Finite Group Databasebeta

← #31917812

Subgroups — #31917812

Order 18176.

318 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590025C212yesmaximal
2#22586513C214yesmaximal, complemented
3C71×#3135C211yesmaximal, characteristic
4#17985983—22no—
5#17985798C2214yes—
6#17985796—22no—
7#533×C71C411yescharacteristic
8#532×C71C2211yescharacteristic
9#491×C71C2212yes—
10#490×C71C412yes—
11#489×C71—22no—
12#416×C71—22no—
13C2×#97×C71C411yescharacteristic
14C2×#11765849—42no—
15#14556047—24no—
16#14556045—42no—
17C2×#41×C71D412yes—
18C22×C8×C71—21no—
19C4⋊Q8×C71—21no—
20C4⋊Q8×C71C2311yescharacteristic
21#127×C71C2×C411yescharacteristic
22#125×C71C2×C412yes—
23#124×C71D412yes—
24C22⋊Q8×C71—41no—
25#121×C71—41no—
26C4×Q8×C71—41no—
27C4×D4×C71—41no—
28#117×C71—21no—
29C2×C42×C71C2×C411yescharacteristic
30#103×C71—24no—
31#102×C71—24no—
32#97×C71—21no—
33#97×C71C2×C412yes—
34#11765849—44no—
35#11765849—82no—
36C2×C71⋊C8—82no—
37#41×C71—42no—
38#41×C71—42no—
39C2×Q8×C71—41no—
40C2×Q8×C71C2×D412yes—
41C2×D4×C71—22no—
42C2×D4×C71—41no—
43C2×D4×C71C22⋊C412yes—
44C22×C4×C71—21no—
45C22×C4×C71—22no—
46C22×C4×C71C22⋊C411yescharacteristic
47C2×C8×C71—22no—
48C2×C8×C71—41no—
49C2×C8×C71—41no—
50C4⋊C4×C71—22no—
51C4⋊C4×C71—41no—
52C4⋊C4×C71—42no—
53C4⋊C4×C71—41no—
54C22⋊C4×C71—42no—
55C22⋊C4×C71—41no—
56C42×C71—21no—
57C42×C71—41no—
58C42×C71C22⋊C411yescharacteristic
59C42×C71C22×C411yescharacteristic
60C23×C71—22no—
61C23×C71#9911yescharacteristic
62Q8×C71—42no—
63Q8×C71—82no—
64D4×C71—42no—
65D4×C71—82no—
66D4×C71—82no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—81no—
76C2×C4×C71#9911yescharacteristic
77C2×C4×C71C2×C22⋊C411yescharacteristic
78C8×C71—42no—
79C71⋊C8—84no—
80C22×C71—21no—
81C22×C71—41no—
82C22×C71—42no—
83C22×C71—82no—
84C22×C71C2×#9911yescharacteristic
85C4×C71—21no—
86C4×C71—41no—
87C4×C71—41no—
88C4×C71—42no—
89C4×C71—82no—
90C4×C71#35411yescharacteristic
91C4×C71#35411yescharacteristic
92#13598—711nomaximal, Sylow
93C2×C71—21no—
94C2×C71—41no—
95C2×C71—82no—
96C2×C71C2×#35411yescharacteristic
97#3135D7111yescharacteristic
98#2132—712no—
99#1413—714no—
100C71#1359811yescharacteristic, complemented, Sylow
101#533Dic7111yescharacteristic
102#532C2×D7111yescharacteristic
103#491C2×D7112yes—
104#490Dic7112yes—
105#489—22no—
106#416—22no—
107#413—1422no—
108C2×#97Dic7111yescharacteristic
109#356—714no—
110#355—1422no—
111C2×#41C22⋊D7112yes—
112C2×M16—2842no—
113C22×C8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C22×D7111yescharacteristic
116#127C2×Dic7111yescharacteristic
117#125C2×Dic7112yes—
118#124C22⋊D7112yes—
119C22⋊Q8—41no—
120#121—41no—
121C4×Q8—41no—
122C4×D4—41no—
123#117—21no—
124C2×C42C2×Dic7111yescharacteristic
125#103—24no—
126#102—24no—
127#101—1424no—
128#100—2842no—
129#97—21no—
130#97C2×Dic7112yes—
131#41—42no—
132#41—42no—
133C2×Q8—41no—
134C2×Q8C2×C22⋊D7112yes—
135C2×D4—22no—
136C2×D4—41no—
137C2×D4C22⋊Dic7112yes—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4C22⋊Dic7111yescharacteristic
141M16—2844no—
142M16—5682no—
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—5682no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C22⋊C4—42no—
152C22⋊C4—41no—
153C42—21no—
154C42—41no—
155C42C22⋊Dic7111yescharacteristic
156C42C22×Dic7111yescharacteristic
157C23—22no—
158C23#1455604311yescharacteristic
159Q8—42no—
160Q8—82no—
161D4—42no—
162D4—82no—
163D4—82no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—81no—
173C2×C4#1455604311yescharacteristic
174C2×C4C2×C22⋊Dic7111yescharacteristic
175C8—42no—
176C8—5684no—
177C22—21no—
178C22—41no—
179C22—42no—
180C22—82no—
181C22C2×#1455604311yescharacteristic
182C4—21no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—82no—
187C4#1798579411yescharacteristic
188C4#1798579411yescharacteristic
189C2—21no—
190C2—41no—
191C2—82no—
192C2C2×#1798579411yescharacteristic
subgroupG/H
centerC4#17985794
commutatorC42×C71C22×C4
FrattiniC4⋊Q8C22×D71
FittingC71×#3135C2
socleC2×C71C2×#354
prime radical#532×C71C22
cosocleC4⋊Q8×C71C23
hypercenter#3135D71
hypocenterC71#13598
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3135

Complements

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

NH ≅ G/Ntype
2. #22586513C2semidirect
100. C71#13598semidirect

Maximal subgroups

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

subgroupindex
#225900252
#225865132
C71×#31352
#1359871