Finite Group Databasebeta

← C7×#10224 #13187440

Subgroups — C7×#10224 #13187440

Order 1792.

462 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3230C211yesmaximal, characteristic
2C7×#3209C211yesmaximal, characteristic
3C7×#3162C211yesmaximal, characteristic
4C7×#1993C211yesmaximal, characteristic
5C7×#1518C211yesmaximal, characteristic
6C7×#1518C211yesmaximal, characteristic
7C7×#1281C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C7×#533C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C7×#501—21no—
12C7×#501—21no—
13C7×#501—21no—
14C7×#501—21no—
15C7×#500—21no—
16C7×#494—21no—
17C7×#487—21no—
18C7×#487—21no—
19C7×#484—22no—
20C7×#477—22no—
21C7×#462—22no—
22C7×#451—22no—
23C7×#431C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C7×#427—22no—
26C2×C7×#97C2211yescharacteristic
27C7×#382C2211yescharacteristic
28C7×#340—22no—
29C7×#328—22no—
30C7×#328—22no—
31#10224C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8—21no—
33C2×C7×Q16—24no—
34C22×C7×C8D412yes—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8D412yes—
39C7×C4⋊Q8D411yescharacteristic
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×#125—21no—
44C7×#125—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C7×#117—21no—
54C7×#117—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C42×C7C2311yescharacteristic
59C7×#107—24no—
60C7×#107—22no—
61C7×#107—22no—
62C7×#106—22no—
63C7×#106—22no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×#103—24no—
67C7×#103—24no—
68C7×C22⋊C8D412yes—
69C7×C22⋊C8D412yes—
70C7×#97—21no—
71C7×#97—21no—
72C4×C7×C8—21no—
73C4×C7×C8—21no—
74C7×#95—21no—
75C7×#95—21no—
76#3230C2×C711yescharacteristic
77#3209C2×C711yescharacteristic
78#3162C2×C711yescharacteristic
79#1993C2×C711yescharacteristic
80#1518C2×C711yescharacteristic
81#1518C2×C711yescharacteristic
82#1281C2×C711yescharacteristic
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—41no—
89C2×C7×Q8—82no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C7×Q16—84no—
96C2×C7×C8—22no—
97C2×C7×C8—22no—
98C2×C7×C8—22no—
99C2×C7×C8—22no—
100C2×C7×C8Q1614yescomplemented
101C7×C4⋊C4—21no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—42no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C42×C7—41no—
121C42×C7—41no—
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7#4111yescharacteristic
126#533C22×C711yescharacteristic
127#533C22×C711yescharacteristic
128C2×C4⋊Q8C22×C711yescharacteristic
129#501—21no—
130#501—21no—
131#501—21no—
132#501—21no—
133#500—21no—
134#494—21no—
135#487—21no—
136#487—21no—
137#484—22no—
138#477—22no—
139#462—22no—
140#451—22no—
141#431C22×C711yescharacteristic
142#431C22×C711yescharacteristic
143#427—22no—
144C2×#97C22×C711yescharacteristic
145#382C22×C711yescharacteristic
146#340—22no—
147#328—22no—
148#328—22no—
149C23×C7#12711yescharacteristic
150C7×Q8—42no—
151C7×Q8—44no—
152C7×Q8—42no—
153C7×Q8—81no—
154C7×Q8—81no—
155C7×Q8—82no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—82no—
164C2×C4×C7—81no—
165C2×C4×C7—81no—
166C2×C4×C7—81no—
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7C2×Q1612yes—
174C2×C4×C7#13712yes—
175C7×C8—24no—
176C7×C8—42no—
177C7×C8—42no—
178C22×Q8—21no—
179C2×Q16—24no—
180C22×C8C7×D412yes—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C7×D412yes—
185C4⋊Q8C7×D411yescharacteristic
186C4⋊Q8C7×D411yescharacteristic
187C4⋊Q8C7×D411yescharacteristic
188C4⋊Q8C7×D411yescharacteristic
189#125—21no—
190#125—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199#117—21no—
200#117—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C42C23×C711yescharacteristic
205#107—22no—
206#107—22no—
207#107—24no—
208#106—22no—
209#106—22no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212#103—24no—
213#103—24no—
214C22⋊C8C7×D412yes—
215C22⋊C8C7×D412yes—
216#97—21no—
217#97—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95—21no—
221#95—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7#39211yescharacteristic
225C22×C7#49411yescharacteristic
226C22×C7#49711yescharacteristic
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—82no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7—81no—
235C4×C7—81no—
236C4×C7—161no—
237C4×C7#45112yes—
238C4×C7#46212yes—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—82no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4C2×C7×D411yescharacteristic
249C22×C4C2×C7×D411yescharacteristic
250C22×C4C2×C7×D411yescharacteristic
251Q16—84no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8C7×Q1614yescomplemented
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—41no—
277C42—41no—
278C42C2×C7×D411yescharacteristic
279C42C2×C7×D411yescharacteristic
280C42C2×C7×D411yescharacteristic
281C42C7×#4111yescharacteristic
282C2×C7—21no—
283C2×C7—21no—
284C2×C7#200811yescharacteristic
285C2×C7#200911yescharacteristic
286C2×C7#201611yescharacteristic
287C23C7×#12711yescharacteristic
288Q8—42no—
289Q8—44no—
290Q8—42no—
291Q8—81no—
292Q8—81no—
293Q8—82no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4C7×#12011yescharacteristic
306C2×C4C7×#12011yescharacteristic
307C2×C4C7×#12011yescharacteristic
308C2×C4C7×#12111yescharacteristic
309C2×C4C7×#12111yescharacteristic
310C2×C4C7×#12111yescharacteristic
311C2×C4C2×C7×Q1612yes—
312C2×C4C7×#13712yes—
313C8—24no—
314C8—42no—
315C8—42no—
316C7#1022411yescharacteristic, direct product factor, Sylow
317C22—21no—
318C22—21no—
319C22C7×#39211yescharacteristic
320C22C7×#49411yescharacteristic
321C22C7×#49711yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4C7×#45112yes—
333C4C7×#46212yes—
334C2—21no—
335C2—21no—
336C2C7×#200811yescharacteristic
337C2C7×#200911yescharacteristic
338C2C7×#201611yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
31. #10224C7direct (N × H)
100. C2×C7×C8Q16semidirect
256. C2×C8C7×Q16semidirect
316. C7#10224direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32302
C7×#32092
C7×#31622
C7×#19932
C7×#15182
C7×#12812
#102247