Finite Group Databasebeta

← #31886783

Subgroups — #31886783

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#22589903C211yesmaximal, characteristic
2#22589903C211yesmaximal, characteristic
3#22589878C211yesmaximal, characteristic
4#22586166C211yesmaximal, characteristic
5#22586163C211yesmaximal, characteristic
6#22586163C211yesmaximal, characteristic
7C71×#2805C211yesmaximal, characteristic
8C2×#14556061—21no—
9C2×#14556052—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18#400×C71—21no—
19#400×C71—21no—
20#395×C71—21no—
21#391×C71C2211yescharacteristic
22#389×C71—21no—
23#389×C71C2211yescharacteristic
24#389×C71C2211yescharacteristic
25#387×C71—21no—
26#387×C71—21no—
27C22×C71⋊C8—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33#14556061—42no—
34#14556052—42no—
35C22×Q8×C71D411yescharacteristic
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C42×C71—21no—
46C2×C42×C71C2311yescharacteristic
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56C22×Dic71—41no—
57C2×C71⋊C8—22no—
58C2×C71⋊C8—41no—
59C2×C71⋊C8—41no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C4⋊Dic71—41no—
67C4⋊Dic71—41no—
68C4⋊Dic71—42no—
69C4⋊Dic71—42no—
70C2×Q8×C71—21no—
71C2×Q8×C71—21no—
72C2×Q8×C71—41no—
73C2×Q8×C71—41no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—21no—
84C4⋊C4×C71—21no—
85C4⋊C4×C71—21no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—81no—
94C4⋊C4×C71—81no—
95C4⋊C4×C71Q1612yes—
96C42×C71—41no—
97C42×C71—81no—
98C23×C71#12011yescharacteristic
99Q8×C71—42no—
100Q8×C71—81no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71C2×Q1611yescharacteristic
126C2×C4×C71#13511yescharacteristic
127C2×C4×C71#13511yescharacteristic
128C2×C4×C71#13711yescharacteristic
129C2×C4×C71#13711yescharacteristic
130C2×C4×C71#13711yescharacteristic
131C2×Dic71—81no—
132C2×Dic71—81no—
133C2×Dic71—81no—
134C71⋊C8—42no—
135C71⋊C8—81no—
136C71⋊C8—81no—
137C22×C71#45111yescharacteristic
138C22×C71#45211yescharacteristic
139C22×C71#45211yescharacteristic
140C22×C71#45311yescharacteristic
141C22×C71#45411yescharacteristic
142C22×C71#45411yescharacteristic
143C22×C71#45811yescharacteristic
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—21no—
150C4×C71—42no—
151C4×C71—81no—
152C4×C71—81no—
153C4×C71—81no—
154C4×C71—81no—
155C4×C71—81no—
156Dic71—161no—
157#7835—711nomaximal, Sylow
158C2×C71#161511yescharacteristic
159C2×C71#161711yescharacteristic
160C2×C71#161811yescharacteristic
161C2×C71#163211yescharacteristic
162C2×C71#163611yescharacteristic
163C2×C71#163811yescharacteristic
164C2×C71#165711yescharacteristic
165#2805D7111yescharacteristic
166#2100—711no—
167#2100—711no—
168#2095—711no—
169#1272—711no—
170#1271—711no—
171#1271—711no—
172C71#783511yescharacteristic, complemented, Sylow
173C2×#107—1421no—
174C2×#103—1421no—
175#400—21no—
176#400—21no—
177#395—21no—
178#391C2×D7111yescharacteristic
179#389—21no—
180#389C2×D7111yescharacteristic
181#389C2×D7111yescharacteristic
182#387—21no—
183#387—21no—
184#384—711no—
185#340—1421no—
186#340—1421no—
187#340—1421no—
188#340—1421no—
189#336—711no—
190#336—711no—
191#336—711no—
192C22×Q8C22⋊D7111yescharacteristic
193C22×C8—1421no—
194C22×C8—1421no—
195C22×C8—1421no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—1421no—
201C2×C4⋊C4—1421no—
202C2×C4⋊C4—1421no—
203C2×C4⋊C4C22⋊D7111yescharacteristic
204C2×C4⋊C4C22⋊D7111yescharacteristic
205C2×C4⋊C4C22⋊D7111yescharacteristic
206C2×C4⋊C4C22⋊D7111yescharacteristic
207C2×C4⋊C4C22⋊D7111yescharacteristic
208C2×C42—21no—
209C2×C42C22×D7111yescharacteristic
210#107—2842no—
211#103—2842no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—41no—
224C2×Q8—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—2841no—
232C22×C4C2×C22⋊D7111yescharacteristic
233C22×C4C2×C22⋊D7111yescharacteristic
234C22×C4C2×C22⋊D7111yescharacteristic
235C2×C8—2841no—
236C2×C8—2841no—
237C2×C8—2841no—
238C2×C8—2841no—
239C2×C8—2841no—
240C2×C8—2841no—
241C2×C8—2841no—
242C2×C8—2841no—
243C2×C8—1422no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—2841no—
257C4⋊C4—2841no—
258C4⋊C4—2842no—
259C4⋊C4—2842no—
260C4⋊C4#1176585512yes—
261C42—41no—
262C42—81no—
263C23#1455609711yescharacteristic
264Q8—42no—
265Q8—81no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—5681no—
291C2×C4—5681no—
292C2×C4—5681no—
293C2×C4C2×#1176585511yescharacteristic
294C2×C4#1455614811yescharacteristic
295C2×C4#1455614811yescharacteristic
296C2×C4#1455615811yescharacteristic
297C2×C4#1455615811yescharacteristic
298C2×C4#1455615911yescharacteristic
299C8—2842no—
300C8—5681no—
301C8—5681no—
302C22#1798615511yescharacteristic
303C22#1798616111yescharacteristic
304C22#1798616111yescharacteristic
305C22#1798616711yescharacteristic
306C22#1798617211yescharacteristic
307C22#1798617211yescharacteristic
308C22#1798618911yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—11361no—
322C2#2258737111yescharacteristic
323C2#2258738211yescharacteristic
324C2#2258738811yescharacteristic
325C2#2258747411yescharacteristic
326C2#2258749811yescharacteristic
327C2#2258751211yescharacteristic
328C2#2258763811yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2805C2
socle, prime radicalC23×C71#120
hypercenter#2805D71
hypocenterC71#7835
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
172. C71#7835semidirect

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×#28052
#783571