Finite Group Databasebeta

← #13262931

Subgroups — #13262931

Order 1792.

392 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#89611C211yesmaximal, characteristic
2#11726860C211yesmaximal, characteristic
3D7×C22⋊C16C211yesmaximal, characteristic
4C7×#1322C211yesmaximal, characteristic, complemented
5#11725515C211yesmaximal, characteristic, complemented
6#11725330C211yesmaximal, characteristic, complemented
7#11725330C211yesmaximal, characteristic, complemented
8C22×#1131C411yescharacteristic
9C22×C8×D7C2211yescharacteristic
10C2×#6435C411yescharacteristic
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×Dic7⋊C8C411yescharacteristic
13C2×Dic7⋊C8C2211yescharacteristic
14C2×C4⋊(C7⋊C8)C411yescharacteristic
15C2×#6372—21no—
16C2×D7×C16—21no—
17#89611—21no—
18#89611—21no—
19#89611—21no—
20#89611—21no—
21C7×C22⋊C16C2211yescharacteristic
22C7×C22⋊C16C2211yescharacteristic
23C22⋊(C7⋊C16)C2211yescharacteristic
24C22⋊(C7⋊C16)C2211yescharacteristic
25(C2×D7)⋊C16—21no—
26(C2×D7)⋊C16—21no—
27#11441—71nomaximal, Sylow
28C22×C4×D7C2×C411yescharacteristic
29C22×C7×C8C2×C411yescharacteristic
30C22×C7×C8C2311yescharacteristic
31C2×C42×C7C2×C411yescharacteristic
32C2×(C2×D7)⋊C4C811yescharacteristic
33C2×(C2×D7)⋊C4C811yescharacteristic
34C2×Dic7⋊C4C811yescharacteristic
35C2×Dic7⋊C4C811yescharacteristic
36C2×C4×Dic7C2×C411yescharacteristic
37C22×C7⋊C8C2×C411yescharacteristic
38C22×C7⋊C8C2×C411yescharacteristic
39C2×#1131—21no—
40C2×#1131—21no—
41C2×#1131—42no—
42C2×C8×D7—42no—
43C2×C8×D7D411yescharacteristic
44C2×C8×D7D411yescharacteristic
45C2×C8×D7D411yescharacteristic
46C2×C8×D7D411yescharacteristic
47#6435—21no—
48#6435—21no—
49#6435—21no—
50#6435—21no—
51C2×C7×C16—21no—
52C2×C7×C16—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55Dic7⋊C8—21no—
56Dic7⋊C8—21no—
57Dic7⋊C8—21no—
58Dic7⋊C8—21no—
59C2×C7⋊C16—21no—
60C2×C7⋊C16—21no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63#6372—42no—
64D7×C16—42no—
65C2×#424—71no—
66#2112—71no—
67C2×C22⋊C16—71no—
68#1322—71no—
69#1322—71no—
70#1322—71no—
71#1322D711yescharacteristic, complemented
72C23×D7C2×C811yescharacteristic
73C22×C4×C7C2×C811yescharacteristic
74C22×C4×C7C2×C811yescharacteristic
75C22×C4×C7C22×C411yescharacteristic
76C22×Dic7C2×C811yescharacteristic
77C22×Dic7C2×C811yescharacteristic
78C22×Dic7C2×C811yescharacteristic
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7C22⋊C411yescharacteristic
82C2×C4×D7C22⋊C411yescharacteristic
83C2×C4×D7M1611yescharacteristic
84C2×C4×D7M1611yescharacteristic
85C2×C7×C8—21no—
86C2×C7×C8—41no—
87C2×C7×C8—41no—
88C2×C7×C8C2×D411yescharacteristic
89C2×C7×C8C2×D411yescharacteristic
90C42×C7—21no—
91C42×C7—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C2×C7⋊C8—21no—
103C2×C7⋊C8—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8C22⋊C411yescharacteristic
106C2×C7⋊C8C22⋊C411yescharacteristic
107#1131—42no—
108#1131—42no—
109C8×D7—22no—
110C8×D7—22no—
111C8×D7—41no—
112C8×D7—41no—
113C7×C16—41no—
114C7×C16—41no—
115C7⋊C16—41no—
116C7⋊C16—41no—
117C22×M16—71no—
118C23×C8—71no—
119C2×#117—71no—
120C2×M32—141no—
121C22×C16—141no—
122#424—141no—
123#424—141no—
124#424—141no—
125#424—141no—
126C2×C4⋊C8—71no—
127C2×C4⋊C8—71no—
128C2×C4⋊C8—71no—
129C2×C4⋊C8C2×D711yescharacteristic
130C22⋊C16—71no—
131C22⋊C16—71no—
132C22⋊C16—141no—
133C22⋊C16—141no—
134C22⋊C16C2×D711yescharacteristic
135C22⋊C16C2×D711yescharacteristic
136C23×C7C22×C811yescharacteristic
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—22no—
140C22×D7—41no—
141C22×D7C22⋊C811yescharacteristic
142C22×D7C22⋊C811yescharacteristic
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—41no—
150C2×C4×C7C2×C22⋊C411yescharacteristic
151C2×C4×C7C2×M1611yescharacteristic
152C2×Dic7—22no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—41no—
