Finite Group Databasebeta

← #13247194

Subgroups — #13247194

Order 1792.

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

normal subgroups only

This group has 554 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
1D7×#382C211yesmaximal, characteristic
2#11727561C212yesmaximal
3C7×#1295C211yesmaximal, characteristic, complemented
4#11725465C212yesmaximal, complemented
5#11725436C211yesmaximal, characteristic, complemented
6C2×D7×C4⋊C4C411yescharacteristic
7C2×D7×C4⋊C4C411yescharacteristic
8C2×C42×D7C2211yescharacteristic
9C2×C7×#97C2212yes—
10C7×#382C2211yescharacteristic
11C2×(C2×D7)⋊C8—22no—
12#89749C411yescharacteristic
13#89749C411yescharacteristic
14#89708C2211yescharacteristic
15C2×#6377C2212yes—
16D7×#95—22no—
17#89422—22no—
18#89345—22no—
19#8718—71nomaximal, Sylow
20C22×C4×D7—21no—
21C22×C4×D7—21no—
22C22×C4×D7C2×C411yescharacteristic
23C22×C4×D7D412yes—
24C22×C7×C8—22no—
25C2×C7×C4⋊C4C2×C411yescharacteristic
26C2×C7×C4⋊C4C2×C411yescharacteristic
27C2×C42×C7C2311yescharacteristic
28C2×C4⋊Dic7C2×C411yescharacteristic
29C2×C4⋊Dic7C2×C411yescharacteristic
30C2×Dic7⋊C4—21no—
31C2×Dic7⋊C4—21no—
32C2×C4×Dic7C2×C411yescharacteristic
33C2×C4×Dic7D412yes—
34C22×C7⋊C8—22no—
35D7×C4⋊C4—42no—
36D7×C4⋊C4—42no—
37C42×D7—22no—
38C42×D7—21no—
39C42×D7—21no—
40C7×#97—22no—
41C7×#97—22no—
42C7×#95—22no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46(C2×D7)⋊C8—28no—
47#6377—22no—
48#6377—22no—
49C2×#382—71no—
50#1749—72no—
51#1295—72no—
52#1295—71no—
53#1295D711yescharacteristic, complemented
54C23×D7C22⋊C411yescharacteristic
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7C22×C411yescharacteristic
58C22×C4×C7C2×D412yes—
59C22×Dic7—21no—
60C22×Dic7—21no—
61C22×Dic7C22⋊C412yes—
62C22×Dic7C22⋊C411yescharacteristic
63C2×C4×D7—22no—
64C2×C4×D7—21no—
65C2×C4×D7—21no—
66C2×C4×D7—22no—
67C2×C4×D7—24no—
68C2×C4×D7—22no—
69C2×C4×D7—22no—
70C2×C4×D7—42no—
71C2×C4×D7—44no—
72C2×C4×D7—42no—
73C2×C4×D7—42no—
74C2×C4×D7—44no—
75C2×C7×C8—24no—
76C2×C7×C8—42no—
77C2×C7×C8—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C42×C7#4112yes—
81C42×C7#4111yescharacteristic
82C42×C7#4111yescharacteristic
83C4⋊Dic7—41no—
84C4⋊Dic7—41no—
85Dic7⋊C4—42no—
86Dic7⋊C4—42no—
87C4×Dic7—21no—
88C4×Dic7—21no—
89C4×Dic7—22no—
90C4×Dic7—22no—
91C2×C7⋊C8—24no—
92C2×C7⋊C8—42no—
93C2×C7⋊C8—42no—
94C22×C4⋊C4—71no—
95C22×C4⋊C4—71no—
96C22×C42—71no—
97C2×C22⋊C8—142no—
98C2×#97—72no—
99C2×#97C2×D712yes—
100#382—71no—
101#382—71no—
102#382—71no—
103#382—142no—
104#382C2×D711yescharacteristic
105C2×#95—142no—
106#336—142no—
107C23×C7C2×C22⋊C411yescharacteristic
108C22×D7—21no—
109C22×D7—21no—
110C22×D7—22no—
111C22×D7#10012yes—
112C22×D7#10414yes—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7#11711yescharacteristic
122C2×C4×C7#11711yescharacteristic
123C2×C4×C7#12312yes—
124C2×C4×C7#12312yes—
125C2×Dic7—21no—
126C2×Dic7—21no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7—22no—
130C2×Dic7—41no—
131C2×Dic7—41no—
132C2×Dic7—42no—
133C2×Dic7—42no—
134C2×Dic7#10112yes—
135C4×D7—24no—
136C4×D7—24no—
137C4×D7—44no—
138C4×D7—44no—
139C4×D7—84no—
140C4×D7—84no—
141C7×C8—44no—
142C7⋊C8—44no—
143C23×C4—71no—
144C23×C4—72no—
145C23×C4—141no—
