Finite Group Databasebeta

← C7×#10161 #13187377

Subgroups — C7×#10161 #13187377

Order 1792.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3208C211yesmaximal, characteristic
2C7×#3165C211yesmaximal, characteristic, complemented
3C2×C7×#486C211yesmaximal, characteristic
4C7×#1965C211yesmaximal, characteristic
5C7×#1548C211yesmaximal, characteristic, complemented
6C7×#1493C211yesmaximal, characteristic, complemented
7C7×#1278C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C7×#501—22no—
12C7×#500—21no—
13C7×#498—21no—
14C7×#492—21no—
15C7×#486—21no—
16C7×#486—22no—
17C7×#486—22no—
18C7×#460—22no—
19C7×#450—22no—
20C7×#431C2211yescharacteristic
21C7×#428—22no—
22C7×#423C2211yescharacteristic
23C2×C7×#102—22no—
24C2×C7×#100—22no—
25C4×C7×M16C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27C7×#333—22no—
28C7×Q8⋊C8—22no—
29#10161C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4D412yes—
34C2×C7×SD16—24no—
35C2×C7×M16—21no—
36C2×C7×M16D412yes—
37C22×C7×C8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D412yes—
43C7×C4⋊Q8D411yescharacteristic
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127—21no—
48C7×#125—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C7×#107—24no—
58C7×#106—24no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8D412yes—
62C7×#102—28no—
63C7×#102—24no—
64C7×#100—44no—
65C7×C22⋊C8—21no—
66C7×#97—21no—
67C4×C7×C8—21no—
68C4×C7×C8—21no—
69C4×C7×C8—21no—
70C4×C7×C8D412yes—
71#3208C2×C711yescharacteristic
72#3165C2×C711yescharacteristic, complemented
73C2×#486C2×C711yescharacteristic
74#1965C2×C711yescharacteristic
75#1548C2×C711yescharacteristic, complemented
76#1493C2×C711yescharacteristic, complemented
77#1278C2×C711yescharacteristic, complemented
78C24×C7—22no—
79C2×C7×Q8—21no—
80C2×C7×Q8—21no—
81C2×C7×Q8—22no—
82C2×C7×Q8—41no—
83C2×C7×Q8—82no—
84C2×C7×D4—22no—
85C2×C7×D4—22no—
86C2×C7×D4—44no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—42no—
92C2×C7×D4—42no—
93C2×C7×D4—82no—
94C22×C4×C7—41no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7#4111yescharacteristic
98C7×SD16—84no—
99C7×M16—22no—
100C7×M16—42no—
101C7×M16SD1614yescomplemented
102C2×C7×C8—22no—
103C2×C7×C8—22no—
104C2×C7×C8—22no—
105C2×C7×C8—42no—
106C2×C7×C8—41no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C22⋊C4—41no—
118C42×C7—41no—
119C42×C7C2×D411yescharacteristic
120C42×C7C2×D411yescharacteristic
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123#533C22×C711yescharacteristic
124C2×C4⋊Q8C22×C711yescharacteristic
125C2×#127C22×C711yescharacteristic
126#501—22no—
127#500—21no—
128#498—21no—
129#492—21no—
130#486—22no—
131#486—21no—
132#486—22no—
133#460—22no—
134#450—22no—
135#431C22×C711yescharacteristic
136#428—22no—
137#423C22×C711yescharacteristic
138C2×#102—22no—
139C2×#100—22no—
140C4×M16C22×C711yescharacteristic
141C2×C4×C8C22×C711yescharacteristic
142#333—22no—
143Q8⋊C8—22no—
144C23×C7—42no—
145C23×C7—44no—
146C23×C7—82no—
147C23×C7—82no—
148C23×C7#12111yescharacteristic
149C7×Q8—42no—
150C7×Q8—42no—
151C7×Q8—82no—
152C7×Q8—82no—
153C7×D4—44no—
154C7×D4—44no—
155C7×D4—82no—
156C7×D4—82no—
157C7×D4—82no—
158C7×D4—81no—
159C7×D4—81no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—82no—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12711yescharacteristic
172C2×C4×C7C2×SD1612yes—
173C2×C4×C7#13712yes—
174C7×C8—24no—
175C7×C8—42no—
176C7×C8—42no—
177C7×C8—81no—
