Finite Group Databasebeta

← #13263299

Subgroups — #13263299

Order 1792.

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

normal subgroups only

This group has 334 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
1#11736101C211yesmaximal, characteristic
2#11726869C212yesmaximal, complemented
3C7×#1321C211yesmaximal, characteristic, complemented
4#11725496C211yesmaximal, characteristic, complemented
5#11725336C212yesmaximal, complemented
6#90519C411yescharacteristic
7C2×#6511C411yescharacteristic
8C22×#1131C411yescharacteristic
9C7×#408C2211yescharacteristic
10#89898C2211yescharacteristic
11#89894C2211yescharacteristic
12#89874C411yescharacteristic
13#89502—22no—
14#89501—22no—
15C7×#349C2212yes—
16#89348C2212yes—
17#89307—22no—
18#89306—22no—
19#11538—71nomaximal, Sylow
20C2×C7×#41C2×C411yescharacteristic
21C22×C4×D7C2×C411yescharacteristic
22C2×C7×M16C2311yescharacteristic
23C22×C7×C8C2×C411yescharacteristic
24#6511—42no—
25#6509—21no—
26#6501—21no—
27#6500—21no—
28#6498—21no—
29C4×C22⋊D7—22no—
30#6489C2×C411yescharacteristic
31C2×#1136—21no—
32C2×#1136C2×C411yescharacteristic
33C22×C7⋊C8—21no—
34C22×C7⋊C8C2×C411yescharacteristic
35C2×#1131—21no—
36C2×#1131—21no—
37C2×#1131—42no—
38C7×M32—22no—
39C7×C22⋊C8—22no—
40(C2×D7)⋊C8—22no—
41#6389D412yes—
42Dic7⋊C8—22no—
43C8×Dic7D412yes—
44#6386—22no—
45#2930—71no—
46#2166—72no—
47#1321—71no—
48#1321D711yescharacteristic, complemented
49#1320—72no—
50C23×D7M1611yescharacteristic
51C7×#41—41no—
52C7×#41—41no—
53C2×C7×Q8M1611yescharacteristic
54C2×C7×D4—21no—
55C2×C7×D4M1611yescharacteristic
56C22×C4×C7—21no—
57C22×C4×C7C22×C411yescharacteristic
58C2×C22⋊D7—22no—
59C22×Dic7M1611yescharacteristic
60C2×C4×D7—21no—
61C2×C4×D7—21no—
62C2×C4×D7—41no—
63C2×C4×D7—41no—
64C7×M16—41no—
65C2×C7×C8—21no—
66C2×C7×C8—41no—
67C2×C7×C8C2×D412yes—
68C22⋊Dic7—22no—
69C22⋊Dic7—21no—
70(C2×D7)⋊C4—22no—
71C4⋊Dic7—21no—
72Dic7⋊C4—22no—
73C4×Dic7C22⋊C412yes—
74#1136—42no—
75#1136—41no—
76C2×C7⋊C8—22no—
77C2×C7⋊C8—21no—
78C2×C7⋊C8—42no—
79C2×C7⋊C8C22⋊C411yescharacteristic
80C2×C7⋊C8C22⋊C411yescharacteristic
81#1131—42no—
82#1131—42no—
83C7×C16—42no—
84C7⋊C16—42no—
85C2×#131—71no—
86C22×M16—71no—
87#525—71no—
88#436—142no—
89#435—142no—
90#424—71no—
91#408C2×D711yescharacteristic
92#407—71no—
93#405—71no—
94#364—142no—
95#349—72no—
96#349C2×D712yes—
97#347—142no—
98C23×C7—21no—
99C23×C7C2×M1611yescharacteristic
100C22×D7—21no—
101C22×D7—41no—
102C22×D7—81no—
103C7×Q8—41no—
104C7×D4—41no—
105C7×D4—42no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—41no—
110C2×C4×C7—41no—
111C2×C4×C7C2×C22⋊C411yescharacteristic
112C2×C4×C7C2×M1611yescharacteristic
113C22⋊D7—82no—
114C2×Dic7—22no—
115C2×Dic7—21no—
116C2×Dic7—41no—
117C4×D7—42no—
118C4×D7—41no—
119C4×D7—41no—
120C7×C8—22no—
121C7×C8—41no—
