Subgroups — D₄×#13108083 #27187868

Order 12928.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108083	C₂	1	2	yes	maximal, complemented
2	#20266212	C₂	1	2	yes	maximal, complemented
3	#20266208	C₂	1	4	yes	maximal, complemented
4	D₄×#11717581	C₂	1	2	yes	maximal, complemented
5	#20266180	C₂	1	1	yes	maximal, characteristic, complemented
6	#20266147	C₂	1	2	yes	maximal, complemented
7	C₄×#13108083	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#13108083	—	2	8	no	—
10	C₂×#13108083	C₂²	1	2	yes	—
11	C₂×#13108083	C₂²	1	1	yes	characteristic
12	C₂×#13108083	C₂²	1	1	yes	characteristic
13	C₂×#13108083	C₂²	1	4	yes	complemented
14	C₂²×#11717581	C₂²	1	4	yes	complemented
15	#15692322	C₂²	1	2	yes	complemented
16	C₄⋊#11717581	C₂²	1	2	yes	complemented
17	C₂²⋊#11717581	C₂²	1	4	yes	complemented
18	C₂²⋊#11717581	C₂²	1	4	yes	complemented
19	#15692310	C₂²	1	2	yes	complemented
20	C₄×#11717581	C₂²	1	2	yes	complemented
21	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
22	D₄×Dic₁₀₁	C₄	1	2	yes	—
23	D₄×Dic₁₀₁	C₄	1	1	yes	characteristic
24	D₄×Dic₁₀₁	C₄	1	1	yes	characteristic
25	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₀₁	C₂²	1	2	yes	—
27	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	—
30	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₂²×D₄×C₁₀₁	C₄	1	1	yes	characteristic
32	#13108083	—	2	4	no	—
33	#13108083	—	2	2	no	—
34	#13108083	—	2	8	no	—
35	#13108083	D₄	1	4	yes	direct product factor
36	C₂×#11717581	—	2	8	no	—
37	C₂×#11717581	C₂³	1	2	yes	—
38	C₂×#11717581	C₂³	1	4	yes	—
39	C₂×#11717581	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	—	2	4	no	—
41	C₂²×Dic₁₀₁	—	2	4	no	—
42	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
53	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
54	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
59	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
60	C₂⁴×C₁₀₁	C₂×C₄	1	2	yes	—
61	C₂×D₄×C₁₀₁	—	2	4	no	—
62	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
63	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
66	C₂³×C₁₀₁	—	2	4	no	—
67	C₂³×C₁₀₁	—	2	4	no	—
68	C₂³×C₁₀₁	—	4	2	no	—
69	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
72	D₄×C₁₀₁	—	2	4	no	—
73	D₄×C₁₀₁	—	2	2	no	—
74	D₄×C₁₀₁	M₁₆	1	4	yes	complemented
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₀₁	—	2	4	no	—
79	C₂×Dic₁₀₁	—	2	4	no	—
80	C₂×Dic₁₀₁	—	2	4	no	—
81	C₂×Dic₁₀₁	—	2	2	no	—
82	C₂×Dic₁₀₁	—	2	2	no	—
83	C₂×Dic₁₀₁	—	2	2	no	—
84	C₂×Dic₁₀₁	—	2	2	no	—
85	C₂×Dic₁₀₁	—	4	1	no	—
86	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
92	C₂×Dic₁₀₁	#41	1	2	yes	—
93	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
94	#11717581	—	2	4	no	—
95	#11717581	—	2	2	no	—
96	#11717581	C₂×D₄	1	4	yes	complemented
97	C₂²×C₁₀₁	—	2	2	no	—
98	C₂²×C₁₀₁	—	2	1	no	—
99	C₂²×C₁₀₁	—	2	1	no	—
100	C₂²×C₁₀₁	—	2	2	no	—
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	2	no	—
103	C₂²×C₁₀₁	—	2	2	no	—
104	C₂²×C₁₀₁	—	4	8	no	—
105	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
106	C₂²×C₁₀₁	C₂×M₁₆	1	4	yes	complemented
107	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₁₀₁	—	2	1	no	—
109	C₄×C₁₀₁	C₂×M₁₆	1	2	yes	complemented
110	Dic₁₀₁	—	2	2	no	—
111	Dic₁₀₁	—	2	2	no	—
112	Dic₁₀₁	—	2	1	no	—
113	Dic₁₀₁	—	2	1	no	—
114	Dic₁₀₁	C₄×D₄	1	2	yes	—
115	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
116	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₀₁	—	2	2	no	—
118	C₂×C₁₀₁	—	2	4	no	—
119	C₂×C₁₀₁	—	4	2	no	—
