Finite Group Databasebeta

← #13267409

Subgroups — #13267409

Order 1792.

318 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 71 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×#3341C211yesmaximal, characteristic, complemented
2#11729135C211yesmaximal, characteristic, complemented
3#11728643C211yesmaximal, characteristic, complemented
4#11728601C211yesmaximal, characteristic, complemented
5#11725608C211yesmaximal, characteristic, complemented
6#11725569C211yesmaximal, characteristic, complemented
7D4⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
8C7×#555—21no—
9C7×#500C2211yescharacteristic
10C7×#482—22no—
11C7×#476—21no—
12C7×#466—21no—
13C4×C7×D8C2211yescharacteristic
14C7×C8×D4C2211yescharacteristic
15C7×#427—21no—
16C2×#6400—21no—
17#89787—21no—
18#89467C2211yescharacteristic
19#89286—21no—
20#89286—22no—
21#89280—21no—
22C4⋊(C7⋊C16)C2211yescharacteristic
23C4×C7⋊C16C2211yescharacteristic
24#89253C2211yescharacteristic
25#12343—71nomaximal, Sylow
26C2×C7×D8D411yescharacteristic
27C22×C7×C8—21no—
28C7×C4⋊Q8—21no—
29C7×#124—42no—
30C7×#121—22no—
31C7×#121—41no—
32C4×C7×D4—21no—
33C4×C7×D4D411yescharacteristic
34C7×#117—22no—
35C7×#117—41no—
36C2×#1141—41no—
37C2×D56—21no—
38C2×D56—21no—
39#6437—21no—
40C7×#107—21no—
41C7×#107D412yes—
42C7×#107D411yescharacteristic
43C7×#106—22no—
44C7×C4⋊C8D411yescharacteristic
45C7×#102—22no—
46C7×#102—21no—
47C7×C22⋊C8—21no—
48C4×C7×C8C2311yescharacteristic
49#6400—42no—
50C2×C7⋊C16—21no—
51C2×C7⋊C16—21no—
52#6383—41no—
53C4⋊(C7⋊C8)—21no—
54#3341D711yescharacteristic, complemented
55#2240—71no—
56#2206—71no—
57#1667—71no—
58#1365—71no—
59#1331—71no—
60D4⋊C16—71no—
61C2×C7×Q8—41no—
62C2×C7×D4—21no—
63C2×C7×D4—42no—
64C2×C7×D4D811yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—41no—
67C2×D28—41no—
68C2×D28—41no—
69C2×D28—41no—
70C7×D8—21no—
71C7×D8—21no—
72C2×C7×C8—21no—
73C2×C7×C8—41no—
74C2×C7×C8—41no—
75C2×C7×C8C2×D411yescharacteristic
76C2×C7×C8C2×D411yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4D811yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—42no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—41no—
85C42×C7—42no—
86C42×C7C2×D411yescharacteristic
87#1141—82no—
88C2×C7⋊C8—41no—
89D56—42no—
90D56—41no—
91D56—41no—
92C7⋊C16—41no—
93C7⋊C16—41no—
94#555—21no—
95C2×D16—141no—
96#500C2×D711yescharacteristic
97#493—71no—
98#482—22no—
99#476—21no—
100#466—21no—
101#459—141no—
102C4×D8C2×D711yescharacteristic
103C8×D4C2×D711yescharacteristic
104#427—21no—
105C4⋊C16—71no—
106#357—141no—
107#357—142no—
108C4×C16—71no—
109#335—71no—
110#331—141no—
111C23×C7—21no—
112C23×C7—41no—
113C22×D7—81no—
114C7×Q8—161no—
115C7×D4—21no—
116C7×D4—21no—
117C7×D4—41no—
118C7×D4—41no—
119C7×D4—162no—
120C2×C4×C7—21no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—42no—
125C2×C4×C7—81no—
126C2×C4×C7—81no—
127C2×C4×C7#12011yescharacteristic
128C2×C4×C7C2×D811yescharacteristic
129C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
130D28—81no—
131D28—81no—
132D28—81no—
133D28—81no—
134D28—161no—
135C7×C8—21no—
136C7×C8—41no—
137C7×C8#13511yescharacteristic
138C7×C8AGL(1,ℤ/8)11yescharacteristic
139C7⋊C8—81no—