146C23×C4—141no—
147C22×C8—22no—
148C22×C8—142no—
149C2×C4⋊C4—71no—
150C2×C4⋊C4—71no—
151C2×C4⋊C4—141no—
152C2×C4⋊C4—141no—
153C2×C4⋊C4—282no—
154C2×C4⋊C4—282no—
155C2×C4⋊C4C4×D711yescharacteristic
156C2×C4⋊C4C4×D711yescharacteristic
157C2×C42—71no—
158C2×C42—72no—
159C2×C42—142no—
160C2×C42—141no—
161C2×C42—141no—
162C2×C42C22×D711yescharacteristic
163C22⋊C8—148no—
164#97—22no—
165#97—22no—
166#97—142no—
167#97—142no—
168#95—22no—
169#95—142no—
170#95—142no—
171#95—142no—
172C22×C7#38111yescharacteristic
173C22×C7C2×#10011yescharacteristic
174C22×C7C2×#10111yescharacteristic
175C22×C7C2×#10412yes—
176C22×C7#42112yes—
177C2×D7—22no—
178C2×D7—24no—
179C2×D7—24no—
180C2×D7—22no—
181C2×D7#32914yes—
182C4×C7—21no—
183C4×C7—21no—
184C4×C7—22no—
185C4×C7—22no—
186C4×C7—81no—
187C4×C7—81no—
188Dic7—22no—
189Dic7—42no—
190Dic7—81no—
191Dic7—81no—
192C24—71no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—71no—
196C22×C4—72no—
197C22×C4—142no—
198C22×C4—142no—
199C22×C4—144no—
200C22×C4—141no—
201C22×C4—141no—
202C22×C4—142no—
203C22×C4—142no—
204C22×C4—141no—
205C22×C4—141no—
206C22×C4—282no—
207C22×C4—282no—
208C22×C4—282no—
209C22×C4—284no—
210C22×C4—284no—
211C22×C4C2×C4×D711yescharacteristic
212C22×C4D4×D712yes—
213C2×C8—24no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—144no—
217C2×C8—282no—
218C2×C8—282no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—281no—
222C4⋊C4—281no—
223C4⋊C4—282no—
224C4⋊C4—282no—
225C42—141no—
226C42—142no—
227C42—141no—
228C42—142no—
229C42#115712yes—
230C42#115911yescharacteristic
231C42#116111yescharacteristic
232C2×C7C2×#32911yescharacteristic
233C2×C7#152812yes—
234C2×C7#153311yescharacteristic
235C2×C7#182111yescharacteristic
236C2×C7#182612yes—
237D7—24no—
238C23—72no—
239C23—74no—
240C23—141no—
241C23—141no—
242C23—142no—
243C23D7×C22⋊C411yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—72no—
253C2×C4—144no—
254C2×C4—144no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—284no—
261C2×C4—284no—
262C2×C4—281no—
263C2×C4—281no—
264C2×C4—282no—
265C2×C4—282no—
266C2×C4—564no—
267C2×C4—564no—
268C2×C4#643511yescharacteristic
269C2×C4#644612yes—
270C2×C4#645511yescharacteristic
271C2×C4#645712yes—
272C8—44no—
273C8—284no—
274C7#871811yescharacteristic, complemented, Sylow
275C22—74no—
276C22—142no—
277C22—144no—
278C22—142no—
279C22—144no—
280C22#8943611yescharacteristic
281C22D7×#10011yescharacteristic
282C22D7×#10111yescharacteristic
283C22D7×#10412yes—
284C22#8959312yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—142no—
292C4—282no—
293C4—561no—
294C4—561no—
295C2—144no—
296C2D7×#32911yescharacteristic
297C2#1172610111yescharacteristic
298C2#1172611712yes—
299C2#1172650812yes—
300C2#1172674411yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1295C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1295D7
hypocenterC7#8718
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. C7×#1295C2semidirect
4. #11725465C2semidirect
5. #11725436C2semidirect
53. #1295D7semidirect
274. C7#8718semidirect

Maximal subgroups

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

subgroupindex
D7×#3822
#117275612
C7×#12952
#117254652
#117254362
#87187