Finite Group Databasebeta

← C2×#22586285 #31906605

Subgroups — C2×#22586285 #31906605

Order 18176.

598 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#17985868C211yesmaximal, characteristic
2C2×#17985725C211yesmaximal, characteristic
3#22586285C214yesmaximal, direct product factor
4C2×#324×C71C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic
6C22×Dic71⋊C4—21no—
7C22×Dic71⋊C4—21no—
8C22×C4×Dic71C411yescharacteristic
9C2×C22⋊(C71⋊C8)—21no—
10#17985868—21no—
11#17985868—21no—
12#17985868C412yes—
13#17985868C2212yes—
14#17985725—21no—
15#17985725C2212yes—
16C22×C4⋊C4×C71C2211yescharacteristic
17C2×C22⋊C8×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C22×C71⋊C8—41no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—41no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—22no—
30C2×Dic71⋊C4—22no—
31C2×Dic71⋊C4—44no—
32C2×Dic71⋊C4—44no—
33C2×C4×Dic71—21no—
34C2×C4×Dic71—22no—
35C2×C4×Dic71C814yescomplemented
36C2×C4×Dic71C2×C412yes—
37C22⋊(C71⋊C8)—22no—
38C22⋊(C71⋊C8)—41no—
39C23×C4×C71C2×C411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C23×C4×C71D411yescharacteristic
42C22×C8×C71—41no—
43C2×C4⋊C4×C71—41no—
44C2×C4⋊C4×C71—41no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71C2311yescharacteristic
49C22⋊C8×C71—22no—
50C22⋊C8×C71—41no—
51C22×Dic71—24no—
52C22×Dic71—24no—
53C22×Dic71—24no—
54C22×Dic71—22no—
55C22×Dic71—22no—
56C22×Dic71—42no—
57C22×Dic71—42no—
58C22×Dic71—41no—
59C22×Dic71—41no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C4⋊Dic71—42no—
63C4⋊Dic71—42no—
64Dic71⋊C4—44no—
65Dic71⋊C4—44no—
66Dic71⋊C4—44no—
67Dic71⋊C4—44no—
68C4×Dic71—24no—
69C4×Dic71—22no—
70C4×Dic71C2×C814yescomplemented
71C24×C71C22⋊C411yescharacteristic
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C2×C812yes—
85C22×C4×C71M1612yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C2×C8×C71—42no—
90C2×C8×C71—82no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C23×C71—22no—
96C23×C71—22no—
97C23×C71C22⋊C812yes—
98C23×C71#9911yescharacteristic
99C23×C71#9911yescharacteristic
100C23×C71#10112yes—
101C23×C71C2×C22⋊C411yescharacteristic
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71C22⋊C812yes—
123C2×C4×C71C22×C811yescharacteristic
124C2×C4×C71C2×M1611yescharacteristic
125C2×Dic71—24no—
126C2×Dic71—216no—
127C2×Dic71—24no—
128C2×Dic71—24no—
129C2×Dic71—42no—
130C2×Dic71—42no—
131C2×Dic71—42no—
132C2×Dic71—82no—
133C2×Dic71—82no—
134C8×C71—82no—
135C71⋊C8—82no—
136C22×C71—24no—
137C22×C71—24no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71#32412yes—
143C22×C71#32412yes—
144C22×C71#35312yes—
145C22×C71#35612yes—
146C22×C71C2×C22⋊C811yescharacteristic
147C22×C71C2×#9911yescharacteristic
148C22×C71C2×#10111yescharacteristic
149C4×C71—22no—
150C4×C71—21no—
151C4×C71—21no—
152C4×C71—41no—
153C4×C71—41no—
154C4×C71—41no—
155C4×C71—41no—
156Dic71—28no—
157Dic71—82no—
158C2×#1325—711nomaximal, Sylow
159C2×C71—21no—
160C2×C71—22no—
161C2×C71—21no—
162C2×C71#132514yescomplemented
163C2×C71C2×#32411yescharacteristic
164C2×C71C2×#35311yescharacteristic
165C2×C71C2×#35611yescharacteristic
166C2×#384—711no—
167C2×#324—711no—
168C2×#324D7111yescharacteristic
169#1325—714no—
170C71C2×#132511yescharacteristic, complemented, Sylow
171C22×C4⋊C4—711no—
172C22×C4⋊C4—1421no—
