Finite Group Databasebeta

← #31906893

Subgroups — #31906893

Order 18176.

288 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 69 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#22596980C211yesmaximal, characteristic
2#22590149C212yesmaximal, complemented
3#22586292C212yesmaximal, complemented
4#22586278C211yesmaximal, characteristic
5C71×#1321C211yesmaximal, characteristic
6#41×Dic71C411yescharacteristic
7#17986057—22no—
8#17986049—22no—
9#17986027C2211yescharacteristic
10#17986025C411yescharacteristic
11#17985966C411yescharacteristic
12M16×Dic71C2211yescharacteristic
13C2×C8×Dic71C411yescharacteristic
14#17985819—22no—
15#17985781C2212yes—
16#17985776—22no—
17#408×C71C2211yescharacteristic
18#349×C71C2212yes—
19C2×#11765849C2×C411yescharacteristic
20C22×C71⋊C8C2×C411yescharacteristic
21Q8×Dic71—21no—
22D4×Dic71—22no—
23D4×Dic71—21no—
24#14556087—21no—
25#14556087C2×C411yescharacteristic
26C2×C4×Dic71—21no—
27C2×C4×Dic71C2×C411yescharacteristic
28#14556068—22no—
29Dic71⋊C8—22no—
30Dic71⋊C8—22no—
31C22⋊(C71⋊C8)—22no—
32#14556038D412yes—
33C8×Dic71—21no—
34C8×Dic71—21no—
35C8×Dic71D412yes—
36C2×#41×C71C2×C411yescharacteristic
37C2×M16×C71C2311yescharacteristic
38C22×C8×C71C2×C411yescharacteristic
39M32×C71—22no—
40C22⋊C8×C71—22no—
41C22×Dic71—22no—
42C22×Dic71—21no—
43#11765849—41no—
44C2×C71⋊C8—21no—
45C2×C71⋊C8—41no—
46C2×C71⋊C8C22⋊C412yes—
47C4⋊Dic71—22no—
48C4⋊Dic71—21no—
49C22⋊Dic71—22no—
50C22⋊Dic71M1612yes—
51C4×Dic71—22no—
52C4×Dic71—21no—
53C4×Dic71—21no—
54C4×Dic71C22⋊C411yescharacteristic
55C4×Dic71C22⋊C411yescharacteristic
56C71⋊C16—42no—
57#41×C71—41no—
58#41×C71—41no—
59C2×Q8×C71M1611yescharacteristic
60C2×D4×C71—21no—
61C2×D4×C71M1611yescharacteristic
62C22×C4×C71—21no—
63C22×C4×C71C22×C411yescharacteristic
64M16×C71—41no—
65C2×C8×C71—21no—
66C2×C8×C71—41no—
67C2×C8×C71C2×D412yes—
68C16×C71—42no—
69C23×C71—21no—
70C23×C71C2×M1611yescharacteristic
71Q8×C71—41no—
72D4×C71—41no—
73D4×C71—42no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—41no—
78C2×C4×C71—41no—
79C2×C4×C71C2×C22⋊C411yescharacteristic
80C2×C4×C71C2×M1611yescharacteristic
81C2×Dic71—21no—
82C2×Dic71—22no—
83C2×Dic71—21no—
84C2×Dic71—42no—
85C2×Dic71—44no—
86C8×C71—22no—
87C8×C71—41no—
88C71⋊C8—22no—
89C71⋊C8—41no—
90C22×C71—21no—
91C22×C71—41no—
92C22×C71—42no—
93C22×C71—41no—
94C22×C71#40711yescharacteristic
95C4×C71—21no—
96C4×C71—41no—
97C4×C71#41011yescharacteristic
98C4×C71#41311yescharacteristic
99Dic71—22no—
100Dic71—42no—
101Dic71—41no—
102#11524—711nomaximal, Sylow
103C2×C71—21no—
104C2×C71—41no—
105C2×C71—81no—
106C2×C71#146411yescharacteristic
107#2922—711no—
108#2165—712no—
109#1328—712no—
110#1321—711no—
111#1321D7111yescharacteristic
112C71#1152411yescharacteristic, complemented, Sylow
113C4×#41—711no—
114#433—1422no—
115#431—1422no—
116#424—711no—
117#424—711no—
118#408—711no—
119#408C2×D7111yescharacteristic
120C4×M16—711no—
121C2×C4×C8—711no—
122#364—1422no—
123#349—712no—
124#349C2×D7112yes—
125#347—1422no—
126C2×#41C4×D7111yescharacteristic
127C2×M16—711no—
128C2×M16C22×D7111yescharacteristic
129C22×C8—711no—
130C22×C8C4×D7111yescharacteristic
131C4×Q8—1421no—
132C4×D4—1422no—
133C4×D4—1421no—
134#117—711no—
135#117—1421no—
136C2×C42—711no—
137C2×C42—1421no—
138M32—22no—
139M32—1422no—
140C4⋊C8—1422no—
141C4⋊C8—1422no—
142C22⋊C8—22no—
143C22⋊C8—1422no—
144#97—712no—
145C4×C8—712no—
146C4×C8—1421no—
147C4×C8—1421no—
148#41—41no—
149#41—41no—
150C2×Q8#1176584811yescharacteristic
151C2×D4—21no—
152C2×D4#1176584811yescharacteristic
153C22×C4—21no—
154C22×C4—1422no—
155C22×C4—1421no—
156C22×C4C2×C4×D7111yescharacteristic
157M16—41no—
158M16—2841no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—2841no—
162C2×C8—712no—
163C2×C8—1421no—
164C2×C8D4×D7112yes—
165C4⋊C4—1422no—
166C4⋊C4—1421no—
167C22⋊C4—712no—
168C22⋊C4—1422no—
169C42—711no—
170C42—711no—
171C42—1422no—
172C42—1421no—
173C42—1421no—
174C16—42no—
175C16—2842no—
176C23—21no—
177C23M16×D7111yescharacteristic
178Q8—41no—
179D4—41no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—2842no—
187C2×C4—2844no—
188C2×C4—1421no—
189C2×C4—1422no—
190C2×C4—1421no—
191C2×C4C22⋊C4×D7111yescharacteristic
192C2×C4C2×#1176584811yescharacteristic
193C8—22no—
194C8—41no—
195C8—2841no—
196C8—1422no—
197C22—21no—
198C22—41no—
199C22—42no—
200C22—41no—
201C22#1798596111yescharacteristic
202C4—21no—
203C4—41no—
204C4—2841no—
205C4—2842no—
206C4—1422no—
207C4#1798597111yescharacteristic
208C4#1798598011yescharacteristic
209C2—21no—
210C2—41no—
211C2—81no—
212C2#2258667111yescharacteristic
subgroupG/H
centerC4#17985971
commutatorC22×C4×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#1321C2
socleC2×C71#1464
prime radicalC2×#41×C71C2×C4
cosocleC2×M16×C71C23
hypercenter#1321D71
hypocenterC71#11524
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #22590149C2semidirect
3. #22586292C2semidirect
112. C71#11524semidirect

Maximal subgroups

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

subgroupindex
#225969802
#225901492
#225862922
#225862782
C71×#13212
#1152471