Finite Group Databasebeta

← #33006768

Subgroups — #33006768

Order 18688.

220 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#17988426C211yesmaximal, characteristic, complemented
2C2×#17988425C211yesmaximal, characteristic, complemented
3#22612179C211yesmaximal, characteristic
4#22611789C211yesmaximal, characteristic, complemented
5#22608126C211yesmaximal, characteristic, complemented
6#22608108C211yesmaximal, characteristic, complemented
7C73×#1415C211yesmaximal, characteristic, complemented
8C2×Dic584C2211yescharacteristic
9C2×D1168C2211yescharacteristic
10C2×#14556532—21no—
11#17988962—21no—
12#17988681—21no—
13#17988426—22no—
14#17988425—22no—
15C2×#14556497C2211yescharacteristic
16#17988390—21no—
17C16×Dic73C2211yescharacteristic
18C2×D16×C73C2211yescharacteristic
19C2×C32×C73C2211yescharacteristic
20#366×C73C2211yescharacteristic
21C2×Dic292—21no—
22C2×D584—21no—
23C2×#11924013—41no—
24#14556587—41no—
25#14556556—41no—
26Dic584—21no—
27Dic584D412yescomplemented
28D1168—21no—
29D1168—21no—
30#14556532—42no—
31C2×#11924003D411yescharacteristic
32#14556514—41no—
33C8×Dic73D411yescharacteristic
34#14556497—21no—
35C2×D8×C73—21no—
36D16×C73—21no—
37D16×C73—21no—
38C2×C16×C73C2311yescharacteristic
39#107×C73—21no—
40C32×C73—21no—
41C2×Dic146—41no—
42C2×D292—41no—
43C2×C22⋊D73—81no—
44C2×C22⋊D73—81no—
45Dic292—22no—
46Dic292—41no—
47D584—41no—
48D584—41no—
49#11924013—82no—
50C2×#10596831D811yescharacteristic
51#11924007—81no—
52C4×Dic73—81no—
53C4×Dic73D811yescharacteristic
54#11924003—21no—
55C2×D4×C73—41no—
56D8×C73—41no—
57D8×C73—41no—
58C2×C8×C73C2×D411yescharacteristic
59C4⋊C4×C73—41no—
60C16×C73C2×D411yescharacteristic
61C16×C73#4111yescharacteristic
62C23×C73—81no—
63C22×D73—81no—
64D4×C73—81no—
65D4×C73—81no—
66C2×C4×C73—81no—
67C2×C4×C73C2×D811yescharacteristic
68C22⋊D73—162no—
69C22⋊D73—162no—
70C2×Dic73—81no—
71C2×Dic73D1611yescharacteristic
72C2×Dic73D1611yescharacteristic
73D292—81no—
74D292—81no—
75Dic146—42no—
76Dic146—81no—
77C8×C73#12111yescharacteristic
78C8×C73AGL(1,ℤ/8)11yescharacteristic
79#10596831—21no—
80C22×C73—161no—
81C22×C73—161no—
82C22×C73—161no—
83C22×C73C2×D1611yescharacteristic
84C2×D73—161no—
85C2×D73—161no—
86C2×D73—161no—
87C4×C73—161no—
88C4×C73#45911yescharacteristic
89C4×C73#50911yescharacteristic
90Dic73—21no—
91Dic73—82no—
92Dic73SD6412yescomplemented
93#13686—731nomaximal, Sylow
94C2×C73—321no—
95C2×C73#220611yescharacteristic
96C2×C73C2×SD6411yescharacteristic
97C2×C73#226311yescharacteristic
98D73—321no—
99C2×SD64—731no—
100C2×SD64—731no—
101#2246—731no—
102C4×Q32—731no—
103C4⋊C32—731no—
104#1415—731no—
105#1415D7311yescharacteristic, complemented
106C73#1368611yescharacteristic, complemented, Sylow
107C2×Q32—731no—
108C2×D16—731no—
109C2×D16—1461no—
110C2×D16C2×D7311yescharacteristic
111#493—1461no—
112C4×Q16—1461no—
113SD64—1462no—
114SD64—1462no—
115C2×C32—731no—
116C2×C32C2×D7311yescharacteristic
117#366C2×D7311yescharacteristic
118#358—1461no—
119C4×C16—731no—
120C2×Q16—1461no—
121C2×D8—21no—
122C2×D8—2921no—
123C2×D8—1461no—
124#127—2921no—
125C4×Q8—2921no—
126Q32—732no—
127Q32—1461no—
128D16—21no—
129D16—21no—
130D16—2922no—
131D16—1461no—
132D16—1461no—
133C2×C16—731no—
134C2×C16C22×D7311yescharacteristic
135#107—21no—
136#103—2921no—
137C4×C8—731no—
138C32—21no—
139C32—1461no—
140C2×Q8—2921no—
141C2×D4—41no—
142C2×D4—2921no—
143C2×D4—5841no—
144C2×D4—5841no—
145Q16—2921no—
146Q16—1462no—
147D8—41no—
148D8—41no—
149D8—2921no—
150D8—2921no—
151D8—5842no—
152C2×C8—731no—
153C2×C8D4×D7311yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—5841no—
156C42—5841no—
157C42—731no—
158C16—1461no—
159C16D4×D7311yescharacteristic
160C16#1192402811yescharacteristic
161C23—81no—
162C23—5841no—
163Q8—2922no—
164Q8—5841no—
165D4—81no—
166D4—81no—
167D4—5841no—
168D4—5841no—
169D4—11682no—
170D4—11682no—
171C2×C4—81no—
172C2×C4—5841no—
173C2×C4—731no—
174C2×C4—731no—
175C2×C4D8×D7311yescharacteristic
176C8—1461no—
177C8#1455656411yescharacteristic
178C8#1455661811yescharacteristic
179C22—161no—
180C22—161no—
181C22—161no—
182C22—11681no—
183C22—11681no—
184C22—11681no—
185C22D16×D7311yescharacteristic
186C4—161no—
187C4—5842no—
188C4—732no—
189C4—1461no—
190C4#1798878111yescharacteristic
191C4#1798902411yescharacteristic
192C2—321no—
193C2—23361no—
194C2#2261196511yescharacteristic
195C2SD64×D7311yescharacteristic
196C2#2261224211yescharacteristic
subgroupG/H
centerC22D16×D73
commutator, cosocleC2×C16×C73C23
FrattiniC2×C16C22×D73
FittingC73×#1415C2
socleC22×C73C2×D16
prime radical#22612179C2
hypercenter#1415D73
hypocenterC73#13686
LCSG ▷ C2×C16×C73 ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1415

Complements

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

NH ≅ G/Ntype
1. C2×#17988426C2semidirect
2. C2×#17988425C2semidirect
4. #22611789C2semidirect
5. #22608126C2semidirect
6. #22608108C2semidirect
7. C73×#1415C2semidirect
27. Dic584D4semidirect
92. Dic73SD64semidirect
105. #1415D73semidirect
106. C73#13686semidirect

Maximal subgroups

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

subgroupindex
C2×#179884262
C2×#179884252
#226121792
#226117892
#226081262
#226081082
C73×#14152
#1368673