Finite Group Databasebeta

← C2×#11731162 #13525616

Subgroups — C2×#11731162 #13525616

Order 1792.

2446 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6453C212yesmaximal
2C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
3C2×#89757C212yesmaximal
4C2×#89700C212yesmaximal
5C2×#89707C212yesmaximal
6C2×#89707C211yesmaximal, characteristic
7C2×C4×C4⋊Dic7C211yesmaximal, characteristic
8#11731162C2116yesmaximal, direct product factor
9C2×#89433C214yesmaximal
10C22×C7×C4⋊C4C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C42×C7C2212yes—
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2214yes—
15C22×C4⋊Dic7C2212yes—
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2212yes—
20C22×C4⋊Dic7C2214yes—
21C22×Dic7⋊C4C2214yes—
22C22×C4×Dic7C2212yes—
23C2×#6453—216no—
24C2×#6453C22116yes—
25C4×C7×C4⋊C4C2218yes—
26C2×C7×#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2214yes—
29#89757C22116yes—
30#89707C22116yes—
31#89707C2218yes—
32C4×C4⋊Dic7C2218yes—
33#89700C22116yes—
34#89433C22132yes—
35C2×#2552—71nomaximal, Sylow
36C23×C4×C7C2312yes—
37C23×C4×C7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7C2312yes—
42C23×Dic7C2312yes—
43C23×Dic7C2314yes—
44C2×C7×C4⋊C4D418yes—
45C2×C7×C4⋊C4Q818yes—
46C2×C7×C4⋊C4C2314yes—
47C2×C7×C4⋊C4C2314yes—
48C2×C42×C7—28no—
49C2×C42×C7Q818yes—
50C2×C42×C7C2318yes—
51C2×C42×C7C2314yes—
52C2×C4⋊Dic7—216no—
53C2×C4⋊Dic7—28no—
54C2×C4⋊Dic7—24no—
55C2×C4⋊Dic7—24no—
56C2×C4⋊Dic7—28no—
57C2×C4⋊Dic7—44no—
58C2×C4⋊Dic7—48no—
59C2×C4⋊Dic7C23116yes—
60C2×C4⋊Dic7C2318yes—
61C2×C4⋊Dic7C2314yes—
62C2×C4⋊Dic7C2318yes—
63C2×C4⋊Dic7C2314yes—
64C2×C4⋊Dic7C2318yes—
65C2×C4⋊Dic7C23116yes—
66C2×Dic7⋊C4—216no—
67C2×Dic7⋊C4C23116yes—
68C2×C4×Dic7—28no—
69C2×C4×Dic7C2318yes—
70#6453—264no—
71C7×#95C2318yes—
72C7⋊#95C2318yes—
73C7⋊#95C23116yes—
74C22×#125—72no—
75#2552—716no—
76C2×#398—74no—
77C2×#389—72no—
78C2×#389—72no—
79C2×#384—72no—
80C2×#384—71no—
81C2×C4×C4⋊C4—71no—
82C2×C4×C4⋊C4D711yescharacteristic
83C24×C7C2411yescharacteristic
84C22×C4×C7—24no—
85C22×C4×C7—24no—
86C22×C4×C7—22no—
87C22×C4×C7—24no—
88C22×C4×C7—22no—
89C22×C4×C7—24no—
90C22×C4×C7C2×D418yes—
91C22×C4×C7C2×D414yes—
92C22×C4×C7C2×Q814yes—
93C22×C4×C7C2×Q818yes—
94C22×C4×C7C2×Q814yes—
95C22×C4×C7C2×Q814yes—
96C22×C4×C7C2×Q814yes—
97C22×C4×C7#4114yes—
98C22×C4×C7#4114yes—
99C22×C4×C7C2414yes—
100C22×C4×C7C2412yes—
101C22×C4×C7C2414yes—
102C22×C4×C7C2412yes—
103C22×C4×C7C2412yes—
104C22×Dic7—24no—
105C22×Dic7—24no—
106C22×Dic7—28no—
107C22×Dic7—24no—
108C22×Dic7—24no—
109C22×Dic7—28no—
110C22×Dic7—24no—
111C22×Dic7—28no—
112C22×Dic7—24no—
113C22×Dic7C2414yes—
114C22×Dic7C2414yes—
115C22×Dic7C2418yes—
116C7×C4⋊C4C2×D4116yes—
117C7×C4⋊C4C2×Q8116yes—
118C42×C7—216no—
119C42×C7C2×Q8116yes—
120C4⋊Dic7—232no—
121C4⋊Dic7—216no—
122C4⋊Dic7—28no—
123C4⋊Dic7—216no—
124C4⋊Dic7—28no—
125C4⋊Dic7—48no—
126C4⋊Dic7—416no—
127Dic7⋊C4—232no—
128C4×Dic7—216no—
129C2×#125—716no—