173C22×C4⋊C4—1421no—
174C22×C4⋊C4C2×D7111yescharacteristic
175C22×C42—711no—
176C2×C22⋊C8—21no—
177C2×C22⋊C8—1421no—
178#384—712no—
179#384—712no—
180#384—1421no—
181#384—1421no—
182#324—21no—
183#324—712no—
184#324—1421no—
185#324C2×D7112yes—
186C23×C4—1421no—
187C23×C4—1421no—
188C23×C4C4×D7111yescharacteristic
189C23×C4D28411yescharacteristic
190C23×C4C22⋊D7111yescharacteristic
191C22×C8—41no—
192C22×C8—2841no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—2841no—
196C2×C4⋊C4—2841no—
197C2×C4⋊C4—2844no—
198C2×C4⋊C4—2844no—
199C2×C4⋊C4—712no—
200C2×C4⋊C4—1421no—
201C2×C4⋊C4—1422no—
202C2×C4⋊C4—1422no—
203C2×C4⋊C4—1422no—
204C2×C4⋊C4—1422no—
205C2×C4⋊C4C4×D7111yescharacteristic
206C2×C4⋊C4D28411yescharacteristic
207C2×C4⋊C4C22⋊D7111yescharacteristic
208C2×C4⋊C4C22×D7111yescharacteristic
209C2×C42—714no—
210C2×C42—712no—
211C2×C42—1421no—
212C2×C42—1422no—
213C22⋊C8—22no—
214C22⋊C8—41no—
215C22⋊C8—2841no—
216C22⋊C8—1422no—
217C24(C2×D71)⋊C411yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—2842no—
228C22×C4—2842no—
229C22×C4—2841no—
230C22×C4—2841no—
231C22×C4—1422no—
232C22×C4—1424no—
233C22×C4—1424no—
234C22×C4—1424no—
235C22×C4—1422no—
236C22×C4(C2×D71)⋊C411yescharacteristic
237C22×C4(C2×D71)⋊C411yescharacteristic
238C22×C4(C2×D71)⋊C411yescharacteristic
239C22×C4C8×D7112yes—
240C22×C4#1176584812yes—
241C22×C4C2×C4×D7111yescharacteristic
242C22×C4C2×C22⋊D7111yescharacteristic
243C22×C4C2×D28411yescharacteristic
244C2×C8—42no—
245C2×C8—82no—
246C2×C8—2842no—
247C2×C8—5682no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—2842no—
253C4⋊C4—2842no—
254C4⋊C4—2844no—
255C4⋊C4—2844no—
256C4⋊C4—2844no—
257C4⋊C4—2844no—
258C42—714no—
259C42—1424no—
260C42—1422no—
261C23—22no—
262C23—22no—
263C23(C2×D71)⋊C812yes—
264C23#1455604211yescharacteristic
265C23#1455604211yescharacteristic
266C23#1455604612yes—
267C23C2×(C2×D71)⋊C411yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—2842no—
289C2×C4—2842no—
290C2×C4—2842no—
291C2×C4—5682no—
292C2×C4—5682no—
293C2×C4—1424no—
294C2×C4—1424no—
295C2×C4—14216no—
296C2×C4—1424no—
297C2×C4(C2×D71)⋊C812yes—
298C2×C4C2×C8×D7111yescharacteristic
299C2×C4C2×#1176584811yescharacteristic
300C8—82no—
301C8—5682no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—24no—
307C22—24no—
308C22#1798572412yes—
309C22#1798572412yes—
310C22#1798579112yes—
311C22#1798579712yes—
312C22C2×(C2×D71)⋊C811yescharacteristic
313C22C2×#1455604211yescharacteristic
314C22C2×#1455604611yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—5682no—
323C4—1428no—
324C2—21no—
325C2—21no—
326C2—22no—
327C2#2258628514yesdirect product factor
328C2C2×#1798572411yescharacteristic
329C2C2×#1798579111yescharacteristic
330C2C2×#1798579711yescharacteristic
subgroupG/H
centerC23#14556042
commutatorC2×C4×C71C22×C8
FrattiniC2×C4⋊C4C22×D71
FittingC2×#324×C71C2
socleC23×C71#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenterC2×#324D71
hypocenterC71C2×#1325
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #22586285C2direct (N × H)
35. C2×C4×Dic71C8semidirect
70. C4×Dic71C2×C8semidirect
162. C2×C71#1325semidirect
170. C71C2×#1325semidirect
327. C2#22586285direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179858682
C2×#179857252
#225862852
C2×#324×C712
C2×#132571