Finite Group Databasebeta

← C2×#11740468 #13512628

Subgroups — C2×#11740468 #13512628

Order 1792.

2398 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#125C211yesmaximal, characteristic
2#11740468C2116yesmaximal, direct product factor
3C2×#89757C212yesmaximal
4C2×#89757C214yesmaximal
5C2×#89753C214yesmaximal
6C2×C4⋊C4×Dic7C212yesmaximal
7C2×#89707C211yesmaximal, characteristic
8C2×#89704C211yesmaximal, characteristic
9C2×C7×#125C4116yescomplemented
10C2×C7×#125C2218yes—
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2214yes—
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2212yes—
15C22×C4⋊Dic7C2214yes—
16C22×C4⋊Dic7C2214yes—
17C22×C4×Dic7C2211yescharacteristic
18C22×C4×Dic7C2214yes—
19C22×C4×Dic7C2211yescharacteristic
20C2×C7⋊#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2214yes—
23C2×C7⋊#95C2214yes—
24#89757C22132yes—
25#89757C22116yes—
26#89753C22132yes—
27C4⋊C4×Dic7C22116yes—
28#89707C2218yes—
29#89704C2218yes—
30C2×#2330—71nomaximal, Sylow
31C23×C4×C7C2314yes—
32C23×C4×C7C2312yes—
33C23×C4×C7C2311yescharacteristic
34C23×Dic7C2311yescharacteristic
35C23×Dic7C2314yes—
36C23×Dic7C2311yescharacteristic
37C23×Dic7C2312yes—
38C7×#125C2×C4164yescomplemented
39C2×C7×C4⋊C4C2×C4132yes—
40C2×C7×C4⋊C4C2×C4116yes—
41C2×C7×C4⋊C4C2318yes—
42C2×C7×C4⋊C4C23116yes—
43C2×C42×C7C2×C418yes—
44C2×C42×C7C2314yes—
45C2×C4⋊Dic7—28no—
46C2×C4⋊Dic7—48no—
47C2×C4⋊Dic7—48no—
48C2×C4⋊Dic7C2318yes—
49C2×C4⋊Dic7C23116yes—
50C2×C4⋊Dic7C23116yes—
51C2×C4×Dic7—24no—
52C2×C4×Dic7—48no—
53C2×C4×Dic7Q818yes—
54C2×C4×Dic7C2314yes—
55C2×C4×Dic7C23116yes—
56C2×C4×Dic7C2314yes—
57C7⋊#95C2318yes—
58C7⋊#95C2318yes—
59C7⋊#95C23116yes—
60C7⋊#95C23116yes—
61C22×#125D711yescharacteristic
62#2330—716no—
63C2×#389—72no—
64C2×#389—74no—
65C2×#387—74no—
66C2×#384—71no—
67C2×C4×C4⋊C4—72no—
68C2×#376—71no—
69C24×C7C2411yescharacteristic
70C22×C4×C7—28no—
71C22×C4×C7—24no—
72C22×C4×C7—28no—
73C22×C4×C7—24no—
74C22×C4×C7C22×C418yes—
75C22×C4×C7C22×C4116yes—
76C22×C4×C7C22×C414yes—
77C22×C4×C7C2×Q814yes—
78C22×C4×C7#4114yes—
79C22×C4×C7C2418yes—
80C22×C4×C7C2414yes—
81C22×C4×C7C2412yes—
82C22×Dic7—24no—
83C22×Dic7—24no—
84C22×Dic7—22no—
85C22×Dic7—22no—
86C22×Dic7—28no—
87C22×Dic7—28no—
88C22×Dic7—24no—
89C22×Dic7—28no—
90C22×Dic7—24no—
91C22×Dic7C2×Q818yes—
92C22×Dic7C2412yes—
93C22×Dic7C2412yes—
94C22×Dic7C2418yes—
95C22×Dic7C2414yes—
96C7×C4⋊C4C22×C4164yes—