130C2×#125—1416no—
131C22×C4⋊C4—74no—
132C22×C4⋊C4—74no—
133C22×C4⋊C4—72no—
134C22×C4⋊C4—71no—
135C22×C4⋊C4—71no—
136C22×C4⋊C4—72no—
137C22×C4⋊C4—72no—
138C22×C4⋊C4—74no—
139C22×C4⋊C4C2×D711yescharacteristic
140C22×C4⋊C4C2×D711yescharacteristic
141C22×C42—72no—
142C22×C42C2×D712yes—
143C22×C42C2×D711yescharacteristic
144#398—732no—
145#389—716no—
146#389—716no—
147#384—716no—
148#384—78no—
149C4×C4⋊C4—78no—
150C4×C4⋊C4C2×D718yes—
151C2×#95—72no—
152C2×#95—74no—
153C2×#95C2×D712yes—
154C23×C7C22×D412yes—
155C23×C7C22×Q812yes—
156C23×C7C22×Q812yes—
157C23×C7C2×#4112yes—
158C23×C7C2×#4112yes—
159C23×C7H(5,𝔽2)12yes—
160C23×C72−1+412yes—
161C23×C7C2511yescharacteristic
162C2×C4×C7—28no—
163C2×C4×C7—216no—
164C2×C4×C7—216no—
165C2×C4×C7—24no—
166C2×C4×C7—24no—
167C2×C4×C7—22no—
168C2×C4×C7—24no—
169C2×C4×C7—22no—
170C2×C4×C7—24no—
171C2×C4×C7#125116yescomplemented
172C2×C4×C7C4⋊Q8116yes—
173C2×C4×C7C22×D418yes—
174C2×C4×C7C22×D414yes—
175C2×C4×C7C22×Q814yes—
176C2×C4×C7C22×Q818yes—
177C2×C4×C7C22×Q814yes—
178C2×C4×C7C22×Q814yes—
179C2×C4×C7C22×Q814yes—
180C2×C4×C7C2×#4114yes—
181C2×C4×C7C2×#4114yes—
182C2×Dic7—24no—
183C2×Dic7—24no—
184C2×Dic7—28no—
185C2×Dic7—24no—
186C2×Dic7—24no—
187C2×Dic7—28no—
188C2×Dic7—24no—
189C2×Dic7—28no—
190C2×Dic7—24no—
191C2×Dic7—48no—
192C2×Dic7—48no—
193C2×Dic7—416no—
194C23×C4—74no—
195C23×C4—72no—
196C23×C4—72no—
197C23×C4C22×D712yes—
198C23×C4C22×D711yescharacteristic
199C23×C4C22×D712yes—
200C23×C4C22×D711yescharacteristic
201C23×C4C22×D711yescharacteristic
202#125—1464no—
203C2×C4⋊C4—716no—
204C2×C4⋊C4—78no—
205C2×C4⋊C4—716no—
206C2×C4⋊C4—74no—
207C2×C4⋊C4—74no—
208C2×C4⋊C4—78no—
209C2×C4⋊C4—716no—
210C2×C4⋊C4—78no—
211C2×C4⋊C4—1416no—
212C2×C4⋊C4—1416no—
213C2×C4⋊C4—148no—
214C2×C4⋊C4—144no—
215C2×C4⋊C4—144no—
216C2×C4⋊C4—148no—
217C2×C4⋊C4—284no—
218C2×C4⋊C4—288no—
219C2×C4⋊C4Dic1418yes—
220C2×C4⋊C4D2818yes—
221C2×C4⋊C4C22×D714yes—
222C2×C4⋊C4C22×D714yes—
223C2×C42—28no—
224C2×C42—78no—
225C2×C42—148no—
226C2×C42Dic1418yes—
227C2×C42C22×D718yes—
228C2×C42C22×D714yes—
229#95—78no—
230#95—716no—
231#95C22×D718yes—
232C22×C7C2×#12514yes—
233C22×C7C2×C4⋊Q814yes—
234C22×C7#54414yes—
235C22×C7#54714yes—
236C22×C7#55014yes—
237C22×C7#55414yes—
238C22×C7#55714yes—
239C22×C7C23×D411yescharacteristic
240C22×C7C23×Q811yescharacteristic
241C22×C7C23×Q811yescharacteristic
242C22×C7C22×#4111yescharacteristic
243C22×C7C22×#4111yescharacteristic
244C22×C7C2×H(5,𝔽2)11yescharacteristic
245C22×C7C2×2−1+411yescharacteristic
246C4×C7—24no—
247C4×C7—28no—
248C4×C7—28no—
249C4×C7C2×#12518yescomplemented
250C4×C7C2×C4⋊Q818yes—
251Dic7—44no—
252Dic7—44no—
253Dic7—48no—
254C24C23×D711yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—74no—
262C22×C4—74no—
263C22×C4—78no—
264C22×C4—144no—
265C22×C4—144no—
266C22×C4—148no—
267C22×C4—144no—
268C22×C4—148no—
269C22×C4—144no—
270C22×C4—148no—
271C22×C4—144no—
272C22×C4—144no—
273C22×C4C2×Dic1414yes—
