Finite Group Databasebeta

← #13241908

Subgroups — #13241908

Order 1792.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#11733737C211yesmaximal, characteristic
2#11726359C212yesmaximal, complemented
3C7×#1270C211yesmaximal, characteristic
4#11725445C211yesmaximal, characteristic
5#11725286C212yesmaximal, complemented
6C2×#6499C411yescharacteristic
7C2×D4×Dic7C411yescharacteristic
8C2×#6450C411yescharacteristic
9C7×#390C2211yescharacteristic
10C2×Dic7⋊C8—22no—
11#89749C411yescharacteristic
12#89716—22no—
13C22⋊C4×Dic7—22no—
14C42×Dic7C2211yescharacteristic
15#89701C2211yescharacteristic
16C7×#336C2212yes—
17#89345—22no—
18#89319C2212yes—
19#7601—71nomaximal, Sylow
20C22×C7×D4C2×C411yescharacteristic
21C23×Dic7—21no—
22C22×Dic14C2×C411yescharacteristic
23C22×C7×C8—22no—
24C2×C7×C4⋊C4C2×C411yescharacteristic
25C2×C7×C22⋊C4—22no—
26C2×C42×C7C2311yescharacteristic
27C2×C22⋊Dic7—22no—
28C2×C22⋊Dic7—21no—
29#6499—22no—
30#6499C814yescomplemented
31D4×Dic7—42no—
32C2×C4⋊Dic7C2×C411yescharacteristic
33C2×Dic7⋊C4—21no—
34C2×C4×Dic7—21no—
35C2×C4×Dic7C2×C411yescharacteristic
36C2×C4×Dic7C2×C411yescharacteristic
37C2×C4×Dic7D412yes—
38C2×C4×Dic7D412yes—
39C22×C7⋊C8—22no—
40#6450—42no—
41C7⋊#95—22no—
42C7⋊#95—22no—
43Dic7⋊C8—44no—
44C4×#124—71no—
45#1831—72no—
46#1272—72no—
47#1270—71no—
48#1270D711yescharacteristic
49C24×C7—21no—
50C2×C7×D4—21no—
51C2×C7×D4—41no—
52C2×C7×D4—41no—
53C2×C7×D4C2×C812yes—
54C22×C4×C7—21no—
55C22×C4×C7C22×C411yescharacteristic
56C22×C4×C7C2×D412yes—
57C22×Dic7—21no—
58C22×Dic7—21no—
59C22×Dic7—44no—
60C22×Dic7—42no—
61C22×Dic7C22⋊C411yescharacteristic
62C22×Dic7C22⋊C412yes—
63C22×Dic7C22⋊C411yescharacteristic
64C2×Dic14—21no—
65C2×Dic14—41no—
66C2×Dic14—41no—
67C2×Dic14C2×C812yes—
68C2×C7×C8—42no—
69C2×C7×C8—42no—
70C2×C7×C8—42no—
71C7×C4⋊C4—41no—
72C7×C22⋊C4—42no—
73C7×C22⋊C4—42no—
74C42×C7—41no—
75C22⋊Dic7—28no—
76C22⋊Dic7—42no—
77C4⋊Dic7—41no—
78Dic7⋊C4—42no—
79C4×Dic7—22no—
80C4×Dic7—22no—
81C4×Dic7—22no—
82C4×Dic7—42no—
83C4×Dic7—42no—
84C4×Dic7C2×C812yes—
85C4×Dic7M1612yes—
86C2×C7⋊C8—42no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C2×#124—71no—
90C2×C4×Q8—71no—
91C2×C4×D4—71no—
92C2×C4⋊C8—142no—
93#391—71no—
94#390C2×D711yescharacteristic
95#388—142no—
96#382—71no—
97C4×C22⋊C4—142no—
98C43—71no—
99#336—72no—
100#336—142no—
101#336C2×D712yes—
102C23×C7—22no—
103C23×C7—41no—
104C23×C7—41no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7C2×C22⋊C411yescharacteristic
108C7×D4—42no—
109C7×D4—42no—
110C2×C4×C7—22no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—22no—
114C2×C4×C7—41no—
115C2×C4×C7—42no—
116C2×C4×C7C22×C811yescharacteristic
117C2×C4×C7C2×M1611yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)12yes—
119C2×C4×C7#13712yes—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—21no—
125C2×Dic7—22no—
126C2×Dic7—41no—
127C2×Dic7—41no—
128C2×Dic7—41no—
129C2×Dic7—42no—
130C2×Dic7—84no—
131C2×Dic7C22⋊C814yes—
132C2×Dic7#10012yes—
133Dic14—42no—
134Dic14—42no—
135C7×C8—82no—
136C7⋊C8—82no—
137C22×Q8—71no—
138C22×D4C4×D711yescharacteristic
139C23×C4—141no—
140C22×C8—22no—
141C22×C8—142no—