158C2×Dic7C22⋊C811yescharacteristic
159C2×Dic7C22⋊C811yescharacteristic
160C4×D7—22no—
161C4×D7—21no—
162C4×D7—21no—
163C4×D7—22no—
164C4×D7#9912yes—
165C4×D7#10112yes—
166C7×C8—21no—
167C7×C8—21no—
168C7×C8—41no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C7⋊C8—41no—
172C23×C4—71no—
173C2×M16—141no—
174C2×M16—141no—
175C2×M16—282no—
176C22×C8—71no—
177C22×C8—71no—
178C22×C8—71no—
179C22×C8—71no—
180C22×C8—71no—
181C22×C8—71no—
182C22×C8—282no—
183C22×C8C4×D711yescharacteristic
184C22×C8C22×D711yescharacteristic
185#117—141no—
186#117—141no—
187#117—141no—
188#117—141no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C22⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C42—71no—
194C2×C42C4×D711yescharacteristic
195M32—282no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—141no—
199C2×C16—141no—
200C2×C16—282no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—141no—
204C4⋊C8—141no—
205C4⋊C8—141no—
206C4⋊C8—141no—
207C4⋊C8—141no—
208C4⋊C8—141no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7C2×C22⋊C811yescharacteristic
212C22×C7#50411yescharacteristic
213C22×C7#50411yescharacteristic
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—21no—
219C2×D7—42no—
220C2×D7—42no—
221C2×D7#32412yes—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#9911yescharacteristic
226C4×C7C2×#10111yescharacteristic
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230Dic7#32412yes—
231C24—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—142no—
240C22×C4—142no—
241C22×C4C8×D711yescharacteristic
242C22×C4C8×D711yescharacteristic
243C22×C4C2×C4×D711yescharacteristic
244M16—282no—
245M16—282no—
246C2×C8—21no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—71no—
250C2×C8—71no—
251C2×C8—142no—
252C2×C8—142no—
253C2×C8—141no—
254C2×C8—281no—
255C2×C8—281no—
256C2×C8—281no—
257C2×C8—281no—
258C2×C8D4×D711yescharacteristic
259C2×C8D4×D711yescharacteristic
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—21no—
269C42—21no—
270C42—141no—
271C42—141no—
272C16—41no—
273C16—41no—
274C16—281no—
275C16—281no—
276C2×C7—21no—
277C2×C7—21no—
278C2×C7C2×#32411yescharacteristic
279C2×C7#211311yescharacteristic
280C2×C7#211511yescharacteristic
281D7—22no—
282D7—41no—
283C23—71no—
284C23—71no—
285C23—142no—
286C23—141no—
287C23—141no—
288C23—281no—
289C23C2×C8×D711yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—71no—
298C2×C4—72no—
299C2×C4—72no—
300C2×C4—71no—
301C2×C4—142no—
302C2×C4—142no—
303C2×C4—142no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—281no—
311C2×C4D7×C22⋊C411yescharacteristic
312C2×C4D7×M1611yescharacteristic
313C8—21no—
314C8—21no—
315C8—41no—
316C8—141no—
317C8—141no—
318C8—281no—
319C7#1144111yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—72no—
323C22—141no—
324C22—141no—
325C22—141no—
326C22—141no—
327C22—141no—
328C22—282no—
329C22—282no—
330C22D7×C22⋊C811yescharacteristic
331C22#8967311yescharacteristic
332C22#8967911yescharacteristic
333C4—21no—
334C4—41no—
335C4—41no—
336C4—72no—
337C4—141no—
338C4—281no—
339C4—281no—
340C4D7×#9911yescharacteristic
341C4D7×#10111yescharacteristic
342C2—21no—
343C2—21no—
344C2—142no—
345C2—281no—
346C2D7×#32411yescharacteristic
347C2#1172686111yescharacteristic
348C2#1172687311yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC23×C7C22×C8
FrattiniC22×C8C22×D7
FittingC7×#1322C2
socleC22×C7C2×C22⋊C8
prime radicalC23×D7C2×C8
cosocleC22×C7×C8C23
hypercenter#1322D7
hypocenterC7#11441
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
4. C7×#1322C2semidirect
5. #11725515C2semidirect
6. #11725330C2semidirect
7. #11725330C2semidirect
71. #1322D7semidirect
319. C7#11441semidirect

Maximal subgroups

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

subgroupindex
C2×#896112
#117268602
D7×C22⋊C162
C7×#13222
#117255152
#117253302
#114417