Finite Group Databasebeta

← #13250898

Subgroups — #13250898

Order 1792.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#11740654C211yesmaximal, characteristic
2#11736738C211yesmaximal, characteristic
3C7×#1548C211yesmaximal, characteristic, complemented
4#11729544C211yesmaximal, characteristic, complemented
5#11728251C211yesmaximal, characteristic
6#11725549C211yesmaximal, characteristic, complemented
7C7⋊#1309C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×#6505C412yes—
10C4×#1157C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13#89997C412yes—
14#89862—21no—
15#89862—21no—
16#89855—21no—
17#89855—21no—
18#89696C2211yescharacteristic
19#89696C2211yescharacteristic
20C4×#1136C2211yescharacteristic
21#89607—21no—
22D7×C4⋊C8—21no—
23#89500—22no—
24C8×D28—21no—
25#89451—21no—
26C7×#333—21no—
27C7×#333—21no—
28#89325—22no—
29#89279—21no—
30#89278—21no—
31#9353—71nomaximal, Sylow
32C22×C7×Q8C2×C412yes—
33C2×#1157C2×C411yescharacteristic
34C2×C7×M16—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C42×C7C2311yescharacteristic
42C2×#1144—22no—
43#6505—44no—
44C2×#1143—22no—
45C4×C22⋊D7—21no—
46C4×C22⋊D7—21no—
47#6489C2×C411yescharacteristic
48C2×#1136—21no—
49C2×#1136C2×C412yes—
50C2×#1136D412yes—
51C2×C8×D7—22no—
52C4×D28D411yescharacteristic
53C4×D28D411yescharacteristic
54#6435—21no—
55C42×D7—21no—
56C4×Dic14D411yescharacteristic
57C4×Dic14D411yescharacteristic
58C7×C4⋊C8—21no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8D411yescharacteristic
61C7×C4⋊C8D411yescharacteristic
62C7×C22⋊C8—21no—
63C4×C7×C8—21no—
64#6409—22no—
65#6409—22no—
66(C2×D7)⋊C8—22no—
67#6389—22no—
68Dic7⋊C8—22no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)—21no—
71C4⋊(C7⋊C8)—21no—
72C4⋊(C7⋊C8)—21no—
73#6377—21no—
74C4×C7⋊C8—21no—
75#3039—71no—
76#2990—71no—
77#1997—71no—
78#1548D711yescharacteristic, complemented
79#1489—71no—
80#1309—71no—
81#1299—71no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8—42no—
85C2×C7×Q8—42no—
86C22×C4×C7—22no—
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C2×C22⋊D7—21no—
91#1157—22no—
92#1157—41no—
93C2×D28C22⋊C411yescharacteristic
94C2×C4×D7—21no—
95C2×C4×D7—21no—
96C2×C4×D7—21no—
97C2×Dic14C22⋊C411yescharacteristic
98C7×M16—81no—
99C2×C7×C8—21no—
100C2×C7×C8—21no—
101C2×C7×C8—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C42×C7C2×D411yescharacteristic
107C42×C7C2×D411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7#4111yescharacteristic
110C22⋊Dic7—21no—
111#1144—44no—
112#1143—44no—
113(C2×D7)⋊C4—21no—
114(C2×D7)⋊C4—21no—
115C4⋊Dic7C22⋊C411yescharacteristic
116C4⋊Dic7C22⋊C411yescharacteristic
117Dic7⋊C4—21no—
118Dic7⋊C4—21no—
119C4×Dic7—21no—
120C4×Dic7—21no—
121C4×Dic7—21no—
122#1136—22no—
123#1136—41no—
124#1136—42no—
125#1136—42no—
126C2×C7⋊C8—22no—
127C2×C7⋊C8—22no—
128C2×C7⋊C8—41no—
129C8×D7—82no—
130C2×#137—72no—
131C2×C4⋊Q8C2×D711yescharacteristic
132C4×#41—71no—
133#462—141no—
134#462—141no—
135#461—141no—
136#461—141no—
137#446—141no—
138#435—142no—
139C8×D4—141no—
140#432—141no—
141#431C2×D711yescharacteristic
142#423—71no—
143#423—71no—
144#423C2×D711yescharacteristic
145C2×C4⋊C8—141no—
146#419—72no—
147C4×M16—71no—
148#343—142no—
149#333—21no—
150#333—21no—
151Q8⋊C8—141no—
152D4⋊C8—141no—
153C23×C7C4×D411yescharacteristic
154C22×D7—21no—
155C7×Q8—42no—
156C7×Q8—42no—
157C7×Q8—82no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—42no—
165C2×C4×C7—82no—
166C2×C4×C7C2×C22⋊C411yescharacteristic
167C2×C4×C7C4×D411yescharacteristic
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12311yescharacteristic
172C22⋊D7—42no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176D28—21no—
177D28#10012yes—
178C4×D7—42no—
179C4×D7—41no—
180C4×D7—41no—
181C4×D7—41no—
182C4×D7—41no—
183Dic14—21no—
184Dic14—21no—
185C7×C8—41no—
