Finite Group Databasebeta

← #13267014

Subgroups — #13267014

Order 1792.

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

normal subgroups only

This group has 396 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×#3186C211yesmaximal, characteristic
2#11728823C212yesmaximal, complemented
3#11727693C211yesmaximal, characteristic, complemented
4#11725580C212yesmaximal, complemented
5#11725479C211yesmaximal, characteristic, complemented
6C2×C7×#121—21no—
7C7×#480—22no—
8C7×#466—21no—
9C7×#466C2212yes—
10C7×#447—22no—
11C2×C7×C22⋊C8C2211yescharacteristic
12#89984—21no—
13C2×#6392C2211yescharacteristic, complemented
14#89357—22no—
15#89346—21no—
16#89346C2211yescharacteristic, complemented
17C22⋊(C7⋊C16)C2212yescomplemented
18#89285—22no—
19#12281—71nomaximal, Sylow
20C22×C7×D4—21no—
21C22×C7×D4—41no—
22C23×C4×C7D411yescharacteristic
23C2×C7×D8—21no—
24C2×C7×D8D412yes—
25C22×C7×C8D411yescharacteristic
26C22×C7×C8C2311yescharacteristic
27C7×#121—22no—
28C7×#121—22no—
29C7×#121—41no—
30C7×#121—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C22⋊C4—41no—
33C2×C22⋊Dic7—41no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C7×#107—21no—
37C7×#107D412yes—
38C7×#102—22no—
39C7×#102—22no—
40C7×C22⋊C8—22no—
41C7⋊#95—41no—
42#6392—22no—
43C2×C7⋊C16—22no—
44#3186D711yescharacteristic
45#2233—72no—
46#2076—71no—
47#1347—72no—
48#1314—71no—
49C24×C7—41no—
50C24×C7D811yescharacteristic
51C2×C7×D4—22no—
52C2×C7×D4—41no—
53C2×C7×D4—42no—
54C2×C7×D4—42no—
55C2×C7×D4—81no—
56C2×C7×D4—81no—
57C2×C7×D4—82no—
58C2×C7×D4—82no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7D811yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×Dic7—41no—
67C22×Dic7—41no—
68C7×D8—42no—
69C7×D8—81no—
70C2×C7×C8—21no—
71C2×C7×C8—41no—
72C2×C7×C8—41no—
73C2×C7×C8C2×D412yes—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—41no—
76C7×C22⋊C4—42no—
77C7×C22⋊C4—42no—
78C22⋊Dic7—82no—
79C4⋊Dic7—41no—
80C4⋊Dic7—41no—
81C4⋊Dic7—41no—
82C4⋊Dic7—41no—
83C7⋊C16—42no—
84C2×#121—21no—
85#480—22no—
86#466—21no—
87#466C2×D712yes—
88#447—22no—
89C2×#107—71no—
90C2×C22⋊C8C2×D711yescharacteristic
91#380—141no—
92#367—142no—
93#357—142no—
94C22⋊C16—72no—
95#340—71no—
96#340—141no—
97C23×C7—21no—
98C23×C7—21no—
99C23×C7—41no—
100C23×C7—41no—
101C23×C7—42no—
102C23×C7—82no—
103C23×C7—82no—
104C23×C7—82no—
105C23×C7SD3212yes—
106C23×C7C2×D811yescharacteristic
107C7×D4—42no—
108C7×D4—42no—
109C7×D4—82no—
110C7×D4—84no—
111C7×D4—82no—
112C7×D4—82no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—41no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7—81no—
121C2×C4×C7—82no—
122C2×C4×C7#12011yescharacteristic
123C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
124C2×Dic7—81no—
125C2×Dic7—81no—
126C2×Dic7—81no—
127C2×Dic7—81no—
128C2×Dic7—81no—
129C2×Dic7—81no—
130C7×C8—22no—
131C7×C8—41no—
132C22×D4—21no—
133C22×D4—41no—
134C23×C4C22⋊D711yescharacteristic
135C2×D8—21no—
136C2×D8C22⋊D712yes—
137C22×C8C22⋊D711yescharacteristic
138C22×C8C22×D711yescharacteristic
139#121—22no—
140#121—22no—
141#121—41no—
142#121—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—141no—
145C2×C4⋊C4—141no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—281no—
148C2×C16—142no—
149#107—21no—
150#107—142no—
151#107C22⋊D712yes—
152#102—22no—
153#102—22no—
154C22⋊C8—22no—
155#95—281no—
156C22×C7—21no—
157C22×C7—21no—
158C22×C7—22no—
159C22×C7—42no—
160C22×C7—41no—
161C22×C7—41no—
162C22×C7—81no—
163C22×C7—82no—
164C22×C7—82no—
165C22×C7—82no—
166C22×C7—164no—
167C22×C7#44711yescharacteristic
168C22×C7C2×SD3211yescharacteristic
169C22×C7#50911yescharacteristic
170C4×C7—21no—
171C4×C7—41no—
172C4×C7—82no—
173C4×C7#45511yescharacteristic
174C4×C7#45811yescharacteristic
175Dic7—161no—
176Dic7—161no—
177C24—41no—
178C24#114111yescharacteristic
179C2×D4—22no—
180C2×D4—41no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—82no—
186C2×D4—82no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—281no—
193C22×C4—281no—
194C22×C4#114111yescharacteristic
195C22×C4C2×C22⋊D711yescharacteristic
196D8—42no—
197D8—81no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8C2×C22⋊D712yes—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—281no—
205C4⋊C4—281no—
206C4⋊C4—281no—
207C4⋊C4—281no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—562no—
211C16—282no—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7—42no—
215C2×C7—162no—
216C2×C7#160711yescharacteristic
217C2×C7#218711yescharacteristic
218C2×C7#219011yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23#640112yes—
228C23C2×#114111yescharacteristic
229D4—42no—
230D4—42no—
231D4—82no—
232D4—84no—
233D4—82no—
234D4—82no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—82no—
244C2×C4—561no—
245C2×C4—561no—
246C2×C4—561no—
247C2×C4—561no—
248C2×C4—561no—
249C2×C4—561no—
250C2×C4#649511yescharacteristic
251C2×C4#651611yescharacteristic
252C8—22no—
253C8—41no—
254C7#1228111yescharacteristic, complemented, Sylow
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—164no—
266C22#8991911yescharacteristic
267C22C2×#640111yescharacteristic
268C22#8998911yescharacteristic
269C4—21no—
270C4—41no—
271C4—82no—
272C4—1121no—
273C4—1121no—
274C4#9001411yescharacteristic
275C4#9001511yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—162no—
280C2#1172959711yescharacteristic
281C2#1172979011yescharacteristic
282C2#1172986611yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
Fitting, prime radicalC7×#3186C2
socleC22×C7#447
hypercenter#3186D7
hypocenterC7#12281
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3186

Complements

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

NH ≅ G/Ntype
2. #11728823C2semidirect
3. #11727693C2semidirect
4. #11725580C2semidirect
5. #11725479C2semidirect
13. C2×#6392C22semidirect
16. #89346C22semidirect
17. C22⋊(C7⋊C16)C22semidirect
254. C7#12281semidirect

Maximal subgroups

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

subgroupindex
C7×#31862
#117288232
#117276932
#117255802
#117254792
#122817