Finite Group Databasebeta

← #31906666

Subgroups — #31906666

Order 18176.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#22596983C211yesmaximal, characteristic
2#22590140C212yesmaximal, complemented
3#22586282C211yesmaximal, characteristic, complemented
4#22586265C212yesmaximal
5C71×#1323C211yesmaximal, characteristic, complemented
6#17986467C411yescharacteristic
7#17986059—22no—
8#17986051—22no—
9#17986026C411yescharacteristic
10#17985967C411yescharacteristic
11#17985965C2211yescharacteristic
12M16×Dic71C411yescharacteristic
13C2×C8×Dic71C2211yescharacteristic
14Dic71⋊C16—22no—
15C22⋊(C71⋊C16)C2212yes—
16#17985774—22no—
17#424×C71C2211yescharacteristic
18C22⋊C16×C71C2212yes—
19C2×#11765865C2×C411yescharacteristic
20C2×#11765849C2×C411yescharacteristic
21C22×C71⋊C8C2×C411yescharacteristic
22#14556093C812yes—
23#14556090C812yes—
24(C22⋊D71)⋊C4—22no—
25#14556085—21no—
26C2×C4×Dic71C2×C411yescharacteristic
27C42×D71—21no—
28C2×C71⋊C16—22no—
29C4⋊(C71⋊C8)—22no—
30(C2×D71)⋊C8—22no—
31(C2×D71)⋊C8—22no—
32#14556038—21no—
33C8×Dic71—21no—
34C8×Dic71D412yes—
35C8×Dic71D412yes—
36C2×M16×C71C2×C411yescharacteristic
37C22×C8×C71C2311yescharacteristic
38#117×C71C2×C411yescharacteristic
39C2×C16×C71—22no—
40C4⋊C8×C71—22no—
41#11765865—41no—
42#11765865—41no—
43C2×Dic142C2×C811yescharacteristic
44C2×D284C2×C811yescharacteristic
45C2×C22⋊D71—21no—
46C22×Dic71M1612yes—
47C2×C4×D71—21no—
48C2×C4×D71—22no—
49#11765849—41no—
50C2×C71⋊C8—21no—
51C2×C71⋊C8—41no—
52C2×C71⋊C8C22⋊C412yes—
53Dic71⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71—22no—
56C4×Dic71C22⋊C411yescharacteristic
57C4×Dic71C22⋊C411yescharacteristic
58C4×Dic71C2×C812yes—
59C71⋊C16—42no—
60C22×C4×C71C22×C411yescharacteristic
61M16×C71—41no—
62C2×C8×C71—21no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D412yes—
65C4⋊C4×C71C2×C812yes—
66C22⋊C4×C71—21no—
67C42×C71—21no—
68C16×C71—42no—
69C23×C71C2×M1611yescharacteristic
70C22×D71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71C2×C22⋊C411yescharacteristic
75C2×C4×C71C22×C811yescharacteristic
76C22⋊D71—42no—
77C2×Dic71—21no—
78C2×Dic71—42no—
79C2×Dic71C22⋊C812yes—
80C2×Dic71C22⋊C812yes—
81D284—41no—
82C4×D71—44no—
83C4×D71—41no—
84C4×D71—41no—
85Dic142—41no—
86C8×C71—22no—
87C8×C71—41no—
88C71⋊C8—22no—
89C71⋊C8—41no—
90C22×C71—22no—
91C22×C71—21no—
92C22×C71C2×C22⋊C811yescharacteristic
93C2×D71—42no—
94C2×D71—41no—
95C4×C71—21no—
96C4×C71—42no—
97C4×C71#41011yescharacteristic
98C4×C71#41311yescharacteristic
99Dic71—22no—
100Dic71—22no—
101Dic71—41no—
102#11463—711nomaximal, Sylow
103C2×C71—41no—
104C2×C71#145811yescharacteristic
105C2×C71#211712yes—
106D71—81no—
107#2922—711no—
108#2163—712no—
109#1323—711no—
110#1323D7111yescharacteristic, complemented
111#1315—712no—
112C71#1146311yescharacteristic, complemented, Sylow
113C4×#41—711no—
114#433—1422no—
115#431—1422no—
116#424—711no—
117#424C2×D7111yescharacteristic
118#408—711no—
119#408—711no—
120C4×M16—711no—
121C2×C4×C8—711no—
122C4⋊C16—1422no—
123C22⋊C16—712no—
124C22⋊C16C2×D7112yes—
125#346—1422no—
126C2×#41—711no—
127C2×M16—711no—
128C2×M16C4×D7111yescharacteristic
129C22×C8—711no—
130C22×C8C22×D7111yescharacteristic
131C4×Q8—712no—
132C4×D4—712no—
133C4×D4—1422no—
134#117—1421no—
135#117C4×D7111yescharacteristic
136C2×C42—711no—
137C2×C42—1421no—
138C2×C16—22no—
139C2×C16—1422no—
140C4⋊C8—22no—
141C4⋊C8—1422no—
142C22⋊C8—1422no—
143C22⋊C8—1422no—
144#97—1421no—
145C4×C8—712no—
146C4×C8—712no—
147C4×C8—1421no—
148#41—2841no—
149#41—2841no—
150C2×Q8—711no—
151C2×D4—711no—
152C2×D4—1421no—
153C22×C4—712no—
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⋊C4C8×D7112yes—
167C22⋊C4—21no—
168C22⋊C4—1422no—
169C42—21no—
170C42—711no—
171C42—711no—
172C42—712no—
173C42—1422no—
174C16—42no—
175C16—2842no—
176C23—1421no—
177C23M16×D7111yescharacteristic
178Q8—2841no—
179D4—2841no—
180D4—2842no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—2844no—
185C2×C4—2841no—
186C2×C4—2841no—
187C2×C4—2842no—
188C2×C4—712no—
189C2×C4—712no—
190C2×C4—1421no—
191C2×C4C22⋊C4×D7111yescharacteristic
192C2×C4C2×C8×D7111yescharacteristic
193C8—22no—
194C8—41no—
195C8—2841no—
196C8—1422no—
197C22—22no—
198C22—21no—
199C22—2842no—
200C22—2841no—
201C22C22⋊C8×D7111yescharacteristic
202C4—21no—
203C4—42no—
204C4—2841no—
205C4—1422no—
206C4—1422no—
207C4#1798597111yescharacteristic
208C4#1798598211yescharacteristic
209C2—41no—
210C2—5681no—
211C2#2258665311yescharacteristic
212C2#2258996612yes—
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC22×C8C22×D71
FittingC71×#1323C2
socleC22×C71C2×C22⋊C8
prime radicalC2×#11765865C2×C4
cosocleC22×C8×C71C23
hypercenter#1323D71
hypocenterC71#11463
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #22590140C2semidirect
3. #22586282C2semidirect
5. C71×#1323C2semidirect
110. #1323D71semidirect
112. C71#11463semidirect

Maximal subgroups

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

subgroupindex
#225969832
#225901402
#225862822
#225862652
C71×#13232
#1146371