Finite Group Databasebeta

← C7×#9995 #13187211

Subgroups — C7×#9995 #13187211

Order 1792.

510 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C7×#3195C211yesmaximal, characteristic
2C2×C7×#484C211yesmaximal, characteristic
3C2×C7×#483C211yesmaximal, characteristic
4C7×#1821C211yesmaximal, characteristic
5C7×#1472C211yesmaximal, characteristic
6C7×#1472C211yesmaximal, characteristic
7C7×#1298C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×C22⋊Q8C2211yescharacteristic
10C22×C7×C4⋊C4C2211yescharacteristic
11C7×#484—22no—
12C7×#484—22no—
13C7×#483—21no—
14C7×#483—21no—
15C7×#483—22no—
16C7×#483—22no—
17C7×#470—22no—
18C7×#451—21no—
19C7×#451—21no—
20C2×C7×#107—21no—
21C2×C7×#106—21no—
22C2×C7×#103—21no—
23C2×C7×#103—21no—
24C7×#407C2211yescharacteristic
25C7×#407C2211yescharacteristic
26C2×C7×C22⋊C8C2211yescharacteristic
27C2×C7×C22⋊C8C2211yescharacteristic
28C7×#343—22no—
29C7×#324—22no—
30C7×#324—22no—
31#9995C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D411yescharacteristic
34C23×C4×C7—21no—
35C23×C4×C7C2311yescharacteristic
36C2×C7×Q16—22no—
37C2×C7×M16—21no—
38C2×C7×M16D412yes—
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4D412yes—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C7×#107—24no—
62C7×#106—22no—
63C7×#106—24no—
64C7×#103—22no—
65C7×#103—22no—
66C7×#103—24no—
67C7×#103—24no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8—21no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8D412yes—
75C7×C22⋊C8D412yes—
76#3195C2×C711yescharacteristic
77C2×#484C2×C711yescharacteristic
78C2×#483C2×C711yescharacteristic
79#1821C2×C711yescharacteristic
80#1472C2×C711yescharacteristic
81#1472C2×C711yescharacteristic
82#1298C2×C711yescharacteristic
83C24×C7#4111yescharacteristic
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—42no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7—82no—
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C7×Q16—82no—
106C7×M16—42no—
107C7×M16—42no—
108C2×C7×C8—22no—
109C2×C7×C8—22no—
110C2×C7×C8—22no—
111C2×C7×C8—42no—
112C2×C7×C8—42no—
113C2×C7×C8—41no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C2×C22⋊Q8C22×C711yescharacteristic
131C2×C22⋊Q8C22×C711yescharacteristic
132C22×C4⋊C4C22×C711yescharacteristic
133#484—22no—
134#484—22no—
135#483—22no—
136#483—21no—
137#483—21no—
138#483—22no—
139#470—22no—
140#451—21no—
141#451—21no—
142C2×#107—21no—
143C2×#106—21no—
144C2×#103—21no—
145C2×#103—21no—
146#407C22×C711yescharacteristic
147#407C22×C711yescharacteristic
148C2×C22⋊C8C22×C711yescharacteristic
149C2×C22⋊C8C22×C711yescharacteristic
150#343—22no—
151#324—22no—
152#324—22no—
153C23×C7—22no—
154C23×C7—22no—
155C23×C7—41no—
156C23×C7#12111yescharacteristic
157C23×C7#12111yescharacteristic
158C23×C7#12111yescharacteristic
159C7×Q8—42no—
160C7×Q8—42no—
161C7×Q8—42no—
162C7×Q8—42no—
163C2×C4×C7—22no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—82no—
177C2×C4×C7—82no—
178C2×C4×C7—82no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12011yescharacteristic
184C2×C4×C7#12711yescharacteristic
185C7×C8—42no—
186C7×C8—42no—
187C7×C8—42no—
188C7×C8—81no—
189C22×Q8C7×D411yescharacteristic
190C22×Q8C7×D411yescharacteristic
191C23×C4—21no—
192C23×C4C23×C711yescharacteristic
193C2×Q16—22no—
194C2×M16—21no—
195C2×M16C7×D412yes—
196C22×C8—21no—
197C22×C8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C7×D412yes—
214C2×C4⋊C4C7×D411yescharacteristic
215C2×C4⋊C4C7×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218#107—24no—
219#106—24no—
220#106—22no—
221#103—24no—
222#103—24no—
223#103—22no—
224#103—22no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8C7×D412yes—
232C22⋊C8C7×D412yes—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—44no—
241C22×C7#39211yescharacteristic
242C22×C7#47112yes—
243C22×C7#47312yes—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—81no—
248C4×C7—82no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7#45812yes—
253C24C7×#4111yescharacteristic
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—82no—
269C22×C4C2×C7×D411yescharacteristic
270C22×C4C2×C7×D411yescharacteristic
271C22×C4C2×C7×D411yescharacteristic
272C22×C4C2×C7×D411yescharacteristic
273C22×C4C2×C7×D411yescharacteristic
274C22×C4C2×C7×D411yescharacteristic
275Q16—82no—
276M16—42no—
277M16—42no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C2×C7—22no—
301C2×C7—22no—
302C2×C7—41no—
303C2×C7#201911yescharacteristic
304C2×C7#202011yescharacteristic
305C2×C7#202211yescharacteristic
306C23—22no—
307C23—22no—
308C23—41no—
309C23C7×#12111yescharacteristic
310C23C7×#12111yescharacteristic
311C23C7×#12111yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—42no—
315Q8—42no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4C7×#12011yescharacteristic
335C2×C4C7×#12011yescharacteristic
336C2×C4C7×#12011yescharacteristic
337C2×C4C7×#12711yescharacteristic
338C8—42no—
339C8—42no—
340C8—42no—
341C8—81no—
342C7#999511yescharacteristic, direct product factor, Sylow
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22C7×#39211yescharacteristic
352C22C7×#47112yes—
353C22C7×#47312yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4C7×#45812yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2C7×#201911yescharacteristic
367C2C7×#202011yescharacteristic
368C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
31. #9995C7direct (N × H)
342. C7#9995direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31952
C2×C7×#4842
C2×C7×#4832
C7×#18212
C7×#14722
C7×#12982
#99957