Finite Group Databasebeta

← #13256464

Subgroups — #13256464

Order 1792.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C7×#2862C211yesmaximal, characteristic, complemented
2#11740661C211yesmaximal, characteristic
3#11740660C211yesmaximal, characteristic
4#11729529C211yesmaximal, characteristic, complemented
5#11729527C211yesmaximal, characteristic, complemented
6#11727149C211yesmaximal, characteristic
7#11725417C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10#90168C2211yescharacteristic
11C7×#391—22no—
12C7×#391—22no—
13C7×#382—22no—
14C43×C7C2211yescharacteristic
15#89860—21no—
16#89860—21no—
17#89859—22no—
18#89856—22no—
19#89851—21no—
20#89850—21no—
21#89740—22no—
22#89739—22no—
23C2×#6379—22no—
24#89697C2211yescharacteristic
25#89697C2211yescharacteristic
26#89696C2211yescharacteristic
27#89342—22no—
28#89342—22no—
29#10286—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8—21no—
32C2×#1157—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4D412yes—
44C2×C7×C4⋊C4D412yes—
45C2×C42×C7—21no—
46C2×C42×C7—21no—
47C2×C42×C7D412yes—
48C2×C42×C7C2311yescharacteristic
49#6493—21no—
50#6493—21no—
51C2×#1136—21no—
52#6438—21no—
53#6437D411yescharacteristic
54C4×D28—21no—
55C4×D28—21no—
56#6432D411yescharacteristic
57C7×#95—28no—
58C22⋊(C7⋊C8)—21no—
59C22⋊(C7⋊C8)—21no—
60#6383—24no—
61#6383—24no—
62#6382—24no—
63#6381—24no—
64#6379—44no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)D412yes—
68C4⋊(C7⋊C8)D412yes—
69C4×C7⋊C8—21no—
70C4×C7⋊C8—21no—
71#3227—71no—
72#3226—71no—
73#2862D711yescharacteristic, complemented
74#1962—71no—
75#1549—71no—
76#1546—71no—
77#1276—71no—
78C2×C7×Q8—44no—
79C2×C7×Q8—41no—
80C2×C7×Q8—41no—
81C2×C7×Q8—41no—
82C2×C7×Q8—41no—
83C2×C7×Q8—42no—
84C2×C7×Q8—42no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C2×C22⋊D7—41no—
92#1157—42no—
93#1157—42no—
94C2×D28—21no—
95C2×D28—21no—
96C2×D28—21no—
97C2×C4×D7—41no—
98C2×Dic14—21no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C42×C7—24no—
108C42×C7—22no—
109C42×C7—22no—
110C42×C7—42no—
111C42×C7C2×D411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7C2×D411yescharacteristic
114C42×C7#4111yescharacteristic
115(C2×D7)⋊C4—41no—
116(C2×D7)⋊C4—41no—
117C4⋊Dic7—21no—
118C4⋊Dic7—21no—
119#1136—42no—
120C2×C7⋊C8—22no—
121C2×C7⋊C8—22no—
122C2×C7⋊C8—41no—
123#532—71no—
124C2×C4⋊Q8C2×D711yescharacteristic
125C2×C4⋊Q8C2×D711yescharacteristic
126#500—141no—
127#498—141no—
128#486—141no—
129#486—141no—
130#481—142no—
131#480—142no—
132#476—142no—
133#474—142no—
134#431—71no—
135#431—71no—
136#423—71no—
137C2×#104—142no—
138#391—22no—
139#391—22no—
140#382—22no—
141C43C2×D711yescharacteristic
142#332—142no—
143#332—142no—
144C23×C7#12011yescharacteristic
145C22×D7—41no—
146C7×Q8—82no—
147C7×Q8—82no—
148C7×Q8—82no—
149C7×Q8—81no—
150C7×Q8—81no—
151C2×C4×C7—22no—
152C2×C4×C7—22no—
153C2×C4×C7—24no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—44no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—44no—
160C2×C4×C7#12011yescharacteristic
161C2×C4×C7#12011yescharacteristic
162C2×C4×C7#12111yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12711yescharacteristic
166C2×C4×C7#13512yes—
167C2×C4×C7AGL(1,ℤ/8)12yes—
168C22⋊D7—82no—
169C2×Dic7—41no—
170D28—42no—
171D28—42no—
172D28—42no—
173C4×D7—81no—
174C4×D7—81no—
175Dic14—42no—
176C7⋊C8—42no—
177C7⋊C8—42no—
