Finite Group Databasebeta

← #13263005

Subgroups — #13263005

Order 1792.

314 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 107 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#11735900C211yesmaximal, characteristic
2C7×#2162C211yesmaximal, characteristic, complemented
3#11727498C211yesmaximal, characteristic, complemented
4#11725331C212yesmaximal, complemented
5#11725329C212yesmaximal
6C4×#1157C411yescharacteristic
7C2×C4×C7×C8C2211yescharacteristic
8#89882C411yescharacteristic
9#89882C2211yescharacteristic
10#89874C411yescharacteristic
11#89874C2211yescharacteristic
12C4×#1136C411yescharacteristic
13#89462—21no—
14#89462—21no—
15#89457—21no—
16#89457—21no—
17C7×C4⋊C16—21no—
18C7×C22⋊C16C2212yes—
19C4×C7×C16—21no—
20C22⋊(C7⋊C16)C2212yes—
21C4⋊(C7⋊C16)—21no—
22C4×C7⋊C16—21no—
23#11462—71nomaximal, Sylow
24C2×#1157C2×C411yescharacteristic
25C22×C7×C8C2×C411yescharacteristic
26C22×C7×C8C2311yescharacteristic
27C2×C42×C7C2×C411yescharacteristic
28C4×C22⋊D7—21no—
29C4×C22⋊D7—21no—
30#6489C2×C411yescharacteristic
31C2×#1136C2×C411yescharacteristic
32C2×#1136C2×C411yescharacteristic
33C4×D28C811yescharacteristic
34C4×D28C811yescharacteristic
35#6435—21no—
36C42×D7—21no—
37C4×Dic14C811yescharacteristic
38C4×Dic14C811yescharacteristic
39C2×C7×C16—22no—
40C4×C7×C8D411yescharacteristic
41C4×C7×C8D411yescharacteristic
42C4×C7×C8D411yescharacteristic
43C4×C7×C8D411yescharacteristic
44(C2×D7)⋊C8—21no—
45(C2×D7)⋊C8—21no—
46(C2×D7)⋊C8—21no—
47(C2×D7)⋊C8—21no—
48Dic7⋊C8—21no—
49Dic7⋊C8—21no—
50Dic7⋊C8—21no—
51Dic7⋊C8—21no—
52C2×C7⋊C16—22no—
53#6377—21no—
54C4×C7⋊C8—21no—
55#2922—71no—
56#2162—71no—
57#2162D711yescharacteristic, complemented
58#1323—72no—
59#1315—72no—
60C22×C4×C7M1611yescharacteristic
61C22×C4×C7M1611yescharacteristic
62C22×C4×C7C22×C411yescharacteristic
63C2×C22⋊D7—21no—
64#1157—41no—
65#1157—41no—
66C2×D28C2×C811yescharacteristic
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×Dic14C2×C811yescharacteristic
71C2×C7×C8—41no—
72C2×C7×C8—41no—
73C2×C7×C8C22⋊C411yescharacteristic
74C2×C7×C8C22⋊C411yescharacteristic
75C2×C7×C8C2×D411yescharacteristic
76C2×C7×C8C2×D411yescharacteristic
77C42×C7C22⋊C411yescharacteristic
78C42×C7C22⋊C411yescharacteristic
79C42×C7C2×C811yescharacteristic
80C42×C7C2×C811yescharacteristic
81C22⋊Dic7—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84C4⋊Dic7C2×C811yescharacteristic
85C4⋊Dic7C2×C811yescharacteristic
86Dic7⋊C4—21no—
87Dic7⋊C4—21no—
88C4×Dic7—21no—
89C4×Dic7—21no—
90C4×Dic7—21no—
91#1136—41no—
92#1136—41no—
93C2×C7⋊C8—21no—
94C2×C7⋊C8—21no—
95C7×C16—42no—
96C7⋊C16—42no—
97C4×#41—71no—
98#433—141no—
99#433—141no—
100#431—141no—
101#431—141no—
102#424—71no—
103#424—71no—
104#408—71no—
105#408—71no—
106C4×M16—71no—
107C2×C4×C8C2×D711yescharacteristic
108C4⋊C16—21no—
109C4⋊C16—141no—
110C22⋊C16—72no—
111C22⋊C16C2×D712yes—
112C4×C16—21no—
113C4×C16—141no—
114C23×C7C2×M1611yescharacteristic
115C22×D7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7C22⋊C811yescharacteristic
121C2×C4×C7C22⋊C811yescharacteristic
122C2×C4×C7C22⋊C811yescharacteristic
123C2×C4×C7C22⋊C811yescharacteristic
124C2×C4×C7C2×C22⋊C411yescharacteristic
125C2×C4×C7C22×C811yescharacteristic
126C22⋊D7—42no—
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—21no—
130D28—41no—
131C4×D7—42no—
132C4×D7—42no—
133C4×D7—41no—
134C4×D7—41no—
135Dic14—41no—
136C7×C8—21no—
137C7×C8—21no—
138C7×C8—21no—
139C7×C8—21no—
140C7⋊C8—41no—
141C7⋊C8—41no—
142C2×#41—71no—
143C2×M16—71no—
