Finite Group Databasebeta

← #33000136

Subgroups — #33000136

Order 18688.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 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#22612084C211yesmaximal, characteristic
2#22611982C211yesmaximal, characteristic
3#22609528C211yesmaximal, characteristic
4C8×Dic292C211yesmaximal, characteristic
5#22607988C211yesmaximal, characteristic
6#22607981C211yesmaximal, characteristic
7C73×#1367C211yesmaximal, characteristic
8C2×Dic584—21no—
9#17988992C2211yescharacteristic
10#17988985—21no—
11#17988967C2211yescharacteristic
12#17988938—21no—
13C8×Dic146—21no—
14C4×Dic292C2211yescharacteristic
15#17988413—21no—
16#17988392—21no—
17#17988392—21no—
18#17988340C2211yescharacteristic
19#17988319—21no—
20C4⋊C16×C73C2211yescharacteristic
21C4⋊C16×C73C2211yescharacteristic
22C82×C73C2211yescharacteristic
23C2×Dic292—21no—
24C2×Dic292—21no—
25C2×Dic292D411yescharacteristic
26#14556591—21no—
27#14556591—21no—
28C4×Dic146—21no—
29Dic584—42no—
30#14556525—21no—
31#14556525—21no—
32#14556525D411yescharacteristic
33#14556523—41no—
34#14556523—41no—
35C4⋊#10596831—21no—
36#14556520—41no—
37#14556516—21no—
38#14556516—42no—
39C8×Dic73—41no—
40C2×C16×C73—21no—
41C2×C16×C73—21no—
42C4×C8×C73D411yescharacteristic
43C4×C8×C73D411yescharacteristic
44C4×C8×C73C2311yescharacteristic
45C2×Dic146—21no—
46C2×Dic146—41no—
47C2×Dic146—41no—
48Dic292—21no—
49Dic292—41no—
50Dic292—41no—
51Dic292—81no—
52Dic292SD1612yes—
53C2×#10596831—41no—
54C4⋊Dic73—21no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58C4⋊Dic73—41no—
59#11924007—41no—
60C4×Dic73—41no—
61C2×C8×C73—21no—
62C2×C8×C73D811yescharacteristic
63C2×C8×C73D811yescharacteristic
64C2×C8×C73C2×D411yescharacteristic
65C2×C8×C73#4111yescharacteristic
66C42×C73C2×D411yescharacteristic
67C16×C73—41no—
68C16×C73—41no—
69C2×C4×C73#12111yescharacteristic
70C2×C4×C73C2×D811yescharacteristic
71C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
72C2×Dic73—41no—
73C2×Dic73—41no—
74C2×Dic73—81no—
75C2×Dic73—81no—
76Dic146—22no—
77Dic146—41no—
78Dic146—82no—
79Dic146—81no—
80Dic146—81no—
81C8×C73—21no—
82C8×C73—21no—
83C8×C73—41no—
84C8×C73Q3212yescomplemented
85C8×C73C2×SD1611yescharacteristic
86C8×C73#13511yescharacteristic
87#10596831—81no—
88C22×C73#45911yescharacteristic
89C4×C73—21no—
90C4×C73#46511yescharacteristic
91C4×C73#47111yescharacteristic
92C4×C73C2×Q3211yescharacteristic
93C4×C73#51011yescharacteristic
94Dic73—42no—
95Dic73—81no—
96Dic73—161no—
97Dic73—161no—
98#12421—731nomaximal, Sylow
99C2×C73#166511yescharacteristic
100C2×C73#220911yescharacteristic
101C2×C73#221211yescharacteristic
102#2227—731no—
103#2209—731no—
104#1707—731no—
105C8×Q16—731no—
106#1367D7311yescharacteristic
107#1365—731no—
108#1362—731no—
109C73#1242111yescharacteristic, complemented, Sylow
110C2×Q32—1461no—
111#500—731no—
112#498—1461no—
113#494—731no—
114#487—1461no—
115C8×Q8—1461no—
116C4×Q16—731no—
117#366—1461no—
118C4⋊C16C2×D7311yescharacteristic
119C4⋊C16C2×D7311yescharacteristic
120#358—1461no—
121#358—1461no—
122#335—731no—
123Q8⋊C8—1461no—
124C82C2×D7311yescharacteristic
125C2×Q16—731no—
126C2×Q16—1461no—
127C2×Q16—1461no—
128C4⋊Q8—1461no—
129C4⋊Q8—1461no—
130C4×Q8—1461no—
131Q32—2922no—
132C2×C16—21no—
133C2×C16—21no—
134#107—731no—
135#107—1461no—
136#107—1461no—
137#106—2921no—
138#106—2921no—
139C4⋊C8—2921no—
140C4⋊C8—1461no—
141#103—2922no—
142#103—1461no—
143C4×C8—2921no—
144C4×C8D29211yescharacteristic
145C4×C8D29211yescharacteristic
146C4×C8C22×D7311yescharacteristic
147C2×Q8—2921no—
148C2×Q8—2921no—
149C2×Q8—1461no—
150Q16—2921no—
151Q16—2921no—
152Q16—5841no—
153Q16—732no—
154Q16—1461no—
155C2×C8—21no—
156C2×C8—2921no—
157C2×C8D58411yescharacteristic
158C2×C8D58411yescharacteristic
159C2×C8D4×D7311yescharacteristic
160C2×C8#1192402911yescharacteristic
161C4⋊C4—2921no—
162C4⋊C4—2921no—
163C4⋊C4—2921no—
164C4⋊C4—2921no—
165C4⋊C4—2921no—
166C4⋊C4—1461no—
167C42—2921no—
168C42C2×D29211yescharacteristic
169C16—41no—
170C16—41no—
171Q8—2921no—
172Q8—5842no—
173Q8—5841no—
174Q8—5841no—
175Q8—1462no—
176C2×C4—2921no—
177C2×C4—2921no—
178C2×C4—5841no—
179C2×C4—5841no—
180C2×C4#1455656611yescharacteristic
181C2×C4C2×D58411yescharacteristic
182C2×C4#1455661911yescharacteristic
183C8—21no—
184C8—21no—
185C8—41no—
186C8—5841no—
187C8Dic58412yescomplemented
188C8SD16×D7311yescharacteristic
189C8#1455661211yescharacteristic
190C22#1798878311yescharacteristic
191C4—21no—
192C4—2922no—
193C4—5841no—
194C4—11681no—
195C4—11681no—
196C4#1798881711yescharacteristic
197C4#1798885111yescharacteristic
198C4C2×Dic58411yescharacteristic
199C4#1798903011yescharacteristic
200C2#2260927311yescharacteristic
201C2#2261198211yescharacteristic
202C2#2261200111yescharacteristic
subgroupG/H
centerC22#17988783
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1367C2
socle, prime radicalC22×C73#459
hypercenter#1367D73
hypocenterC73#12421
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1367

Complements

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

NH ≅ G/Ntype
84. C8×C73Q32semidirect
109. C73#12421semidirect
187. C8Dic584semidirect

Maximal subgroups

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

subgroupindex
#226120842
#226119822
#226095282
C8×Dic2922
#226079882
#226079812
C73×#13672
#1242173