Finite Group Databasebeta

← #13263012

Subgroups — #13263012

Order 1792.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#11736093C211yesmaximal, characteristic
2#11726847C211yesmaximal, characteristic, complemented
3#11726846C211yesmaximal, characteristic, complemented
4C7×#1315C211yesmaximal, characteristic
5#11725495C211yesmaximal, characteristic
6#11725331C211yesmaximal, characteristic, complemented
7#11725331C211yesmaximal, characteristic, complemented
8#41×Dic7C411yescharacteristic
9C7×#408C2211yescharacteristic
10#90006C411yescharacteristic
11#89891C411yescharacteristic
12M16×Dic7C411yescharacteristic
13#89874C2211yescharacteristic
14C2×C8×Dic7C2211yescharacteristic
15#89492—21no—
16#89492—21no—
17#89490—21no—
18#89490—21no—
19C7×C22⋊C16C2211yescharacteristic
20C7×C22⋊C16C2211yescharacteristic
21C22⋊(C7⋊C16)C2211yescharacteristic
22C22⋊(C7⋊C16)C2211yescharacteristic
23#89297—21no—
24Dic7⋊C16—21no—
25Dic7⋊C16—21no—
26C16×Dic7—21no—
27#11464—71nomaximal, Sylow
28C2×C7×#41C2×C411yescharacteristic
29C2×C7×M16C2×C411yescharacteristic
30C22×C7×C8C2311yescharacteristic
31Q8×Dic7C811yescharacteristic
32Q8×Dic7C811yescharacteristic
33D4×Dic7—21no—
34D4×Dic7—21no—
35D4×Dic7C811yescharacteristic
36D4×Dic7C811yescharacteristic
37#6489—21no—
38#6489C2×C411yescharacteristic
39C2×C4×Dic7—21no—
40C2×C4×Dic7C2×C411yescharacteristic
41C2×#1136C2×C411yescharacteristic
42C22×C7⋊C8C2×C411yescharacteristic
43C2×C7×C16—21no—
44C2×C7×C16—21no—
45C7×C22⋊C8—21no—
46C7×C22⋊C8—21no—
47C22⋊(C7⋊C8)—21no—
48C22⋊(C7⋊C8)—21no—
49#6389—21no—
50Dic7⋊C8—21no—
51Dic7⋊C8—21no—
52Dic7⋊C8—21no—
53Dic7⋊C8—21no—
54C8×Dic7—21no—
55C8×Dic7D411yescharacteristic
56C8×Dic7D411yescharacteristic
57C8×Dic7D411yescharacteristic
58C8×Dic7D411yescharacteristic
59C2×C7⋊C16—21no—
60C2×C7⋊C16—21no—
61#2922—71no—
62#2163—71no—
63#2162—71no—
64#1323—71no—
65#1323—71no—
66#1315—71no—
67#1315D711yescharacteristic
68C7×#41—41no—
69C7×#41—41no—
70C2×C7×Q8C2×C811yescharacteristic
71C2×C7×D4—21no—
72C2×C7×D4C2×C811yescharacteristic
73C22×C4×C7—21no—
74C22×C4×C7C22×C411yescharacteristic
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7M1611yescharacteristic
78C22×Dic7M1611yescharacteristic
79C7×M16—41no—
80C2×C7×C8—21no—
81C2×C7×C8—41no—
82C2×C7×C8C2×D411yescharacteristic
83C2×C7×C8C2×D411yescharacteristic
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2×C811yescharacteristic
90C4⋊Dic7C2×C811yescharacteristic
91C4×Dic7—21no—
92C4×Dic7—21no—
93C4×Dic7—21no—
94C4×Dic7C22⋊C411yescharacteristic
95C4×Dic7C22⋊C411yescharacteristic
96C4×Dic7C2×C811yescharacteristic
97C4×Dic7C2×C811yescharacteristic
98#1136—41no—
99C2×C7⋊C8—21no—
100C2×C7⋊C8—41no—
101C2×C7⋊C8C22⋊C411yescharacteristic
102C2×C7⋊C8C22⋊C411yescharacteristic
103C7×C16—41no—
104C7×C16—41no—
105C7⋊C16—41no—
106C7⋊C16—41no—
107C4×#41—71no—
108#433—141no—
109#433—141no—
110#431—141no—
111#431—141no—
112#424—71no—
113#424—71no—
114#408—71no—
115#408C2×D711yescharacteristic
116C4×M16—71no—
117C2×C4×C8—71no—
118C4⋊C16—141no—
119C4⋊C16—141no—
120C22⋊C16—71no—
121C22⋊C16—71no—
122C22⋊C16C2×D711yescharacteristic
123C22⋊C16C2×D711yescharacteristic
124#346—141no—
125C4×C16—141no—
126C23×C7—21no—
127C23×C7C2×M1611yescharacteristic
128C7×Q8—41no—
129C7×D4—41no—
130C7×D4—42no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7C2×C22⋊C411yescharacteristic
137C2×C4×C7C22×C811yescharacteristic
