Finite Group Databasebeta

← #13267059

Subgroups — #13267059

Order 1792.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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×#3245C211yesmaximal, characteristic
2#11728822C212yesmaximal, complemented
3#11728216C211yesmaximal, characteristic, complemented
4#11725580C212yesmaximal, complemented
5#11725515C211yesmaximal, characteristic, complemented
6C7×#532—21no—
7C7×#474—22no—
8C7×#466—21no—
9C7×#466C2212yes—
10C7×#459—22no—
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×#6392C2211yescharacteristic
13#89700—21no—
14#89357—22no—
15#89346—21no—
16#89346C2211yescharacteristic
17C22⋊(C7⋊C16)C2212yes—
18#89284—22no—
19#12288—71nomaximal, Sylow
20C2×C7×#41—41no—
21C2×C7×D8—21no—
22C2×C7×D8D412yes—
23C22×C7×C8D411yescharacteristic
24C22×C7×C8C2311yescharacteristic
25C7×C4⋊Q8—21no—
26C7×#127—21no—
27C7×#124—41no—
28C7×#121—22no—
29C7×#121—41no—
30C4×C7×D4—22no—
31C4×C7×D4—41no—
32C2×C42×C7D411yescharacteristic
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×Dic7⋊C4—41no—
36C7×#107—21no—
37C7×#107D412yes—
38C7×C4⋊C8—22no—
39C7×#102—22no—
40C7×#102—22no—
41C7⋊#95—41no—
42#6392—22no—
43C2×C7⋊C16—22no—
44#3245D711yescharacteristic
45#2232—72no—
46#2086—71no—
47#1347—72no—
48#1322—71no—
49C7×#41—82no—
50C7×#41—82no—
51C2×C7×Q8—41no—
52C2×C7×D4—22no—
53C2×C7×D4—41no—
54C2×C7×D4—41no—
55C2×C7×D4—42no—
56C22×C4×C7—42no—
57C22×C4×C7D811yescharacteristic
58C22×C4×C7D811yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×Dic7—41no—
61C22×Dic7—41no—
62C7×D8—42no—
63C7×D8—81no—
64C2×C7×C8—21no—
65C2×C7×C8—41no—
66C2×C7×C8—41no—
67C2×C7×C8C2×D412yes—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—41no—
70C7×C22⋊C4—42no—
71C7×C22⋊C4—42no—
72C42×C7—21no—
73C42×C7—21no—
74C4⋊Dic7—22no—
75C4⋊Dic7—41no—
76C4⋊Dic7—41no—
77C4⋊Dic7—41no—
78Dic7⋊C4—82no—
79C7⋊C16—42no—
80#532—21no—
81#474—22no—
82#466—21no—
83#466C2×D712yes—
84#459—22no—
85C2×#107—71no—
86C2×C4⋊C8C2×D711yescharacteristic
87#389—141no—
88#366—142no—
89#357—142no—
90C22⋊C16—72no—
91#340—71no—
92#340—141no—
93C23×C7—42no—
94C23×C7C2×D811yescharacteristic
95C7×Q8—82no—
96C7×D4—42no—
97C7×D4—42no—
98C7×D4—82no—
99C7×D4—82no—
100C7×D4—82no—
101C7×D4—82no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—41no—
108C2×C4×C7—42no—
109C2×C4×C7—84no—
110C2×C4×C7D1612yes—
111C2×C4×C7#12011yescharacteristic
112C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
113C2×Dic7—42no—
114C2×Dic7—81no—
115C2×Dic7—81no—
116C2×Dic7—81no—
117C2×Dic7—81no—
118C2×Dic7—81no—
119C7×C8—22no—
120C7×C8—41no—
121C2×#41—41no—
122C2×D8—21no—
123C2×D8C22⋊D712yes—
124C22×C8C22⋊D711yescharacteristic
125C22×C8C22×D711yescharacteristic
126C4⋊Q8—21no—
127#127—21no—
128#124—41no—
129#121—22no—
130#121—41no—
131C4×D4—22no—
132C4×D4—41no—
133C2×C4⋊C4—141no—
134C2×C4⋊C4—141no—
135C2×C4⋊C4—281no—
136C2×C42C22⋊D711yescharacteristic
137C2×C16—142no—
138#107—21no—
139#107—142no—
140#107C22⋊D712yes—
141C4⋊C8—22no—
142#102—22no—
143#102—22no—
144#95—281no—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7—82no—
148C22×C7—82no—
149C22×C7—82no—
150C22×C7#44711yescharacteristic
151C22×C7C2×D1611yescharacteristic
152C22×C7#50911yescharacteristic
153C4×C7—21no—
154C4×C7—22no—
155C4×C7—41no—
156C4×C7—82no—
157C4×C7#45611yescharacteristic
158C4×C7#45811yescharacteristic
159Dic7—82no—
160Dic7—161no—
161#41—82no—
162#41—82no—
163C2×Q8—41no—
164C2×D4—22no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—42no—
168C22×C4—42no—
169C22×C4—281no—
170C22×C4—281no—
171C22×C4#114111yescharacteristic
172C22×C4#114111yescharacteristic
173C22×C4C2×C22⋊D711yescharacteristic
174D8—42no—
175D8—81no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×C22⋊D712yes—
180C4⋊C4—22no—
181C4⋊C4—41no—
182C4⋊C4—142no—
183C4⋊C4—281no—
184C4⋊C4—281no—
185C4⋊C4—281no—
186C4⋊C4—562no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C42—21no—
190C42—21no—
191C16—282no—
192C2×C7—21no—
193C2×C7—21no—
194C2×C7—162no—
195C2×C7#161011yescharacteristic
196C2×C7#218411yescharacteristic
197C2×C7#219111yescharacteristic
198C23—42no—
199C23C2×#114111yescharacteristic
200Q8—82no—
201D4—42no—
202D4—42no—
203D4—82no—
204D4—82no—
205D4—82no—
206D4—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4—282no—
216C2×C4—561no—
217C2×C4—561no—
218C2×C4—561no—
219C2×C4—561no—
220C2×C4—561no—
221C2×C4#640012yes—
222C2×C4#649511yescharacteristic
223C2×C4#651611yescharacteristic
224C8—22no—
225C8—41no—
226C7#1228811yescharacteristic, complemented, Sylow
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22C2×#640011yescharacteristic
233C22#8991911yescharacteristic
234C22#8998911yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—562no—
240C4—1121no—
241C4#9001511yescharacteristic
242C4#9001811yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2#1172962811yescharacteristic
247C2#1172978911yescharacteristic
248C2#1172986711yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
Fitting, prime radicalC7×#3245C2
socleC22×C7#447
hypercenter#3245D7
hypocenterC7#12288
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
2. #11728822C2semidirect
3. #11728216C2semidirect
4. #11725580C2semidirect
5. #11725515C2semidirect
226. C7#12288semidirect

Maximal subgroups

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

subgroupindex
C7×#32452
#117288222
#117282162
#117255802
#117255152
#122887