144C2×M16—71no—
145C22×C8C4×D711yescharacteristic
146C22×C8C22×D711yescharacteristic
147C4×Q8—71no—
148C4×Q8—71no—
149C4×D4—71no—
150C4×D4—71no—
151C4×D4—141no—
152C4×D4—141no—
153#117—71no—
154#117—141no—
155C2×C42—141no—
156C2×C42C4×D711yescharacteristic
157C2×C16—22no—
158C2×C16—142no—
159C4⋊C8—141no—
160C4⋊C8—141no—
161C4⋊C8—141no—
162C4⋊C8—141no—
163C22⋊C8—141no—
164C22⋊C8—141no—
165C22⋊C8—141no—
166C22⋊C8—141no—
167#97—141no—
168C4×C8—141no—
169C4×C8D2811yescharacteristic
170C4×C8D2811yescharacteristic
171C4×C8C22⋊D711yescharacteristic
172C4×C8C22⋊D711yescharacteristic
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7C2×C22⋊C811yescharacteristic
177C2×D7—41no—
178C2×D7—41no—
179C2×D7—41no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—21no—
183C4×C7#35012yes—
184C4×C7#35012yes—
185C4×C7#41011yescharacteristic
186C4×C7#41311yescharacteristic
187Dic7—41no—
188Dic7—41no—
189Dic7—41no—
190#41—281no—
191#41—281no—
192C2×Q8—71no—
193C2×D4—71no—
194C2×D4—141no—
195C22×C4—141no—
196C22×C4—141no—
197C22×C4—141no—
198C22×C4#113111yescharacteristic
199C22×C4#113111yescharacteristic
200C22×C4C2×C4×D711yescharacteristic
201M16—281no—
202M16—281no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—141no—
206C2×C8—141no—
207C2×C8(C2×D7)⋊C411yescharacteristic
208C2×C8(C2×D7)⋊C411yescharacteristic
209C2×C8C2×D2811yescharacteristic
210C2×C8C2×C22⋊D711yescharacteristic
211C4⋊C4—71no—
212C4⋊C4—71no—
213C4⋊C4—141no—
214C4⋊C4—141no—
215C22⋊C4—141no—
216C22⋊C4—141no—
217C22⋊C4—141no—
218C42—141no—
219C42—141no—
220C42—141no—
221C42C8×D711yescharacteristic
222C42C8×D711yescharacteristic
223C42(C2×D7)⋊C411yescharacteristic
224C42(C2×D7)⋊C411yescharacteristic
225C16—42no—
226C16—282no—
227C2×C7—41no—
228C2×C7#145811yescharacteristic
229C2×C7C2×#35011yescharacteristic
230C2×C7C2×#35011yescharacteristic
231D7—81no—
232C23—141no—
233C23C2×#113111yescharacteristic
234Q8—281no—
235D4—281no—
236D4—282no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—141no—
242C2×C4—141no—
243C2×C4—141no—
244C2×C4—281no—
245C2×C4—281no—
246C2×C4—282no—
247C2×C4—282no—
248C2×C4(C2×D7)⋊C811yescharacteristic
249C2×C4(C2×D7)⋊C811yescharacteristic
250C2×C4(C2×D7)⋊C811yescharacteristic
251C2×C4(C2×D7)⋊C811yescharacteristic
252C2×C4C2×C8×D711yescharacteristic
253C2×C4C2×(C2×D7)⋊C411yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—281no—
259C8—281no—
260C7#1146211yescharacteristic, complemented, Sylow
261C22—21no—
262C22—21no—
263C22—21no—
264C22—281no—
265C22—281no—
266C22—281no—
267C22C2×(C2×D7)⋊C811yescharacteristic
268C4—21no—
269C4—21no—
270C4—21no—
271C4—281no—
272C4—281no—
273C4—281no—
274C4#8930312yes—
275C4#8931012yes—
276C4#8977811yescharacteristic
277C4#8990411yescharacteristic
278C2—41no—
279C2—561no—
280C2#1172853611yescharacteristic
281C2C2×#8930311yescharacteristic
282C2C2×#8931011yescharacteristic
subgroupG/H
centerC42(C2×D7)⋊C4
commutatorC2×C4×C7C22×C8
FrattiniC22×C8C22×D7
FittingC7×#2162C2
socleC22×C7C2×C22⋊C8
prime radicalC2×#1157C2×C4
cosocleC22×C7×C8C23
hypercenter#2162D7
hypocenterC7#11462
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C42 ◁ C2×C4×C8 ◁ #2162

Complements

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

NH ≅ G/Ntype
2. C7×#2162C2semidirect
3. #11727498C2semidirect
4. #11725331C2semidirect
57. #2162D7semidirect
260. C7#11462semidirect

Maximal subgroups

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

subgroupindex
#117359002
C7×#21622
#117274982
#117253312
#117253292
#114627