97C7×C4⋊C4C22×C4132yes—
98C42×C7C22×C4116yes—
99C4⋊Dic7—216no—
100C4⋊Dic7—416no—
101C4⋊Dic7—416no—
102C4×Dic7—28no—
103C4×Dic7—416no—
104C4×Dic7C2×Q8116yes—
105C2×#125Dic7116yescomplemented
106C2×#125C2×D718yes—
107C22×C4⋊C4—72no—
108C22×C4⋊C4—74no—
109C22×C4⋊C4—74no—
110C22×C4⋊C4C2×D712yes—
111C22×C4⋊C4C2×D714yes—
112C22×C42—71no—
113C22×C42—71no—
114C22×C42—74no—
115C22×C42C2×D711yescharacteristic
116#389—732no—
117#389—716no—
118#387—732no—
119#384—78no—
120C4×C4⋊C4—716no—
121#376—78no—
122C2×#95—72no—
123C2×#95—72no—
124C2×#95—74no—
125C2×#95—74no—
126C23×C7C23×C412yes—
127C23×C7C22×Q812yes—
128C23×C7C2×#4112yes—
129C23×C7H(5,𝔽2)14yes—
130C23×C72−1+414yes—
131C23×C7C2511yescharacteristic
132C2×C4×C7—232no—
133C2×C4×C7—216no—
134C2×C4×C7—28no—
135C2×C4×C7—24no—
136C2×C4×C7—28no—
137C2×C4×C7—24no—
138C2×C4×C7C4×Q8116yes—
139C2×C4×C7C23×C418yes—
140C2×C4×C7C23×C4116yes—
141C2×C4×C7C23×C414yes—
142C2×C4×C7C22×Q814yes—
143C2×C4×C7C2×#4114yes—
144C2×Dic7—216no—
145C2×Dic7—24no—
146C2×Dic7—24no—
147C2×Dic7—22no—
148C2×Dic7—22no—
149C2×Dic7—28no—
150C2×Dic7—28no—
151C2×Dic7—24no—
152C2×Dic7—28no—
153C2×Dic7—24no—
154C2×Dic7—44no—
155C2×Dic7—416no—
156C2×Dic7—44no—
157C2×Dic7C22×Q818yes—
158C23×C4—72no—
159C23×C4—71no—
160C23×C4—74no—
161C23×C4—71no—
162C23×C4C22×D712yes—
163C23×C4C22×D714yes—
164C23×C4C22×D711yescharacteristic
165#125C2×Dic7164yescomplemented
166C2×C4⋊C4—78no—
167C2×C4⋊C4—716no—
168C2×C4⋊C4—716no—
169C2×C4⋊C4—148no—
170C2×C4⋊C4—288no—
171C2×C4⋊C4—288no—
172C2×C4⋊C4C2×Dic7132yes—
173C2×C4⋊C4C2×Dic7116yes—
174C2×C4⋊C4C22×D718yes—
175C2×C4⋊C4C22×D7116yes—
176C2×C42—78no—
177C2×C42—74no—
178C2×C42—74no—
179C2×C42—716no—
180C2×C42—144no—
181C2×C42—288no—
182C2×C42C2×Dic718yes—
183C2×C42C22×D714yes—
184#95—78no—
185#95—78no—
186#95—716no—
187#95—716no—
188C22×C7C2×C4×Q814yes—
189C22×C7#52018yes—
190C22×C7#54014yes—
191C22×C7#54114yes—
192C22×C7#54418yes—
193C22×C7C24×C411yescharacteristic
194C22×C7C23×Q811yescharacteristic
195C22×C7C22×#4111yescharacteristic
196C22×C7C2×H(5,𝔽2)12yes—
197C22×C7C2×2−1+412yes—
198C4×C7—216no—
199C4×C7—28no—
200C4×C7C2×C4×Q818yes—
201Dic7—28no—
202Dic7—42no—
203Dic7—48no—
204Dic7—42no—
205C24C23×D711yescharacteristic
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—78no—
211C22×C4—74no—
212C22×C4—72no—
213C22×C4—72no—
214C22×C4—78no—
215C22×C4—144no—
216C22×C4—144no—
217C22×C4—142no—
218C22×C4—142no—
