Subgroups — #27187703

Order 12928.

556 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692342	C₂	1	2	yes	maximal
2	#20266142	C₂	1	2	yes	maximal, complemented
3	#20266141	C₂	1	4	yes	maximal, complemented
4	C₄×#13108084	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#13108083	C₂	1	2	yes	maximal, complemented
6	#20266136	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692310	C₂	1	2	yes	maximal, complemented
8	C₄×#13108073	C₂	1	1	yes	maximal, characteristic
9	#15692342	C₄	1	16	yes	complemented
10	C₂×#13108084	C₂²	1	2	yes	—
11	C₂×#13108083	C₂²	1	4	yes	—
12	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	—
13	C₂²×#11717581	C₂²	1	4	yes	—
14	#15692310	C₂²	1	1	yes	characteristic
15	#15692310	C₂²	1	2	yes	—
16	#15692310	C₂²	1	4	yes	—
17	#15692310	C₂²	1	1	yes	characteristic
18	#15692310	C₂²	1	2	yes	—
19	C₄×#11717581	C₂²	1	4	yes	—
20	C₄×#11717581	C₂²	1	2	yes	—
21	C₂×#13108073	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₀₂	C₄	1	1	yes	characteristic
23	#15692220	C₄	1	1	yes	characteristic
24	D₄×Dic₁₀₁	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₀₁	C₄	1	2	yes	—
26	#15692215	C₄	1	2	yes	—
27	#15692213	C₂²	1	1	yes	characteristic
28	#15692211	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
30	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₀₁	C₄	1	1	yes	characteristic
32	#13108084	C₂×C₄	1	8	yes	complemented
33	#13108083	C₂×C₄	1	16	yes	complemented
34	D₁₀₁⋊C₈	C₂×C₄	1	8	yes	complemented
35	C₂×#11717581	C₂×C₄	1	16	yes	complemented
36	C₂×#11717581	C₂³	1	4	yes	—
37	C₂×#11717581	C₂³	1	2	yes	—
38	C₂×#11717581	C₂³	1	2	yes	—
39	#13108073	C₂×C₄	1	8	yes	—
40	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
48	#13108053	C₂×C₄	1	2	yes	—
49	#13108053	C₂×C₄	1	2	yes	—
50	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
51	#13108051	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
56	C₄×Dic₁₀₁	C₂³	1	2	yes	—
57	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
62	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
64	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
65	D₄×C₁₀₁	C₄²	1	4	yes	—
66	C₂×C₄×C₁₀₁	—	4	2	no	—
67	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²⋊D₁₀₁	C₄²	1	8	yes	—
72	C₂×Dic₁₀₁	—	4	2	no	—
73	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
78	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
79	C₄×D₁₀₁	—	4	1	no	—
80	C₄×D₁₀₁	—	4	1	no	—
81	C₄×D₁₀₁	C₂²×C₄	1	4	yes	—
82	Dic₂₀₂	C₄²	1	4	yes	—
83	#11717581	C₂²×C₄	1	4	yes	—
84	#11717581	C₂²×C₄	1	8	yes	—
85	#11717581	C₂²×C₄	1	4	yes	—
86	C₂²×C₁₀₁	—	2	2	no	—
87	C₂²×C₁₀₁	—	2	2	no	—
88	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	—
89	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₁₀₁	—	2	1	no	—
91	C₂×D₁₀₁	—	2	1	no	—
92	C₂×D₁₀₁	C₂×C₄²	1	2	yes	—
93	C₄×C₁₀₁	—	2	1	no	—
94	C₄×C₁₀₁	—	2	1	no	—
95	C₄×C₁₀₁	—	2	2	no	—
96	C₄×C₁₀₁	C₂×C₄²	1	2	yes	—
97	Dic₁₀₁	—	2	1	no	—
98	Dic₁₀₁	—	2	1	no	—
99	Dic₁₀₁	—	2	2	no	—
100	Dic₁₀₁	C₂×C₄²	1	4	yes	—
101	Dic₁₀₁	C₂³×C₄	1	2	yes	—
102	C₂×C₁₀₁	—	2	4	no	—
103	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
104	C₂×C₁₀₁	#568	1	1	yes	characteristic
105	C₂×C₁₀₁	#568	1	1	yes	characteristic
106	D₁₀₁	—	2	2	no	—
107	#2875	—	101	1	no	maximal, Sylow
108	C₁₀₁	#2875	1	1	yes	characteristic, complemented, Sylow
109	C₂×#131	—	101	2	no	—
110	C₄×#41	—	101	1	no	—
111	#405	—	101	4	no	—
112	#405	—	101	2	no	—
113	C₄×M₁₆	—	101	2	no	—
114	C₄×M₁₆	—	101	1	no	—
