Finite Group Databasebeta

← #13250784

Subgroups — #13250784

Order 1792.

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

normal subgroups only

This group has 360 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
1C7×#3236C211yesmaximal, characteristic
2#11740778C211yesmaximal, characteristic
3#11729586C211yesmaximal, characteristic
4#11729491C211yesmaximal, characteristic
5#11729490C211yesmaximal, characteristic
6C7⋊#1308C211yesmaximal, characteristic
7#11725563C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C7×#125C2211yescharacteristic
10C7×#501—21no—
11C7×#491—21no—
12C7×#484C412yes—
13C7×#483C412yes—
14C7×#479—22no—
15C7×#432C2211yescharacteristic
16#89796—22no—
17#89796—22no—
18#89772—22no—
19#89698C2211yescharacteristic
20#89698C2211yescharacteristic
21#89698C2211yescharacteristic
22#89697C2211yescharacteristic
23#89342—21no—
24#89341—21no—
25#89338—22no—
26#89323—22no—
27#9336—71nomaximal, Sylow
28C7×C4⋊Q8D411yescharacteristic
29C7×C4⋊Q8D411yescharacteristic
30C7×#125—21no—
31C7×#125—21no—
32C7×#125—21no—
33C7×#125—21no—
34C7×#125—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8C2×C412yes—
37C4×C7×Q8—21no—
38C4×C7×Q8D412yes—
39C7×#117—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4C2×C411yescharacteristic
43C2×C7×C4⋊C4C2×C411yescharacteristic
44C2×C42×C7C2311yescharacteristic
45C7×#107—22no—
46C7×#106—22no—
47C7×C4⋊C8—21no—
48C7×#103—22no—
49C7×#103—22no—
50C7×C22⋊C8C2×C412yes—
51C7×#97—21no—
52C22⋊(C7⋊C8)—21no—
53C22⋊(C7⋊C8)—21no—
54C22⋊(C7⋊C8)—21no—
55C22⋊(C7⋊C8)D412yes—
56C4⋊(C7⋊C8)—24no—
57C4⋊(C7⋊C8)—21no—
58C4⋊(C7⋊C8)—21no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61#6377—22no—
62#6377—21no—
63#6377—21no—
64#6377D412yes—
65C4×C7⋊C8—22no—
66C4×C7⋊C8—21no—
67#3236D711yescharacteristic
68#2987—71no—
69#1554—71no—
70#1534—71no—
71#1470—71no—
72#1308—71no—
73#1287—71no—
74C2×C7×Q8—21no—
75C2×C7×Q8C22⋊C412yes—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7#4111yescharacteristic
81C2×C7×C8—22no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4C22⋊C412yes—
96C7×C22⋊C4—22no—
97C42×C7—22no—
98C42×C7C2×D411yescharacteristic
99C42×C7C2×D411yescharacteristic
100C42×C7C2×D411yescharacteristic
101C42×C7#4111yescharacteristic
102C2×C7⋊C8—22no—
103C2×C7⋊C8—22no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106#533C2×D711yescharacteristic
107C2×#125C2×D711yescharacteristic
108#501—21no—
109#491—21no—
110#484Dic712yes—
111#483Dic712yes—
112#479—22no—
113#446—142no—
114#446—142no—
115#433—142no—
116#432—71no—
117#432—71no—
118#432—71no—
119#432C2×D711yescharacteristic
120#431—71no—
121#333—141no—
122#332—141no—
123#330—142no—
124#324—142no—
125C23×C7C4×D411yescharacteristic
126C7×Q8—42no—
127C7×Q8—81no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—81no—
137C2×C4×C7C2×C22⋊C411yescharacteristic
138C2×C4×C7C4×D411yescharacteristic
139C2×C4×C7#12011yescharacteristic
140C2×C4×C7#12111yescharacteristic
141C2×C4×C7#12111yescharacteristic
142C2×C4×C7#12311yescharacteristic
143C7×C8—42no—
144C7⋊C8—42no—
145C7⋊C8—42no—
146C7⋊C8—81no—
147C7⋊C8—81no—
148C4⋊Q8C22⋊D711yescharacteristic
149C4⋊Q8C22⋊D711yescharacteristic
150#125—21no—
151#125—21no—
152#125—21no—
153#125—21no—
154#125—21no—
155C22⋊Q8—21no—
156C22⋊Q8C2×Dic712yes—
157C4×Q8—21no—
158C4×Q8C22⋊D712yes—
159#117—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic711yescharacteristic
163C2×C4⋊C4C2×Dic711yescharacteristic
164C2×C42C22×D711yescharacteristic
165#107—22no—
166#106—22no—
167C4⋊C8—21no—
168C4⋊C8—144no—
169C4⋊C8—141no—
170C4⋊C8—141no—
171C4⋊C8—141no—
172C4⋊C8—141no—
173#103—22no—
174#103—22no—
175C22⋊C8—72no—
176C22⋊C8—141no—
177C22⋊C8—141no—
178C22⋊C8—141no—
179C22⋊C8C2×Dic712yes—
180#97—21no—
181#97—72no—
182#97—142no—
183#97—141no—
184#97—141no—
185C4×C8—142no—
186C4×C8—141no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7#38611yescharacteristic
191C4×C7—21no—
192C4×C7—21no—
193C4×C7—21no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—42no—
198C4×C7—42no—
199C4×C7—81no—
200C4×C7—81no—
201C2×Q8—21no—
202C2×Q8C22⋊Dic712yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4D4×D711yescharacteristic
206C22×C4#115911yescharacteristic
207C22×C4C22×Dic711yescharacteristic
208C2×C8—22no—
209C2×C8—142no—
210C2×C8—142no—
211C2×C8—281no—
212C2×C8—281no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4C22⋊Dic712yes—
227C22⋊C4—22no—
228C42—22no—
229C42D4×D711yescharacteristic
230C42#115911yescharacteristic
231C42C2×C22⋊D711yescharacteristic
232C42C2×C22⋊D711yescharacteristic
233C2×C7—41no—
234C2×C7#188111yescharacteristic
235C2×C7#190511yescharacteristic
236C2×C7#191511yescharacteristic
237C23D4×Dic711yescharacteristic
238Q8—42no—
239Q8—81no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—81no—
249C2×C4D4×Dic711yescharacteristic
250C2×C4#649811yescharacteristic
251C2×C4#650011yescharacteristic
252C2×C4#650211yescharacteristic
253C2×C4#650211yescharacteristic
254C2×C4C2×C22⋊Dic711yescharacteristic
255C8—42no—
256C8—282no—
257C8—282no—
258C8—561no—
259C8—561no—
260C7#933611yescharacteristic, complemented, Sylow
261C22—21no—
262C22—21no—
263C22—21no—
264C22#8999211yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—21no—
271C4—42no—
272C4—42no—
273C4—81no—
274C4—81no—
275C2—41no—
276C2#1172962011yescharacteristic
277C2#1172985511yescharacteristic
278C2#1172987011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3236C2
socleC22×C7#386
prime radicalC23×C7C4×D4
cosocleC2×C42×C7C23
hypercenter#3236D7
hypocenterC7#9336
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3236

Complements

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

NH ≅ G/Ntype
260. C7#9336semidirect

Maximal subgroups

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

subgroupindex
C7×#32362
#117407782
#117295862
#117294912
#117294902
C7⋊#13082
#117255632
#93367