140C2×D8—141no—
141C2×D8—141no—
142C2×D8—281no—
143C2×D8C22⋊D711yescharacteristic
144C22×C8—21no—
145C4⋊Q8—21no—
146#127—141no—
147#124—42no—
148#121—22no—
149#121—41no—
150C4×D4—21no—
151C4×D4C22⋊D711yescharacteristic
152#117—22no—
153#117—41no—
154D16—282no—
155C2×C16—141no—
156C2×C16—141no—
157#107—21no—
158#107C22⋊D711yescharacteristic
159#107C22⋊D712yes—
160#106—22no—
161C4⋊C8—141no—
162C4⋊C8C22⋊D711yescharacteristic
163#102—22no—
164#102—21no—
165#102—281no—
166C22⋊C8—21no—
167C4×C8C22×D711yescharacteristic
168C22×C7—41no—
169C22×C7—41no—
170C22×C7—41no—
171C22×C7—81no—
172C22×C7—81no—
173C22×C7—81no—
174C22×C7#44711yescharacteristic
175C2×D7—161no—
176C2×D7—161no—
177C2×D7—161no—
178C4×C7—21no—
179C4×C7—41no—
180C4×C7—81no—
181C4×C7—82no—
182C4×C7#44911yescharacteristic
183C4×C7#45411yescharacteristic
184C4×C7#50811yescharacteristic
185C4×C7#50911yescharacteristic
186C2×Q8—41no—
187C2×D4—21no—
188C2×D4—42no—
189C2×D4—281no—
190C2×D4—281no—
191C2×D4—281no—
192C2×D4#114111yescharacteristic
193C22×C4—21no—
194C22×C4—41no—
195D8—21no—
196D8—21no—
197D8—282no—
198D8—281no—
199D8—281no—
200D8—562no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—281no—
205C2×C8C2×C22⋊D711yescharacteristic
206C2×C8C2×C22⋊D711yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4#114111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—41no—
215C42—42no—
216C42C2×C22⋊D711yescharacteristic
217C16—281no—
218C16—281no—
219C2×C7—81no—
220C2×C7—161no—
221C2×C7#162511yescharacteristic
222C2×C7#218611yescharacteristic
223C2×C7#219211yescharacteristic
224D7—321no—
225C23—21no—
226C23—41no—
227C23—561no—
228Q8—161no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—162no—
234D4—561no—
235D4—561no—
236D4—561no—
237D4—561no—
238D4—1121no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4C2×#114111yescharacteristic
247C2×C4#649511yescharacteristic
248C2×C4#651611yescharacteristic
249C8—21no—
250C8—41no—
251C8—561no—
252C8#651211yescharacteristic
253C8#651311yescharacteristic
254C7#1234311yescharacteristic, complemented, Sylow
255C22—41no—
256C22—41no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—1121no—
262C22—1121no—
263C22—1121no—
264C22#8998911yescharacteristic
265C4—21no—
266C4—41no—
267C4—81no—
268C4—82no—
269C4#8996511yescharacteristic
270C4#8996611yescharacteristic
271C4#9001011yescharacteristic
272C4#9001711yescharacteristic
273C2—81no—
274C2—161no—
275C2—2241no—
276C2#1172966411yescharacteristic
277C2#1172983611yescharacteristic
278C2#1172987611yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#3341C2
socleC22×C7#447
hypercenter#3341D7
hypocenterC7#12343
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3341

Complements

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

NH ≅ G/Ntype
1. C7×#3341C2semidirect
2. #11729135C2semidirect
3. #11728643C2semidirect
4. #11728601C2semidirect
5. #11725608C2semidirect
6. #11725569C2semidirect
7. D4⋊(C7⋊C16)C2semidirect
54. #3341D7semidirect
254. C7#12343semidirect

Maximal subgroups

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

subgroupindex
C7×#33412
#117291352
#117286432
#117286012
#117256082
#117255692
D4⋊(C7⋊C16)2
#123437