178C22×Q8—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C7×D412yes—
182C2×SD16—24no—
183C2×M16—21no—
184C2×M16C7×D412yes—
185C22×C8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8C7×D412yes—
190C4⋊Q8C7×D411yescharacteristic
191C4⋊Q8C7×D411yescharacteristic
192#127—21no—
193#127—21no—
194#127—21no—
195#127—21no—
196#125—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201#117—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C42C23×C711yescharacteristic
205#107—24no—
206#106—24no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8C7×D412yes—
210#102—28no—
211#102—24no—
212#100—44no—
213C22⋊C8—21no—
214#97—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8C7×D412yes—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—82no—
222C22×C7—82no—
223C22×C7—82no—
224C22×C7—164no—
225C22×C7#39211yescharacteristic
226C22×C7#46112yes—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—81no—
230C4×C7—82no—
231C4×C7—81no—
232C4×C7#44812yes—
233C4×C7#45512yes—
234C4×C7#47212yes—
235C4×C7#49211yescharacteristic
236C4×C7#49711yescharacteristic
237C24—22no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—82no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—82no—
253C22×C4—41no—
254C22×C4C2×C7×D411yescharacteristic
255C22×C4C2×C7×D411yescharacteristic
256C22×C4C7×#4111yescharacteristic
257SD16—84no—
258M16—22no—
259M16—42no—
260M16C7×SD1614yescomplemented
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—41no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C22⋊C4—41no—
277C42—41no—
278C42C2×C7×D411yescharacteristic
279C42C2×C7×D411yescharacteristic
280C42C2×C7×D411yescharacteristic
281C42C2×C7×D411yescharacteristic
282C2×C7—22no—
283C2×C7—162no—
284C2×C7#200411yescharacteristic
285C2×C7#201311yescharacteristic
286C2×C7#201811yescharacteristic
287C23—42no—
288C23—44no—
289C23—82no—
290C23—82no—
291C23C7×#12111yescharacteristic
292Q8—42no—
293Q8—42no—
294Q8—82no—
295Q8—82no—
296D4—44no—
297D4—44no—
298D4—82no—
299D4—82no—
300D4—81no—
301D4—81no—
302D4—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4C7×#12011yescharacteristic
310C2×C4C7×#12011yescharacteristic
311C2×C4C7×#12011yescharacteristic
312C2×C4C7×#12111yescharacteristic
313C2×C4C7×#12111yescharacteristic
314C2×C4C7×#12711yescharacteristic
315C2×C4C2×C7×SD1612yes—
316C2×C4C7×#13712yes—
317C8—24no—
318C8—42no—
319C8—42no—
320C8—81no—
321C7#1016111yescharacteristic, direct product factor, Sylow
322C22—21no—
323C22—21no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—164no—
328C22C7×#39211yescharacteristic
329C22C7×#46112yes—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—81no—
335C4C7×#44812yes—
336C4C7×#45512yes—
337C4C7×#47212yes—
338C4C7×#49211yescharacteristic
339C4C7×#49711yescharacteristic
340C2—22no—
341C2—162no—
342C2C7×#200411yescharacteristic
343C2C7×#201311yescharacteristic
344C2C7×#201811yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
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
2. C7×#3165C2semidirect
5. C7×#1548C2semidirect
6. C7×#1493C2semidirect
7. C7×#1278C2semidirect
29. #10161C7direct (N × H)
72. #3165C2×C7semidirect
75. #1548C2×C7semidirect
76. #1493C2×C7semidirect
77. #1278C2×C7semidirect
101. C7×M16SD16semidirect
260. M16C7×SD16semidirect
321. C7#10161direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32082
C7×#31652
C2×C7×#4862
C7×#19652
C7×#15482
C7×#14932
C7×#12782
#101617