Finite Group Databasebeta

← #22622612

Subgroups — #22622612

Order 9344.

336 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 123 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#17989315C211yesmaximal, characteristic, complemented
2#17988941C212yesmaximal
3#17988937C211yesmaximal, characteristic, complemented
4#17988894C212yesmaximal, complemented
5#17988867C211yesmaximal, characteristic
6#17988806C212yesmaximal
7#17988794C211yesmaximal, characteristic, complemented
8Q8×#10596831C211yesmaximal, characteristic, complemented
9C4×#11924018C212yesmaximal, complemented
10C4×#11924016C211yesmaximal, characteristic
11#556×C73C211yesmaximal, characteristic
12C2×#11924018C2212yes—
13C2×#11924016C2211yescharacteristic
14#14556591C2211yescharacteristic
15#14556589—21no—
16#14556583C2212yes—
17#14556581—22no—
18C4×Dic146C2211yescharacteristic
19C4×Dic146C2212yes—
20#14556556—21no—
21#14556524C2212yes—
22#14556522C2211yescharacteristic
23C4⋊#10596831C2211yescharacteristic
24C4⋊#10596831C2212yes—
25#14556515C2212yes—
26#14556515C2212yes—
27#14556515C2212yes—
28#14556514C2211yescharacteristic
29#14556512C2212yes—
30#14556512C2211yescharacteristic
31C4×#10596831C2211yescharacteristic
32C4×#10596831C2212yes—
33C4⋊Q8×C73C2211yescharacteristic
34#126×C73—22no—
35#124×C73C2212yes—
36C22⋊Q8×C73—21no—
37C4×Q8×C73C2212yes—
38C4×Q8×C73C2211yescharacteristic
39C4×D4×C73C2211yescharacteristic
40#117×C73—21no—
41C2×Dic146—21no—
42C2×Dic146C2311yescharacteristic
43#11924018—24no—
44#11924016—22no—
45C2×#10596831C2311yescharacteristic
46C2×#10596831C2312yes—
47C2×#10596831C2311yescharacteristic
48C4⋊Dic73—21no—
49C4⋊Dic73C2311yescharacteristic
50C4⋊Dic73C2312yes—
51#11924007—22no—
52#11924007—21no—
53#11924007—22no—
54#11924007—22no—
55C4×Dic73—22no—
56C4×Dic73—21no—
57C2×Q8×C73D411yescharacteristic
58C2×Q8×C73C2312yes—
59C2×D4×C73C2311yescharacteristic
60C22×C4×C73—21no—
61C4⋊C4×C73—21no—
62C4⋊C4×C73—22no—
63C4⋊C4×C73—21no—
64C4⋊C4×C73D411yescharacteristic
65C4⋊C4×C73D412yes—
66C4⋊C4×C73C2311yescharacteristic
67C22⋊C4×C73—22no—
68C22⋊C4×C73—21no—
69C22⋊C4×C73—22no—
70C42×C73—22no—
71C42×C73C2311yescharacteristic
72C42×C73C2312yes—
73C23×C73—21no—
74Q8×C73—22no—
75Q8×C73—22no—
76Q8×C73—22no—
77D4×C73—21no—
78D4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—22no—
81C2×C4×C73—42no—
82C2×C4×C73C2×D412yes—
83C2×C4×C73C2×D411yescharacteristic
84C2×C4×C73C2×D412yes—
85C2×C4×C73C2×D411yescharacteristic
86C2×C4×C73C2411yescharacteristic
87C2×Dic73—21no—
88C2×Dic73—21no—
89C2×Dic73—22no—
90Dic146—21no—
91Dic146—42no—
92Dic146#4112yes—
93#10596831—21no—
94#10596831—22no—
95#10596831—21no—
96C22×C73—41no—
97C22×C73—41no—
98C22×C73—41no—
99C22×C73C22×D411yescharacteristic
100C4×C73—22no—
101C4×C73—22no—
102C4×C73—21no—
103C4×C73—42no—
104C4×C73—41no—
