Finite Group Databasebeta

← #31886762

Subgroups — #31886762

Order 18176.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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#22589903C212yesmaximal
2#22589878C211yesmaximal, characteristic
3#22586166C211yesmaximal, characteristic
4#22586163C212yesmaximal
5C71×#2809C211yesmaximal, characteristic
6C2×#14556061—21no—
7C2×#14556052—21no—
8#17985869C2211yescharacteristic
9#17985761—22no—
10#17985761—22no—
11#17985754C2211yescharacteristic
12#17985754C2212yes—
13#400×C71—21no—
14#398×C71—22no—
15#391×C71—21no—
16#391×C71C2211yescharacteristic
17#389×C71C2212yes—
18#387×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556061—42no—
24#14556052—42no—
25C22×Q8×C71D411yescharacteristic
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71D412yes—
30C2×C4⋊C4×C71D412yes—
31C2×C4⋊C4×C71D411yescharacteristic
32C2×C42×C71—21no—
33C2×C42×C71C2311yescharacteristic
34#95×C71—21no—
35#95×C71—22no—
36#95×C71—22no—
37#95×C71—22no—
38#95×C71—22no—
39C22×Dic71—41no—
40C2×C71⋊C8—22no—
41C2×C71⋊C8—41no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—42no—
44C2×C71⋊C8—42no—
45C2×C71⋊C8—42no—
46C4⋊Dic71—41no—
47C4⋊Dic71—41no—
48C4⋊Dic71—44no—
49C2×Q8×C71—21no—
50C2×Q8×C71—41no—
51C2×Q8×C71—41no—
52C2×Q8×C71Q1612yes—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71C2×D411yescharacteristic
58C22×C4×C71C2×D412yes—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71—42no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—81no—
66C4⋊C4×C71—81no—
67C42×C71—41no—
68C42×C71—81no—
69C23×C71#12011yescharacteristic
70Q8×C71—42no—
71Q8×C71—81no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—41no—
78C2×C4×C71—41no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—41no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71C2×Q1611yescharacteristic
88C2×C4×C71#13512yes—
89C2×C4×C71#13712yes—
90C2×C4×C71#13711yescharacteristic
91C2×Dic71—81no—
92C2×Dic71—82no—
93C71⋊C8—42no—
94C71⋊C8—82no—
95C22×C71#45111yescharacteristic
96C22×C71#45212yes—
97C22×C71#45311yescharacteristic
98C22×C71#45412yes—
99C22×C71#45811yescharacteristic
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—22no—
103C4×C71—22no—
104C4×C71—42no—
105C4×C71—82no—
106C4×C71—82no—
107C4×C71—81no—
108Dic71—161no—
109#7831—711nomaximal, Sylow
110C2×C71#161411yescharacteristic
111C2×C71#161512yes—
112C2×C71#163411yescharacteristic
113C2×C71#163612yes—
114C2×C71#165711yescharacteristic
115#2809D7111yescharacteristic
116#2100—712no—
117#2095—711no—
118#1272—711no—
119#1271—712no—
120C71#783111yescharacteristic, complemented, Sylow
121C2×#107—1421no—
122C2×#103—1421no—
123#400—21no—
124#398—22no—
125#391—21no—
126#391C2×D7111yescharacteristic
127#389C2×D7112yes—
128#387—22no—
129#384—711no—
130#340—1422no—
131#340—1422no—
132#336—711no—
133#336—712no—
134C22×Q8C22⋊D7111yescharacteristic
135C22×C8—1422no—
136C22×C8—1421no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—1422no—
141C2×C4⋊C4—1421no—
142C2×C4⋊C4C22⋊D7112yes—
143C2×C4⋊C4C22⋊D7112yes—
144C2×C4⋊C4C22⋊D7111yescharacteristic
145C2×C42—21no—
146C2×C42C22×D7111yescharacteristic
147#107—2842no—
148#103—2842no—
149#95—21no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8#1176585512yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—2841no—
163C22×C4C2×C22⋊D7111yescharacteristic
164C22×C4C2×C22⋊D7112yes—
165C2×C8—2841no—
166C2×C8—2841no—
167C2×C8—2842no—
168C2×C8—2842no—
169C2×C8—2842no—
170C2×C8—1422no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—2841no—
180C4⋊C4—2841no—
181C4⋊C4—2844no—
182C42—41no—
183C42—81no—
184C23#1455609711yescharacteristic
185Q8—42no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—5681no—
203C2×C4—5682no—
204C2×C4C2×#1176585511yescharacteristic
205C2×C4#1455614812yes—
206C2×C4#1455615812yes—
207C2×C4#1455615911yescharacteristic
208C8—2842no—
209C8—5682no—
210C22#1798615511yescharacteristic
211C22#1798616112yes—
212C22#1798616711yescharacteristic
213C22#1798617212yes—
214C22#1798618911yescharacteristic
215C4—21no—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—82no—
221C4—82no—
222C4—81no—
223C4—11361no—
224C2#2258736511yescharacteristic
225C2#2258737112yes—
226C2#2258748511yescharacteristic
227C2#2258749812yes—
228C2#2258763811yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2809C2
socle, prime radicalC23×C71#120
hypercenter#2809D71
hypocenterC71#7831
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
120. C71#7831semidirect

Maximal subgroups

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

subgroupindex
#225899032
#225898782
#225861662
#225861632
C71×#28092
#783171