Finite Group Databasebeta

← #33000764

Subgroups — #33000764

Order 18688.

284 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 83 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#22612187C211yesmaximal, characteristic
2#22612184C211yesmaximal, characteristic, complemented
3#22612159C211yesmaximal, characteristic, complemented
4#22612155C211yesmaximal, characteristic
5#22609537C211yesmaximal, characteristic
6#22609528C211yesmaximal, characteristic, complemented
7C73×#1312C211yesmaximal, characteristic, complemented
8C2×Dic584—22no—
9C2×#14556536—22no—
10#17988992C2211yescharacteristic
11#17988985—21no—
12#17988967C2211yescharacteristic
13#17988967C2211yescharacteristic
14#17988964C2211yescharacteristic
15#17988938—21no—
16#17988934—21no—
17#17988934—21no—
18#17988392—22no—
19#17988389—22no—
20C4×C16×C73C2211yescharacteristic
21C4×C16×C73C2211yescharacteristic
22C82×C73C2211yescharacteristic
23C2×Dic292—21no—
24C2×Dic292—21no—
25C2×Dic292—21no—
26C2×Dic292—21no—
27C2×D584—21no—
28C2×D584—21no—
29#14556591—21no—
30#14556591—21no—
31#14556591—21no—
32#14556588—21no—
33Dic584—44no—
34#14556536—44no—
35#14556525—21no—
36#14556525—21no—
37#14556523—42no—
38#14556516—42no—
39#14556513—42no—
40#14556513—42no—
41C2×C16×C73D412yes—
42C2×C16×C73D412yes—
43C4×C8×C73D411yescharacteristic
44C4×C8×C73D411yescharacteristic
45C4×C8×C73C2311yescharacteristic
46C2×Dic146—41no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×D292—41no—
50C2×D292—41no—
51C2×D292—41no—
52Dic292—42no—
53Dic292—42no—
54Dic292—42no—
55Dic292—42no—
56D584—42no—
57D584—42no—
58C4⋊Dic73—41no—
59C4⋊Dic73—41no—
60C4⋊Dic73—41no—
61C4⋊Dic73—41no—
62C4⋊Dic73—41no—
63C4⋊Dic73—41no—
64C2×C8×C73D812yes—
65C2×C8×C73D812yes—
66C2×C8×C73C2×D411yescharacteristic
67C2×C8×C73C2×D411yescharacteristic
68C42×C73C2×D411yescharacteristic
69C16×C73—22no—
70C16×C73—22no—
71C22×D73—81no—
72C2×C4×C73#12711yescharacteristic
73C2×C4×C73C2×D811yescharacteristic
74C2×C4×C73C2×D811yescharacteristic
75C2×Dic73—81no—
76C2×Dic73—81no—
77C2×Dic73—81no—
78D292—81no—
79D292—81no—
80D292—82no—
81D292—82no—
82Dic146—81no—
83Dic146—81no—
84Dic146—81no—
85Dic146—81no—
86Dic146—82no—
87C8×C73—22no—
88C8×C73—22no—
89C8×C73AGL(1,ℤ/8)12yes—
90C8×C73AGL(1,ℤ/8)12yes—
91C22×C73#49311yescharacteristic
92C2×D73—161no—
93C2×D73—161no—
94C2×D73—161no—
95C4×C73#49611yescharacteristic
96C4×C73#49611yescharacteristic
97C4×C73#50812yes—
98C4×C73#51012yes—
99Dic73—161no—
100Dic73—161no—
101Dic73—161no—
102#12530—731nomaximal, Sylow
103C2×C73#172111yescharacteristic
104C2×C73#224911yescharacteristic
105C2×C73#225111yescharacteristic
106D73—321no—
107#2248—731no—
108#2247—731no—
109#2241—731no—
110#2240—731no—
111#1709—731no—
112#1707—731no—
113#1312D7311yescharacteristic, complemented
114C73#1253011yescharacteristic, complemented, Sylow
115C2×Q32—1462no—
116C2×SD32—1462no—
117#500—731no—
118#498—1461no—
119#494—731no—
120#494—731no—
121#493—731no—
122#487—1461no—
123#486—1461no—
124#486—1461no—
125#358—1462no—
126#357—1462no—
127C4×C16C2×D7311yescharacteristic
128C4×C16C2×D7311yescharacteristic
129C82C2×D7311yescharacteristic
130C2×Q16—1461no—
131C2×Q16—1461no—
132C2×Q16—1461no—
133C2×Q16—1461no—
134C2×D8—1461no—
135C2×D8—1461no—
136C4⋊Q8—1461no—
137C4⋊Q8—1461no—
138C4⋊Q8—1461no—
139#127—1461no—
140Q32—2924no—
141SD32—2924no—
142C2×C16D29212yes—
143C2×C16D29212yes—
144#107—1461no—
145#107—1461no—
146#106—2922no—
147#103—2922no—
148#102—2922no—
149#102—2922no—
150C4×C8D29211yescharacteristic
151C4×C8D29211yescharacteristic
152C4×C8C22×D7311yescharacteristic
153C2×Q8—2921no—
154C2×Q8—2921no—
155C2×Q8—2921no—
156C2×D4—2921no—
157C2×D4—2921no—
158C2×D4—2921no—
159Q16—2922no—
160Q16—2922no—
161Q16—2922no—
162Q16—2922no—
163D8—2922no—
164D8—2922no—
165C2×C8D58412yes—
166C2×C8D58412yes—
167C2×C8C2×D29211yescharacteristic
168C2×C8C2×D29211yescharacteristic
169C4⋊C4—2921no—
170C4⋊C4—2921no—
171C4⋊C4—2921no—
172C4⋊C4—2921no—
173C4⋊C4—2921no—
174C4⋊C4—2921no—
175C42C2×D29211yescharacteristic
176C16—22no—
177C16—22no—
178C23—5841no—
179Q8—5841no—
180Q8—5841no—
181Q8—5841no—
182Q8—5841no—
183Q8—5842no—
184D4—5841no—
185D4—5841no—
186D4—5842no—
187D4—5842no—
188C2×C4—5841no—
189C2×C4—5841no—
190C2×C4—5841no—
191C2×C4#1455658811yescharacteristic
192C2×C4C2×D58411yescharacteristic
193C2×C4C2×D58411yescharacteristic
194C8—22no—
195C8—22no—
196C8#1455661912yes—
197C8#1455661912yes—
198C22—11681no—
199C22—11681no—
200C22—11681no—
201C22#1798896411yescharacteristic
202C4—11681no—
203C4—11681no—
204C4—11681no—
205C4#1798897611yescharacteristic
206C4#1798897611yescharacteristic
207C4#1798901912yes—
208C4#1798903012yes—
209C2—23361no—
210C2#2260959711yescharacteristic
211C2#2261219211yescharacteristic
212C2#2261220211yescharacteristic
subgroupG/H
centerC22#17988964
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1312C2
socleC22×C73#493
prime radical#17988964C22
hypercenter#1312D73
hypocenterC73#12530
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1312

Complements

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

NH ≅ G/Ntype
2. #22612184C2semidirect
3. #22612159C2semidirect
6. #22609528C2semidirect
7. C73×#1312C2semidirect
113. #1312D73semidirect
114. C73#12530semidirect

Maximal subgroups

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

subgroupindex
#226121872
#226121842
#226121592
#226121552
#226095372
#226095282
C73×#13122
#1253073