274C22×C4C2×Dic1418yes—
275C22×C4C2×Dic1414yes—
276C22×C4C2×Dic1414yes—
277C22×C4C2×Dic1414yes—
278C22×C4C2×D2818yes—
279C22×C4C2×D2814yes—
280C22×C4#115914yes—
281C22×C4#116114yes—
282C22×C4C23×D712yes—
283C22×C4C23×D714yes—
284C22×C4C23×D714yes—
285C22×C4C23×D712yes—
286C22×C4C23×D712yes—
287C4⋊C4—1432no—
288C4⋊C4—1432no—
289C4⋊C4—1416no—
290C4⋊C4—148no—
291C4⋊C4—148no—
292C4⋊C4—1416no—
293C4⋊C4—2816no—
294C4⋊C4—288no—
295C4⋊C4C2×Dic14116yes—
296C4⋊C4C2×D28116yes—
297C42—216no—
298C42—1416no—
299C42C2×Dic14116yes—
300C2×C7#255218yescomplemented
301C2×C7C22×#12511yescharacteristic
302C2×C7C22×C4⋊Q811yescharacteristic
303C2×C7C2×#54411yescharacteristic
304C2×C7C2×#54711yescharacteristic
305C2×C7C2×#55011yescharacteristic
306C2×C7C2×#55411yescharacteristic
307C2×C7C2×#55711yescharacteristic
308C23C22×Dic1412yes—
309C23C22×Dic1412yes—
310C23C22×D2812yes—
311C23C2×#115912yes—
312C23C2×#116112yes—
313C23#655312yes—
314C23#655412yes—
315C23C24×D711yescharacteristic
316C2×C4—28no—
317C2×C4—216no—
318C2×C4—216no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—144no—
326C2×C4—144no—
327C2×C4—148no—
328C2×C4—144no—
329C2×C4—144no—
330C2×C4—148no—
331C2×C4—144no—
332C2×C4—148no—
333C2×C4—144no—
334C2×C4—288no—
335C2×C4—288no—
336C2×C4—2816no—
337C2×C4#6432116yes—
338C2×C4#6453116yescomplemented
339C2×C4C22×Dic1414yes—
340C2×C4C22×Dic1418yes—
341C2×C4C22×Dic1414yes—
342C2×C4C22×Dic1414yes—
343C2×C4C22×Dic1414yes—
344C2×C4C22×D2818yes—
345C2×C4C22×D2814yes—
346C2×C4C2×#115914yes—
347C2×C4C2×#116114yes—
348C7C2×#255211yescharacteristic, complemented, Sylow
349C22C2×#643214yes—
350C22C2×#645314yes—
351C22#9021014yes—
352C22#9023414yes—
353C22#9024614yes—
354C22#9026114yes—
355C22#9026814yes—
356C22C23×Dic1411yescharacteristic
357C22C23×Dic1411yescharacteristic
358C22C23×D2811yescharacteristic
359C22C22×#115911yescharacteristic
360C22C22×#116111yescharacteristic
361C22C2×#655311yescharacteristic
362C22C2×#655411yescharacteristic
363C4—24no—
364C4—28no—
365C4—28no—
366C4—284no—
367C4—284no—
368C4—288no—
369C4C2×#643218yes—
370C4C2×#645318yescomplemented
371C2#1173116218yesdirect product factor
372C2C22×#643211yescharacteristic
373C2C22×#645311yescharacteristic
374C2C2×#9021011yescharacteristic
375C2C2×#9024611yescharacteristic
376C2C2×#9023411yescharacteristic
377C2C2×#9026811yescharacteristic
378C2C2×#9026111yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C4×C7×C4⋊C4C2
socle, prime radicalC24×C7C24
hypercenterC2×C4×C4⋊C4D7
hypocenterC7C2×#2552
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #11731162C2direct (N × H)
171. C2×C4×C7#125semidirect
249. C4×C7C2×#125semidirect
300. C2×C7#2552semidirect
338. C2×C4#6453semidirect
348. C7C2×#2552semidirect
370. C4C2×#6453semidirect
371. C2#11731162direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64532
C2×C4×C7×C4⋊C42
C2×#897572
C2×#897002
C2×#897072
C2×C4×C4⋊Dic72
#117311622
C2×#894332
C2×#25527