Finite Group Databasebeta

← #31906780

Subgroups — #31906780

Order 18176.

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

normal subgroups only

This group has 356 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#22597048C211yesmaximal, characteristic
2#22590140C212yesmaximal, complemented
3#22586266C211yesmaximal, characteristic, complemented
4#22586265C212yesmaximal, complemented
5C71×#1315C211yesmaximal, characteristic, complemented
6C2×#14556136C411yescharacteristic
7C2×#14556135C411yescharacteristic
8#17986545C411yescharacteristic
9#17986074—22no—
10(C22⋊D71)⋊C8—22no—
11#17986022C411yescharacteristic
12#17985965C2211yescharacteristic
13C2×C8×Dic71C2211yescharacteristic
14Dic71⋊C16—22no—
15C22⋊(C71⋊C16)C2212yes—
16#17985774—22no—
17#408×C71C2211yescharacteristic
18C22⋊C16×C71C2212yes—
19C2×#11765865C2×C411yescharacteristic
20#14556136—42no—
21#14556135—42no—
22C2×#11765849—21no—
23C2×#11765849C2×C411yescharacteristic
24C2×#11765848—21no—
25C22×C71⋊C8—21no—
26C22×C71⋊C8C2×C411yescharacteristic
27C2×C8×D71—21no—
28#14556126C811yescharacteristic
29#14556124C811yescharacteristic
30#14556117C811yescharacteristic
31#14556115C811yescharacteristic
32#14556100—22no—
33C4×C22⋊D71—22no—
34C2×C4×Dic71C2×C411yescharacteristic
35C2×C71⋊C16—22no—
36Dic71⋊C8—22no—
37(C2×D71)⋊C8—22no—
38C8×Dic71D412yes—
39C8×Dic71D412yes—
40C2×#41×C71C2×C411yescharacteristic
41C2×M16×C71C2×C411yescharacteristic
42C22×C8×C71C2311yescharacteristic
43C2×C16×C71—22no—
44C22⋊C8×C71—22no—
45#11765865—41no—
46#11765865—41no—
47C2×Dic142C2×C811yescharacteristic
48C2×D284C2×C811yescharacteristic
49C2×C22⋊D71—21no—
50C2×C22⋊D71—22no—
51C22×Dic71M1612yes—
52C2×C4×D71—21no—
53#11765849—42no—
54#11765849—41no—
55#11765848—42no—
56C2×C71⋊C8—22no—
57C2×C71⋊C8—21no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8C22⋊C411yescharacteristic
60C2×C71⋊C8C22⋊C411yescharacteristic
61C8×D71—42no—
62Dic71⋊C4—22no—
63C22⋊Dic71—22no—
64(C2×D71)⋊C4—22no—
65C4×Dic71C22⋊C412yes—
66C4×Dic71C2×C811yescharacteristic
67C4×Dic71C2×C811yescharacteristic
68C71⋊C16—42no—
69#41×C71—41no—
70#41×C71—41no—
71C2×Q8×C71C2×C811yescharacteristic
72C2×D4×C71—21no—
73C2×D4×C71C2×C811yescharacteristic
74C22×C4×C71—21no—
75C22×C4×C71C22×C411yescharacteristic
76M16×C71—41no—
77C2×C8×C71—21no—
78C2×C8×C71—41no—
79C2×C8×C71C2×D412yes—
80C16×C71—42no—
81C23×C71—21no—
82C23×C71C2×M1611yescharacteristic
83C22×D71—21no—
84Q8×C71—41no—
85D4×C71—41no—
86D4×C71—42no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71C2×C22⋊C411yescharacteristic
93C2×C4×C71C22×C811yescharacteristic
94C22⋊D71—44no—
95C22⋊D71—42no—
96C2×Dic71—21no—
97C2×Dic71—42no—
98C2×Dic71C22⋊C812yes—
99C2×Dic71C22⋊C812yes—
100D284—41no—
101C4×D71—41no—
102C4×D71—41no—
103Dic142—41no—
104C8×C71—22no—
105C8×C71—41no—
106C71⋊C8—21no—
107C71⋊C8—41no—
108C71⋊C8#10012yes—
109C22×C71—22no—
110C22×C71—21no—
111C22×C71—42no—
112C22×C71—41no—
113C22×C71C2×C22⋊C811yescharacteristic
114C2×D71—42no—
115C2×D71—41no—
116C4×C71—21no—
117C4×C71—41no—
118C4×C71#41011yescharacteristic
119C4×C71C2×#10011yescharacteristic