138C2×Dic7—22no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—41no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7C22⋊C811yescharacteristic
145C2×Dic7C22⋊C811yescharacteristic
146C2×Dic7C22⋊C811yescharacteristic
147C2×Dic7C22⋊C811yescharacteristic
148C7×C8—21no—
149C7×C8—21no—
150C7×C8—41no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8—41no—
154C2×#41C4×D711yescharacteristic
155C2×M16—71no—
156C2×M16C4×D711yescharacteristic
157C22×C8—71no—
158C22×C8C22×D711yescharacteristic
159C4×Q8—71no—
160C4×Q8—71no—
161C4×D4—71no—
162C4×D4—71no—
163C4×D4—141no—
164C4×D4—141no—
165#117—71no—
166#117—141no—
167C2×C42—71no—
168C2×C42—141no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—141no—
172C2×C16—141no—
173C4⋊C8—141no—
174C4⋊C8—141no—
175C4⋊C8—141no—
176C4⋊C8—141no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—141no—
180C22⋊C8—141no—
181#97—141no—
182C4×C8—71no—
183C4×C8—71no—
184C4×C8—71no—
185C4×C8—71no—
186C4×C8—141no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7C2×C22⋊C811yescharacteristic
194C4×C7—21no—
195C4×C7—41no—
196C4×C7#41011yescharacteristic
197C4×C7#41311yescharacteristic
198Dic7—21no—
199Dic7—21no—
200Dic7—21no—
201Dic7—41no—
202Dic7—41no—
203Dic7#35012yes—
204#41—41no—
205#41—41no—
206C2×Q8C8×D711yescharacteristic
207C2×D4—21no—
208C2×D4C8×D711yescharacteristic
209C22×C4—21no—
210C22×C4—71no—
211C22×C4—71no—
212C22×C4—141no—
213C22×C4—141no—
214C22×C4C2×C4×D711yescharacteristic
215M16—41no—
216M16—281no—
217C2×C8—21no—
218C2×C8—41no—
219C2×C8—71no—
220C2×C8—71no—
221C2×C8—141no—
222C2×C8—281no—
223C2×C8D4×D711yescharacteristic
224C2×C8D4×D711yescharacteristic
225C4⋊C4—71no—
226C4⋊C4—71no—
227C4⋊C4—141no—
228C4⋊C4—141no—
229C22⋊C4—141no—
230C22⋊C4—141no—
231C22⋊C4—141no—
232C42—71no—
233C42—71no—
234C42—71no—
235C42—71no—
236C42—141no—
237C42—141no—
238C42—141no—
239C16—41no—
240C16—41no—
241C16—281no—
242C16—281no—
243C2×C7—41no—
244C2×C7—81no—
245C2×C7#145811yescharacteristic
246C2×C7C2×#35011yescharacteristic
247C2×C7#211711yescharacteristic
248C23—21no—
249C23D7×M1611yescharacteristic
250Q8—41no—
251D4—41no—
252D4—42no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—71no—
259C2×C4—71no—
260C2×C4—71no—
261C2×C4—71no—
262C2×C4—142no—
263C2×C4—141no—
264C2×C4—141no—
265C2×C4—282no—
266C2×C4—281no—
267C2×C4—282no—
268C2×C4D7×C22⋊C411yescharacteristic
269C2×C4C2×C8×D711yescharacteristic
270C8—21no—
271C8—21no—
272C8—41no—
273C8—141no—
274C8—141no—
275C8—281no—
276C7#1146411yescharacteristic, complemented, Sylow
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22D7×C22⋊C811yescharacteristic
284C4—21no—
285C4—41no—
286C4—72no—
287C4—141no—
288C4—141no—
289C4—141no—
290C4—281no—
291C4—281no—
292C4#8951211yescharacteristic
293C4#8951711yescharacteristic
294C2—41no—
295C2—81no—
296C2#1172585411yescharacteristic
297C2D7×#35011yescharacteristic
298C2#1172687911yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC2×C4×C7C22×C8
FrattiniC22×C8C22×D7
FittingC7×#1315C2
socleC22×C7C2×C22⋊C8
prime radicalC2×C7×#41C2×C4
cosocleC22×C7×C8C23
hypercenter#1315D7
hypocenterC7#11464
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #11726847C2semidirect
3. #11726846C2semidirect
6. #11725331C2semidirect
7. #11725331C2semidirect
276. C7#11464semidirect

Maximal subgroups

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

subgroupindex
#117360932
#117268472
#117268462
C7×#13152
#117254952
#117253312
#114647