Finite Group Databasebeta

← #34057623

Subgroups — #34057623

Order 19072.

326 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic298C211yesmaximal, characteristic
2#23750035C212yesmaximal
3#23750021C211yesmaximal, characteristic
4#23749929C212yesmaximal
5#23749920C211yesmaximal, characteristic
6#23749910C212yesmaximal
7#23749907C211yesmaximal, characteristic
8#23749756C211yesmaximal, characteristic
9#23749730C212yesmaximal
10C4×#14558726C211yesmaximal, characteristic
11#557×C149C211yesmaximal, characteristic
12C2×#14558726C2211yescharacteristic
13#18011911C2212yes—
14#18011911C2211yescharacteristic
15#18011910—21no—
16#18011910—22no—
17Q8×Dic149—21no—
18C4×Dic298C2211yescharacteristic
19C4×Dic298C2212yes—
20#18011844C2212yes—
21#18011844C2211yescharacteristic
22#18011842C2212yes—
23#18011842C2212yes—
24#18011842C2212yes—
25C4⋊#11924746C2211yescharacteristic
26C4⋊#11924746C2212yes—
27#18011835C2211yescharacteristic
28#18011835C2212yes—
29#18011834C2212yes—
30#18011834C2211yescharacteristic
31#18011822C2212yes—
32C4×#11924746C2211yescharacteristic
33C4⋊Q8×C149—21no—
34C4⋊Q8×C149C2211yescharacteristic
35#125×C149—22no—
36#125×C149C2212yes—
37C4×Q8×C149—21no—
38C4×Q8×C149C2212yes—
39C4×Q8×C149C2211yescharacteristic
40C4×Q8×C149C2211yescharacteristic
41C2×Dic298—22no—
42C2×Dic298—21no—
43C2×Dic298C2311yescharacteristic
44#14558726Q814yescomplemented
45C2×#11924746C2311yescharacteristic
46C2×#11924746C2312yes—
47C2×#11924746C2311yescharacteristic
48C4⋊Dic149—22no—
49C4⋊Dic149—21no—
50C4⋊Dic149C2311yescharacteristic
51C4⋊Dic149C2312yes—
52#14558715—22no—
53#14558715—21no—
54C4×Dic149—22no—
55C4×Dic149—21no—
56C2×Q8×C149—21no—
57C2×Q8×C149D411yescharacteristic
58C2×Q8×C149C2311yescharacteristic
59C4⋊C4×C149—22no—
60C4⋊C4×C149—21no—
61C4⋊C4×C149—22no—
62C4⋊C4×C149—21no—
63C4⋊C4×C149—22no—
64C4⋊C4×C149D411yescharacteristic
65C4⋊C4×C149D412yes—
66C4⋊C4×C149C2311yescharacteristic
67C4⋊C4×C149C2312yes—
68C42×C149—22no—
69C42×C149—21no—
70C42×C149C2312yes—
71C42×C149C2311yescharacteristic
72Dic298—21no—
73Dic298—82no—
74Dic298—81no—
75Dic298C2×Q812yes—
76C2×Dic149—21no—
77C2×Dic149—21no—
78C2×Dic149—22no—
79#11924746—21no—
80#11924746—22no—
81#11924746C2×Q812yes—
82Q8×C149—22no—
83Q8×C149—21no—
84Q8×C149—42no—
85Q8×C149C2×Q812yes—
86C2×C4×C149—21no—
87C2×C4×C149—21no—
88C2×C4×C149—22no—
89C2×C4×C149C2×D412yes—
90C2×C4×C149C2×D411yescharacteristic
91C2×C4×C149C2×D412yes—
92C2×C4×C149C2×D411yescharacteristic
93C2×C4×C149C2411yescharacteristic
94C22×C149C22×D411yescharacteristic
95C4×C149—22no—
96C4×C149—22no—
97C4×C149—22no—
98C4×C149—21no—
99C4×C149—42no—
100C4×C149—41no—
101C4×C149#13712yes—
102C4×C149C22×Q811yescharacteristic
103C4×C1492−1+411yescharacteristic