186C7×C8—41no—
187C7×C8—81no—
188C7⋊C8—42no—
189C7⋊C8—42no—
190C7⋊C8—42no—
191C2×#41—71no—
192C22×Q8C4×D712yes—
193#137—284no—
194C2×Q16—142no—
195C2×SD16—142no—
196C2×M16—21no—
197C2×M16—72no—
198C2×M16—72no—
199C2×M16—141no—
200C22×C8—142no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4×Q8—71no—
206C4×Q8—71no—
207C4×D4—71no—
208C4×D4—71no—
209C4×D4—141no—
210C4×D4—141no—
211#117—71no—
212#117—141no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C42—141no—
216C2×C42C22×D711yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—142no—
220C4⋊C8—141no—
221C4⋊C8—141no—
222C4⋊C8—141no—
223C4⋊C8—141no—
224C4⋊C8C22⋊D711yescharacteristic
225C4⋊C8C22⋊D711yescharacteristic
226#103—142no—
227#103—142no—
228C22⋊C8—21no—
229C22⋊C8—142no—
230#97—142no—
231#97—141no—
232C4×C8—21no—
233C4×C8—141no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7#38611yescharacteristic
238C2×D7—22no—
239C2×D7—41no—
240C2×D7—41no—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—82no—
245C4×C7C2×#10011yescharacteristic
246C4×C7#41311yescharacteristic
247C4×C7#45511yescharacteristic
248C4×C7#45611yescharacteristic
249C4×C7#47111yescharacteristic
250C4×C7#47311yescharacteristic
251Dic7—41no—
252Dic7—41no—
253Dic7—41no—
254#41—142no—
255#41—281no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—71no—
261C2×D4—71no—
262C2×D4—141no—
263C22×C4—22no—
264C22×C4—141no—
265C22×C4—141no—
266C22×C4—141no—
267C22×C4C2×C4×D711yescharacteristic
268C22×C4D4×D711yescharacteristic
269C22×C4#115911yescharacteristic
270Q16—284no—
271SD16—284no—
272M16—81no—
273M16—142no—
274M16—281no—
275M16—282no—
276M16—282no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—142no—
281C2×C8—142no—
282C2×C8—281no—
283C2×C8—562no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—71no—
289C4⋊C4—71no—
290C4⋊C4—141no—
291C4⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C42—141no—
296C42—141no—
297C42—141no—
298C42#115711yescharacteristic
299C42D4×D711yescharacteristic
300C42D4×D711yescharacteristic
301C42C2×C22⋊D711yescharacteristic
302C2×C7—41no—
303C2×C7#188211yescharacteristic
304C2×C7#190011yescharacteristic
305C2×C7#190811yescharacteristic
306D7—42no—
307C23—141no—
308C23(C22⋊D7)⋊C411yescharacteristic
309Q8—42no—
310Q8—42no—
311Q8—82no—
312Q8—141no—
313Q8—141no—
314D4—72no—
315D4—141no—
316D4—282no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—282no—
329C2×C4—281no—
330C2×C4—281no—
331C2×C4—281no—
332C2×C4—281no—
333C2×C4D7×C22⋊C411yescharacteristic
334C2×C4#644611yescharacteristic
335C2×C4#644711yescharacteristic
336C2×C4C4×C22⋊D711yescharacteristic
337C2×C4#650011yescharacteristic
338C2×C4#650111yescharacteristic
339C8—41no—
340C8—41no—
341C8—81no—
342C8—282no—
343C8—282no—
344C8—282no—
345C7#935311yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—21no—
349C22—142no—
350C22—281no—
351C22—281no—
352C22#8972811yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—82no—
357C4—281no—
358C4—281no—
359C4—281no—
360C4D7×#10011yescharacteristic
361C4#8951711yescharacteristic
362C4#8994911yescharacteristic
363C4#8996411yescharacteristic
364C4#8997411yescharacteristic
365C4#8997711yescharacteristic
366C2—41no—
367C2—282no—
368C2#1172783511yescharacteristic
369C2#1172804011yescharacteristic
370C2#1172810211yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1548C2
socleC22×C7#386
prime radicalC2×#1157C2×C4
cosocleC2×C42×C7C23
hypercenter#1548D7
hypocenterC7#9353
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C7×#1548C2semidirect
4. #11729544C2semidirect
6. #11725549C2semidirect
7. C7⋊#1309C2semidirect
78. #1548D7semidirect
345. C7#9353semidirect

Maximal subgroups

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

subgroupindex
#117406542
#117367382
C7×#15482
#117295442
#117282512
#117255492
C7⋊#13092
#93537