115	C₂×#97	—	101	2	no	—
116	C₂×#97	—	101	1	no	—
117	C₂×#41	—	101	1	no	—
118	#131	—	101	16	no	—
119	C₂×M₁₆	—	101	4	no	—
120	C₂×M₁₆	—	101	2	no	—
121	C₂²×C₈	—	101	4	no	—
122	C₂²×C₈	—	101	2	no	—
123	C₄×Q₈	—	101	1	no	—
124	C₄×Q₈	—	101	1	no	—
125	C₄×D₄	—	101	1	no	—
126	C₄×D₄	—	101	2	no	—
127	C₄×D₄	—	101	2	no	—
128	C₄×D₄	#87938	1	1	yes	characteristic
129	#117	—	101	1	no	—
130	#117	—	101	2	no	—
131	C₂×C₄²	—	101	2	no	—
132	C₂×C₄²	—	101	1	no	—
133	#97	—	101	1	no	—
134	#97	—	101	2	no	—
135	#97	—	101	1	no	—
136	#97	—	101	4	no	—
137	#97	—	101	2	no	—
138	C₄×C₈	—	101	4	no	—
139	C₄×C₈	—	101	2	no	—
140	#41	—	101	8	no	—
141	C₂×Q₈	—	101	1	no	—
142	C₂×D₄	—	101	2	no	—
143	C₂×D₄	C₂×#87938	1	1	yes	characteristic
144	C₂²×C₄	—	101	2	no	—
145	C₂²×C₄	—	101	1	no	—
146	C₂²×C₄	—	101	2	no	—
147	C₂²×C₄	—	101	1	no	—
148	C₂²×C₄	—	101	1	no	—
149	C₂²×C₄	C₂×#87938	1	2	yes	—
150	M₁₆	—	101	16	no	—
151	M₁₆	—	101	8	no	—
152	C₂×C₈	—	101	4	no	—
153	C₂×C₈	—	101	2	no	—
154	C₂×C₈	—	101	2	no	—
155	C₂×C₈	—	101	16	no	—
156	C₂×C₈	—	101	8	no	—
157	C₄⋊C₄	—	101	2	no	—
158	C₄⋊C₄	—	101	2	no	—
159	C₄⋊C₄	—	101	1	no	—
160	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic
161	C₂²⋊C₄	—	101	2	no	—
162	C₂²⋊C₄	—	101	2	no	—
163	C₂²⋊C₄	C₂×#87938	1	2	yes	—
164	C₄²	—	101	2	no	—
165	C₄²	—	101	2	no	—
166	C₄²	—	101	1	no	—
167	C₄²	—	101	2	no	—
168	C₄²	—	101	1	no	—
169	C₄²	—	101	1	no	—
170	C₄²	C₂×#87938	1	1	yes	characteristic
171	C₂³	—	101	1	no	—
172	C₂³	C₂²×#87938	1	2	yes	—
173	Q₈	—	101	4	no	—
174	D₄	—	101	8	no	—
175	D₄	C₄×#87938	1	4	yes	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	101	8	no	—
178	C₂×C₄	—	101	1	no	—
179	C₂×C₄	—	101	2	no	—
180	C₂×C₄	—	101	4	no	—
181	C₂×C₄	—	101	1	no	—
182	C₂×C₄	—	101	1	no	—
183	C₂×C₄	—	101	2	no	—
184	C₂×C₄	—	404	1	no	—
185	C₂×C₄	—	404	1	no	—
186	C₂×C₄	—	404	2	no	—
187	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
188	C₂×C₄	C₂²×#87938	1	2	yes	—
189	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
190	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
191	C₈	—	101	4	no	—
192	C₈	—	101	8	no	—
193	C₈	—	101	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	101	2	no	—
197	C₂²	—	202	1	no	—
198	C₂²	—	202	1	no	—
199	C₂²	C₂×C₄×#87938	1	4	yes	—
200	C₂²	C₂³×#87938	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	101	2	no	—
205	C₄	—	101	4	no	—
206	C₄	—	202	1	no	—
207	C₄	—	202	1	no	—
208	C₄	—	202	2	no	—
209	C₄	C₂×C₄×#87938	1	2	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	202	2	no	—
212	C₂	C₂²×C₄×#87938	1	1	yes	characteristic
213	C₂	#20266255	1	1	yes	characteristic
214	C₂	#20266256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator	C₂×C₁₀₁	C₂²×C₄²
Fitting	C₄×D₄×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂³×C₄
prime radical	C₂×#13108073	C₂²
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₄×D₄	#87938
hypocenter	C₁₀₁	#2875
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #20266142	C₂	semidirect
3. #20266141	C₂	semidirect
4. C₄×#13108084	C₂	semidirect
5. C₄×#13108083	C₂	semidirect
6. #20266136	C₂	semidirect
7. C₂×#15692310	C₂	semidirect
9. #15692342	C₄	semidirect
32. #13108084	C₂×C₄	semidirect
33. #13108083	C₂×C₄	semidirect
34. D₁₀₁⋊C₈	C₂×C₄	semidirect
35. C₂×#11717581	C₂×C₄	semidirect
108. C₁₀₁	#2875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692342	2
#20266142	2
#20266141	2
C₄×#13108084	2
C₄×#13108083	2
#20266136	2
C₂×#15692310	2
C₄×#13108073	2
#2875	101