219C22×C4—148no—
220C22×C4—148no—
221C22×C4—144no—
222C22×C4—148no—
223C22×C4—144no—
224C22×C4Q8×D714yes—
225C22×C4#116114yes—
226C22×C4C22×Dic718yes—
227C22×C4C22×Dic7116yes—
228C22×C4C22×Dic714yes—
229C22×C4C23×D718yes—
230C22×C4C23×D714yes—
231C22×C4C23×D712yes—
232C4⋊C4—1416no—
233C4⋊C4—2816no—
234C4⋊C4—2816no—
235C4⋊C4C22×Dic7164yes—
236C4⋊C4C22×Dic7132yes—
237C42—716no—
238C42—148no—
239C42—2816no—
240C42C22×Dic7116yes—
241C2×C7#233018yescomplemented
242C2×C7C22×C4×Q811yescharacteristic
243C2×C7C2×#52012yes—
244C2×C7C2×#54011yescharacteristic
245C2×C7C2×#54111yescharacteristic
246C2×C7C2×#54412yes—
247C23C2×Q8×D712yes—
248C23C2×#116112yes—
249C23#655314yes—
250C23#655414yes—
251C23C23×Dic712yes—
252C23C24×D711yescharacteristic
253C2×C4—232no—
254C2×C4—216no—
255C2×C4—28no—
256C2×C4—24no—
257C2×C4—28no—
258C2×C4—24no—
259C2×C4—78no—
260C2×C4—1416no—
261C2×C4—144no—
262C2×C4—144no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—148no—
266C2×C4—148no—
267C2×C4—144no—
268C2×C4—148no—
269C2×C4—144no—
270C2×C4—284no—
271C2×C4—284no—
272C2×C4—2816no—
273C2×C4Q8×Dic7116yes—
274C2×C4C2×Q8×D714yes—
275C2×C4C2×#116114yes—
276C2×C4C23×Dic718yes—
277C2×C4C23×Dic7116yes—
278C2×C4C23×Dic714yes—
279C7C2×#233011yescharacteristic, complemented, Sylow
280C22#9038018yes—
281C22#9038814yes—
282C22#9038914yes—
283C22C2×Q8×Dic714yes—
284C22#9051818yes—
285C22C22×Q8×D711yescharacteristic
286C22C22×#116111yescharacteristic
287C22C2×#655312yes—
288C22C2×#655412yes—
289C22C24×Dic711yescharacteristic
290C4—216no—
291C4—28no—
292C4—148no—
293C4—282no—
294C4—288no—
295C4—282no—
296C4C2×Q8×Dic718yes—
297C2#1174046818yesdirect product factor
298C2C2×#9038911yescharacteristic
299C2C2×#9038811yescharacteristic
300C2C2×#9038012yes—
301C2C22×Q8×Dic711yescharacteristic
302C2C2×#9051812yes—
subgroupG/H
centerC24C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC22×C7×#125C2
socle, prime radicalC24×C7C24
cosocleC23×C7C25
hypercenterC22×#125D7
hypocenterC7C2×#2330
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #11740468C2direct (N × H)
9. C2×C7×#125C4semidirect
38. C7×#125C2×C4semidirect
105. C2×#125Dic7semidirect
165. #125C2×Dic7semidirect
241. C2×C7#2330semidirect
279. C7C2×#2330semidirect
297. C2#11740468direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1252
#117404682
C2×#897572
C2×#897532
C2×C4⋊C4×Dic72
C2×#897072
C2×#897042
C2×#23307