122C7⋊C8—21no—
123C7⋊C8—41no—
124C7⋊C8#10012yes—
125C2×#41C4×D711yescharacteristic
126C23×C4—71no—
127#131—282no—
128C2×M16—71no—
129C2×M16—141no—
130C2×M16—141no—
131C2×M16—141no—
132C2×M16—282no—
133C2×M16C22×D711yescharacteristic
134C22×C8—71no—
135C22×C8—141no—
136C22×C8C4×D711yescharacteristic
137#123—141no—
138C22⋊Q8—141no—
139#121—141no—
140#120—141no—
141C4×D4—142no—
142#117—71no—
143M32—22no—
144M32—142no—
145C4⋊C8—142no—
146C22⋊C8—22no—
147C22⋊C8—142no—
148#97—72no—
149C4×C8—72no—
150C22×C7—21no—
151C22×C7—41no—
152C22×C7—42no—
153C22×C7—41no—
154C22×C7#40711yescharacteristic
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C2×D7—82no—
159C4×C7—21no—
160C4×C7—41no—
161C4×C7#41011yescharacteristic
162C4×C7C2×#10011yescharacteristic
163Dic7—42no—
164Dic7—41no—
165C24—71no—
166#41—41no—
167#41—41no—
168C2×Q8#113111yescharacteristic
169C2×D4—21no—
170C2×D4—142no—
171C2×D4#113111yescharacteristic
172C22×C4—21no—
173C22×C4—71no—
174C22×C4—141no—
175C22×C4—141no—
176C22×C4—281no—
177C22×C4—281no—
178C22×C4C2×C4×D711yescharacteristic
179M16—41no—
180M16—282no—
181M16—282no—
182M16—282no—
183M16—281no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—71no—
187C2×C8—71no—
188C2×C8—142no—
189C2×C8—141no—
190C2×C8—282no—
191C2×C8D4×D712yes—
192C4⋊C4—142no—
193C4⋊C4—141no—
194C22⋊C4—142no—
195C22⋊C4—142no—
196C22⋊C4—141no—
197C42—72no—
198C16—42no—
199C16—282no—
200C2×C7—21no—
201C2×C7—41no—
202C2×C7—81no—
203C2×C7#146611yescharacteristic
204D7—81no—
205C23—21no—
206C23—141no—
207C23—281no—
208C23—561no—
209C23D7×M1611yescharacteristic
210Q8—41no—
211D4—41no—
212D4—42no—
213D4—562no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—142no—
220C2×C4—141no—
221C2×C4—282no—
222C2×C4—281no—
223C2×C4—281no—
224C2×C4—281no—
225C2×C4D7×C22⋊C411yescharacteristic
226C2×C4C2×#113111yescharacteristic
227C8—22no—
228C8—41no—
229C8—72no—
230C8—141no—
231C8—281no—
232C7#1153811yescharacteristic, complemented, Sylow
233C22—21no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—281no—
238C22—281no—
239C22—281no—
240C22—562no—
241C22#8949811yescharacteristic
242C4—21no—
243C4—41no—
244C4—282no—
245C4—281no—
246C4#8951211yescharacteristic
247C4D7×#10011yescharacteristic
248C2—21no—
249C2—41no—
250C2—81no—
251C2—561no—
252C2#1172585711yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1321C2
socleC2×C7#1466
prime radical#90519C4
cosocleC2×C7×M16C23
hypercenter#1321D7
hypocenterC7#11538
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #11726869C2semidirect
3. C7×#1321C2semidirect
4. #11725496C2semidirect
5. #11725336C2semidirect
48. #1321D7semidirect
232. C7#11538semidirect

Maximal subgroups

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

subgroupindex
#117361012
#117268692
C7×#13212
#117254962
#117253362
#115387