105C4×C73#13512yes—
106C4×C73C2×#4111yescharacteristic
107C4×C732−1+411yescharacteristic
108Dic73—22no—
109Dic73—42no—
110Dic73—41no—
111C2×C73—81no—
112C2×C73#54811yescharacteristic
113C2×C73C2×#13511yescharacteristic
114C2×C73#57811yescharacteristic
115#3364—731nomaximal, Sylow
116C73#336411yescharacteristic, complemented, Sylow
117#557—731no—
118#556D7311yescharacteristic
119#488—732no—
120#487—731no—
121#479—732no—
122#474—731no—
123#464—732no—
124#462—731no—
125C8×Q8—731no—
126C4×Q16—732no—
127C4×SD16—731no—
128C2×Q16—732no—
129C2×SD16—731no—
130C4⋊Q8—731no—
131C4⋊Q8—1461no—
132C4⋊Q8C2×D7311yescharacteristic
133#126—22no—
134#125—732no—
135#125—1462no—
136#124C2×D7312yes—
137C22⋊Q8—21no—
138C4×Q8—731no—
139C4×Q8—732no—
140C4×Q8—1461no—
141C4×Q8C2×D7312yes—
142C4×Q8C2×D7311yescharacteristic
143C4×D4C2×D7311yescharacteristic
144#117—21no—
145#107—732no—
146#106—731no—
147C4⋊C8—731no—
148C4⋊C8—732no—
149#103—732no—
150#103—732no—
151#103—732no—
152#103—731no—
153#102—731no—
154#102—732no—
155C4×C8—731no—
156C4×C8—732no—
157C2×Q8—731no—
158C2×Q8—1461no—
159C2×Q8C22⋊D7311yescharacteristic
160C2×Q8C22×D7312yes—
161C2×D4C22×D7311yescharacteristic
162C22×C4—21no—
163Q16—1464no—
164SD16—1462no—
165C2×C8—731no—
166C2×C8—732no—
167C2×C8—731no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—731no—
172C4⋊C4—732no—
173C4⋊C4—1461no—
174C4⋊C4—1462no—
175C4⋊C4—1462no—
176C4⋊C4—1462no—
177C4⋊C4—1461no—
178C4⋊C4C22⋊D7311yescharacteristic
179C4⋊C4C22⋊D7312yes—
180C4⋊C4C22×D7311yescharacteristic
181C22⋊C4—22no—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C42—22no—
185C42—1462no—
186C42—1461no—
187C42C22×D7311yescharacteristic
188C42C22×D7312yes—
189C23—21no—
190Q8—22no—
191Q8—22no—
192Q8—22no—
193Q8—2922no—
194Q8—732no—
195Q8—1461no—
196D4—21no—
197D4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—1461no—
202C2×C4—1462no—
203C2×C4—1461no—
204C2×C4C2×C22⋊D7312yes—
205C2×C4C2×C22⋊D7311yescharacteristic
206C2×C4C2×C22⋊D7312yes—
207C2×C4C2×C22⋊D7311yescharacteristic
208C2×C4C23×D7311yescharacteristic
209C8—1461no—
210C8—1462no—
211C8—1461no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22C22×C22⋊D7311yescharacteristic
216C4—22no—
217C4—21no—
218C4—22no—
219C4—42no—
220C4—41no—
221C4—2921no—
222C4—2922no—
223C4—1462no—
224C4#1455661112yes—
225C4#41×D7311yescharacteristic
226C4#1455663811yescharacteristic
227C2—81no—
228C2#1798924411yescharacteristic
229C2C2×#1455661111yescharacteristic
230C2#1798940211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#556×C73C2
socleC22×C73C22×D4
hypercenter#556D73
hypocenterC73#3364
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #17989315C2semidirect
3. #17988937C2semidirect
4. #17988894C2semidirect
7. #17988794C2semidirect
8. Q8×#10596831C2semidirect
9. C4×#11924018C2semidirect
116. C73#3364semidirect