142#124—74no—
143#124—142no—
144C4×Q8—282no—
145C4×D4—282no—
146C2×C4⋊C4—71no—
147C2×C4⋊C4—141no—
148C2×C4⋊C4C4×D711yescharacteristic
149C2×C22⋊C4—22no—
150C2×C22⋊C4—142no—
151C2×C22⋊C4—141no—
152C2×C42—71no—
153C2×C42—71no—
154C2×C42—72no—
155C2×C42—72no—
156C2×C42—141no—
157C2×C42C22×D711yescharacteristic
158C4⋊C8—284no—
159#95—142no—
160#95—142no—
161C22×C7—42no—
162C22×C7—44no—
163C22×C7—81no—
164C22×C7—82no—
165C22×C7—81no—
166C22×C7C2×C22⋊C811yescharacteristic
167C22×C7#41011yescharacteristic
168C22×C7C2×#10011yescharacteristic
169C22×C7#41712yes—
170C22×C7#42112yes—
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—22no—
174C4×C7—22no—
175C4×C7—81no—
176Dic7—24no—
177Dic7—22no—
178Dic7—41no—
179Dic7—42no—
180Dic7—81no—
181C24—21no—
182C2×Q8—72no—
183C2×Q8—141no—
184C2×Q8—281no—
185C2×Q8—281no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4C8×D712yes—
190C22×C4—21no—
191C22×C4—72no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—141no—
195C22×C4—141no—
196C22×C4—282no—
197C22×C4—284no—
198C22×C4C2×C4×D711yescharacteristic
199C22×C4D4×D712yes—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—282no—
204C2×C8—282no—
205C2×C8—282no—
206C4⋊C4—41no—
207C4⋊C4—281no—
208C4⋊C4—282no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—148no—
212C22⋊C4—282no—
213C42—41no—
214C42—72no—
215C42—72no—
216C42—142no—
217C42—142no—
218C42—142no—
219C42—282no—
220C42—282no—
221C2×C7—82no—
222C2×C7#146011yescharacteristic
223C2×C7#147712yes—
224C2×C7#151212yes—
225C2×C7#153311yescharacteristic
226C2×C7#156011yescharacteristic
227C23—22no—
228C23—41no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23D7×C22⋊C411yescharacteristic
233Q8—282no—
234Q8—282no—
235D4—42no—
236D4—42no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—74no—
244C2×C4—72no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—141no—
250C2×C4—142no—
251C2×C4—281no—
252C2×C4—281no—
253C2×C4—281no—
254C2×C4—282no—
255C2×C4—564no—
256C2×C4C2×C8×D711yescharacteristic
257C2×C4D7×M1611yescharacteristic
258C2×C4#647412yes—
259C2×C4#647812yes—
260C8—82no—
261C8—562no—
262C7#760111yescharacteristic, complemented, Sylow
263C22—42no—
264C22—44no—
265C22—81no—
266C22—81no—
267C22—82no—
268C22D7×C22⋊C811yescharacteristic
269C22#8951211yescharacteristic
270C22D7×#10011yescharacteristic
271C22#8954312yes—
272C22#8959312yes—
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—81no—
278C4—142no—
279C4—144no—
280C4—282no—
281C4—281no—
282C4—561no—
283C2—82no—
284C2#1172585511yescharacteristic
285C2#1172592012yes—
286C2#1172603412yes—
287C2#1172610111yescharacteristic
288C2#1172614311yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC2×C4×C7C22×C8
FrattiniC2×C42C22×D7
FittingC7×#1270C2
socleC23×C7C2×C22⋊C4
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#1270D7
hypocenterC7#7601
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #11726359C2semidirect
5. #11725286C2semidirect
30. #6499C8semidirect
262. C7#7601semidirect

Maximal subgroups

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

subgroupindex
#117337372
#117263592
C7×#12702
#117254452
#117252862
#76017