104Dic149—22no—
105Dic149—41no—
106Dic149—42no—
107C2×C149D4×Q811yescharacteristic
108C2×C149C2×#13711yescharacteristic
109C2×C149#57711yescharacteristic
110C149#338711yescharacteristic, complemented, Sylow
111#3387—1491nomaximal, Sylow
112Q82—1491no—
113#557D14911yescharacteristic
114#501—1492no—
115#498—1491no—
116#479—1492no—
117#477—1491no—
118#475—1491no—
119#475—1492no—
120#446—1491no—
121#441—1492no—
122C4×Q16—1491no—
123C2×Q16—1491no—
124C4⋊Q8—21no—
125C4⋊Q8—2981no—
126C4⋊Q8—2982no—
127C4⋊Q8—1492no—
128C4⋊Q8—1491no—
129C4⋊Q8C2×D14911yescharacteristic
130#125—22no—
131#125C2×D14912yes—
132C4×Q8—21no—
133C4×Q8—2981no—
134C4×Q8—1491no—
135C4×Q8—1492no—
136C4×Q8C2×D14911yescharacteristic
137C4×Q8C2×D14912yes—
138C4×Q8C2×D14911yescharacteristic
139#107—1491no—
140#107—1492no—
141#106—1492no—
142#106—1492no—
143#106—1492no—
144C4⋊C8—1491no—
145C4⋊C8—1492no—
146#103—1491no—
147#103—1492no—
148#103—1491no—
149#103—1492no—
150#97—1492no—
151C4×C8—1491no—
152C2×Q8—21no—
153C2×Q8—2982no—
154C2×Q8—2981no—
155C2×Q8—1491no—
156C2×Q8C22⋊D14911yescharacteristic
157C2×Q8C22×D14911yescharacteristic
158Q16—1494no—
159C2×C8—1491no—
160C2×C8—1492no—
161C2×C8—1491no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—2982no—
168C4⋊C4—2981no—
169C4⋊C4—2982no—
170C4⋊C4—2981no—
171C4⋊C4—1491no—
172C4⋊C4—1492no—
173C4⋊C4C22⋊D14911yescharacteristic
174C4⋊C4C22⋊D14912yes—
175C4⋊C4C22×D14912yes—
176C4⋊C4C22×D14911yescharacteristic
177C42—22no—
178C42—21no—
179C42—2982no—
180C42—2981no—
181C42C22×D14911yescharacteristic
182C42C22×D14912yes—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—2981no—
187Q8—1492no—
188Q8—11922no—
189Q8—11921no—
190Q8Q8×D14912yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—2981no—
195C2×C4—2982no—
196C2×C4—2981no—
197C2×C4C2×C22⋊D14912yes—
198C2×C4C2×C22⋊D14912yes—
199C2×C4C2×C22⋊D14911yescharacteristic
200C2×C4C2×C22⋊D14911yescharacteristic
201C2×C4C23×D14911yescharacteristic
202C8—2981no—
203C8—2982no—
204C8—1492no—
205C22C22×C22⋊D14911yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—22no—
210C4—42no—
211C4—41no—
212C4—2982no—
213C4—5962no—
214C4—5961no—
215C4#1801194112yes—
216C4C2×Q8×D14911yescharacteristic
217C4#1801195911yescharacteristic
218C2Q8×C22⋊D14911yescharacteristic
219C2C2×#1801194111yescharacteristic
220C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#557×C149C2
socle, prime radicalC22×C149C22×D4
hypercenter#557D149
hypocenterC149#3387
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
44. #14558726Q8semidirect
110. C149#3387semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2982
#237500352
#237500212
#237499292
#237499202
#237499102
#237499072
#237497562
#237497302
C4×#145587262
#557×C1492
#3387149