
3. #22624853C2semidirect
129. C73#11490semidirect

Maximal subgroups

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

subgroupindex
#226262892
#226248542
#226248532
#226186412
#1149073