178C7⋊C8—81no—
179C2×#41—141no—
180C22×Q8—21no—
181C22×Q8—21no—
182C2×M16—141no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—71no—
192#127—71no—
193#124—141no—
194#121—141no—
195#121—141no—
196C4×D4—141no—
197C4×D4—141no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22⋊D712yes—
201C2×C4⋊C4C22⋊D712yes—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C22⋊D712yes—
205C2×C42C22×D711yescharacteristic
206#107—144no—
207#106—144no—
208C4⋊C8—72no—
209C4⋊C8—72no—
210C4⋊C8—141no—
211C4⋊C8—141no—
212#104—284no—
213#102—144no—
214#102—144no—
215C22⋊C8—141no—
216C22⋊C8—141no—
217C4×C8—141no—
218C4×C8—141no—
219#95—28no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7#39211yescharacteristic
223C22×C7#44912yes—
224C2×D7—81no—
225C2×D7—81no—
226C2×D7—81no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—44no—
230C4×C7—82no—
231C4×C7—82no—
232C4×C7#45412yes—
233C4×C7#45612yes—
234C4×C7#46312yes—
235C4×C7#49511yescharacteristic
236C4×C7#49611yescharacteristic
237Dic7—81no—
238#41—282no—
239#41—282no—
240C2×Q8—44no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—141no—
248C2×D4—141no—
249C2×D4—141no—
250C2×D4—141no—
251C2×D4—281no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—281no—
256C22×C4C2×C22⋊D711yescharacteristic
257C22×C4C2×C22⋊D711yescharacteristic
258C22×C4C2×C22⋊D711yescharacteristic
259M16—282no—
260C2×C8—142no—
261C2×C8—142no—
262C2×C8—281no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C22⋊C4—281no—
274C22⋊C4—281no—
275C42—24no—
276C42—22no—
277C42—22no—
278C42—42no—
279C42D4×D711yescharacteristic
280C42D4×D711yescharacteristic
281C42D4×D711yescharacteristic
282C42#115911yescharacteristic
283C2×C7—22no—
284C2×C7#200311yescharacteristic
285C2×C7#200511yescharacteristic
286C2×C7#201211yescharacteristic
287D7—161no—
288C23—281no—
289C23#651611yescharacteristic
290Q8—82no—
291Q8—82no—
292Q8—81no—
293Q8—81no—
294Q8—82no—
295Q8—282no—
296D4—282no—
297D4—282no—
298D4—282no—
299D4—562no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—281no—
310C2×C4—561no—
311C2×C4—561no—
312C2×C4#650011yescharacteristic
313C2×C4#650011yescharacteristic
314C2×C4#650111yescharacteristic
315C2×C4#650211yescharacteristic
316C2×C4#650211yescharacteristic
317C2×C4#650311yescharacteristic
318C2×C4#651212yes—
319C2×C4#651312yes—
320C8—282no—
321C8—282no—
322C8—561no—
323C7#1028611yescharacteristic, complemented, Sylow
324C22—21no—
325C22—21no—
326C22—561no—
327C22—561no—
328C22—561no—
329C22#8999511yescharacteristic
330C22#9001012yes—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—82no—
335C4—82no—
336C4—561no—
337C4#8982912yes—
338C4#8983011yescharacteristic
339C4#8983111yescharacteristic
340C4#8983412yes—
341C4#8986712yes—
342C2—22no—
343C2—1121no—
344C2#1172948211yescharacteristic
345C2#1172948411yescharacteristic
346C2#1172954911yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2862C2
socleC22×C7#392
prime radical#90168C22
hypercenter#2862D7
hypocenterC7#10286
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
1. C7×#2862C2semidirect
4. #11729529C2semidirect
5. #11729527C2semidirect
7. #11725417C2semidirect
73. #2862D7semidirect
323. C7#10286semidirect

Maximal subgroups

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

subgroupindex
C7×#28622
#117406612
#117406602
#117295292
#117295272
#117271492
#117254172
#102867