120Dic71—22no—
121Dic71—22no—
122Dic71—41no—
123#11491—711nomaximal, Sylow
124C2×C71—41no—
125C2×C71—81no—
126C2×C71#146011yescharacteristic
127C2×C71#211712yes—
128D71—81no—
129#2931—711no—
130#2163—712no—
131#1315—711no—
132#1315—712no—
133#1315D7111yescharacteristic, complemented
134C71#1149111yescharacteristic, complemented, Sylow
135C2×#131—711no—
136C2×#131—711no—
137#532—711no—
138#436—1422no—
139C8×D4—1422no—
140#423—711no—
141#408—711no—
142#408C2×D7111yescharacteristic
143C2×C4×C8—711no—
144C4⋊C16—1422no—
145C22⋊C16—712no—
146C22⋊C16C2×D7112yes—
147#346—1422no—
148C2×#41—711no—
149C2×#41C4×D7111yescharacteristic
150#131—2842no—
151#131—2842no—
152C2×M16—711no—
153C2×M16—1421no—
154C2×M16—1421no—
155C2×M16C4×D7111yescharacteristic
156C22×C8—711no—
157C22×C8—1421no—
158C22×C8—1421no—
159C22×C8C22×D7111yescharacteristic
160C4⋊Q8—711no—
161#127—711no—
162#124—711no—
163#124—711no—
164#121—1422no—
165C4×D4—1422no—
166C2×C42—711no—
167C2×C16—22no—
168C2×C16—1422no—
169C4⋊C8—1422no—
170C22⋊C8—22no—
171C22⋊C8—1422no—
172C4×C8—712no—
173C4×C8—712no—
174#41—41no—
175#41—41no—
176#41—2841no—
177#41—2841no—
178C2×Q8—711no—
179C2×Q8C8×D7111yescharacteristic
180C2×D4—21no—
181C2×D4—711no—
182C2×D4—1422no—
183C2×D4—1421no—
184C2×D4C8×D7111yescharacteristic
185C22×C4—21no—
186C22×C4—712no—
187C22×C4—1421no—
188C22×C4C2×C4×D7111yescharacteristic
189M16—41no—
190M16—2842no—
191M16—2842no—
192M16—2841no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8—2842no—
196C2×C8—2842no—
197C2×C8—711no—
198C2×C8—711no—
199C2×C8—1421no—
200C2×C8—1422no—
201C2×C8D4×D7112yes—
202C4⋊C4—1422no—
203C22⋊C4—1422no—
204C22⋊C4—1422no—
205C42—712no—
206C42—711no—
207C42—711no—
208C16—42no—
209C16—2842no—
210C23—21no—
211C23—1421no—
212C23M16×D7111yescharacteristic
213Q8—41no—
214Q8—2841no—
215D4—41no—
216D4—42no—
217D4—2841no—
218D4—2842no—
219D4—2844no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2841no—
226C2×C4—2841no—
227C2×C4—2842no—
228C2×C4—712no—
229C2×C4—712no—
230C2×C4—1421no—
231C2×C4C22⋊C4×D7111yescharacteristic
232C2×C4C2×C8×D7111yescharacteristic
233C8—22no—
234C8—41no—
235C8—2841no—
236C8—712no—
237C8—1421no—
238C22—22no—
239C22—21no—
240C22—42no—
241C22—41no—
242C22—2842no—
243C22—2841no—
244C22C22⋊C8×D7111yescharacteristic
245C4—21no—
246C4—41no—
247C4—2841no—
248C4—1422no—
249C4—1422no—
250C4#1798597111yescharacteristic
251C4#100×D7111yescharacteristic
252C2—41no—
253C2—81no—
254C2—5681no—
255C2#2258665911yescharacteristic
256C2#2258996612yes—
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC22×C8C22×D71
FittingC71×#1315C2
socleC22×C71C2×C22⋊C8
prime radical#17986545C4
cosocleC22×C8×C71C23
hypercenter#1315D71
hypocenterC71#11491
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #22590140C2semidirect
3. #22586266C2semidirect
4. #22586265C2semidirect
5. C71×#1315C2semidirect
133. #1315D71semidirect
134. C71#11491semidirect

Maximal subgroups

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

subgroupindex
#225970482
#225901402
#225862662
#225862652
C71×#13152
#1149171