120	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₁₀₁	#568	1	1	yes	characteristic
123	D₄×M₁₆	—	101	1	no	maximal, Sylow
124	C₁₀₁	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	—	101	2	no	—
126	C₂×C₄×D₄	—	101	1	no	—
127	#436	—	101	2	no	—
128	#435	—	101	4	no	—
129	C₈×D₄	—	101	2	no	—
130	#423	—	101	1	no	—
131	#407	—	101	2	no	—
132	C₄×M₁₆	—	101	1	no	—
133	C₂²×D₄	#87938	1	1	yes	characteristic
134	C₂³×C₄	—	101	2	no	—
135	C₂×M₁₆	—	101	1	no	—
136	C₂×M₁₆	—	101	2	no	—
137	C₂×M₁₆	—	101	1	no	—
138	C₂×M₁₆	—	101	4	no	—
139	C₂×M₁₆	—	202	8	no	—
140	C₂²×C₈	—	101	4	no	—
141	C₄×D₄	—	101	1	no	—
142	C₄×D₄	—	101	2	no	—
143	C₄×D₄	—	101	1	no	—
144	C₄×D₄	—	101	2	no	—
145	C₄×D₄	—	101	1	no	—
146	C₄×D₄	—	101	1	no	—
147	C₂×C₄⋊C₄	—	101	1	no	—
148	C₂×C₂²⋊C₄	—	101	2	no	—
149	C₂×C₄²	—	101	1	no	—
150	C₄⋊C₈	—	101	2	no	—
151	C₄⋊C₈	—	101	2	no	—
152	C₂²⋊C₈	—	101	4	no	—
153	C₂²⋊C₈	—	101	4	no	—
154	#97	—	101	2	no	—
155	C₄×C₈	—	101	2	no	—
156	C₂⁴	C₂×#87938	1	2	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	C₂×#87938	1	2	yes	—
159	C₂×D₄	C₂×#87938	1	1	yes	characteristic
160	C₂×D₄	C₂×#87938	1	1	yes	characteristic
161	C₂²×C₄	—	101	1	no	—
162	C₂²×C₄	—	101	1	no	—
163	C₂²×C₄	—	101	2	no	—
164	C₂²×C₄	—	101	2	no	—
165	C₂²×C₄	—	101	2	no	—
166	C₂²×C₄	—	101	2	no	—
167	C₂²×C₄	—	101	2	no	—
168	C₂²×C₄	—	202	4	no	—
169	C₂²×C₄	—	202	4	no	—
170	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
171	M₁₆	—	101	4	no	—
172	M₁₆	—	202	8	no	—
173	M₁₆	—	202	2	no	—
174	M₁₆	—	202	4	no	—
175	C₂×C₈	—	101	2	no	—
176	C₂×C₈	—	101	2	no	—
177	C₂×C₈	—	101	4	no	—
178	C₂×C₈	—	202	8	no	—
179	C₄⋊C₄	—	101	2	no	—
180	C₄⋊C₄	—	101	1	no	—
181	C₄⋊C₄	—	101	1	no	—
182	C₂²⋊C₄	—	101	2	no	—
183	C₂²⋊C₄	—	101	2	no	—
184	C₂²⋊C₄	—	101	2	no	—
185	C₂²⋊C₄	—	101	2	no	—
186	C₄²	—	101	1	no	—
187	C₄²	—	101	1	no	—
188	C₄²	—	101	1	no	—
189	C₄²	—	101	1	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×#87938	1	2	yes	—
194	C₂³	C₂²×#87938	1	1	yes	characteristic
195	C₂³	C₂²×#87938	1	2	yes	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	#13108083	1	4	yes	direct product factor
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	101	2	no	—
201	C₂×C₄	—	101	2	no	—
202	C₂×C₄	—	101	1	no	—
203	C₂×C₄	—	101	1	no	—
204	C₂×C₄	—	101	1	no	—
205	C₂×C₄	—	101	1	no	—
206	C₂×C₄	—	101	2	no	—
207	C₂×C₄	—	101	2	no	—
208	C₂×C₄	—	404	1	no	—
209	C₂×C₄	—	202	2	no	—
210	C₂×C₄	—	202	2	no	—
211	C₂×C₄	—	202	2	no	—
212	C₂×C₄	—	202	2	no	—
213	C₂×C₄	—	202	4	no	—
214	C₂×C₄	—	202	4	no	—
215	C₂×C₄	—	202	4	no	—
216	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
217	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
218	C₈	—	101	4	no	—
219	C₈	—	202	4	no	—
220	C₈	—	202	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	D₄×#87938	1	2	yes	—
230	C₂²	C₂×#13108083	1	4	yes	complemented
231	C₂²	C₂³×#87938	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	101	1	no	—
234	C₄	—	101	1	no	—
235	C₄	—	101	2	no	—
236	C₄	—	202	2	no	—
237	C₄	—	202	2	no	—
238	C₄	—	202	1	no	—
239	C₄	—	202	1	no	—
240	C₄	C₂×#13108083	1	2	yes	complemented
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×D₄×#87938	1	1	yes	characteristic
245	C₂	C₂²×#13108083	1	1	yes	characteristic
246	C₂	#20266255	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	C₂²×D₄×C₁₀₁	C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₂²×D₄	#87938
hypocenter	C₁₀₁	D₄×M₁₆
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108083	C₂	semidirect
2. #20266212	C₂	semidirect
3. #20266208	C₂	semidirect
4. D₄×#11717581	C₂	semidirect
5. #20266180	C₂	semidirect
6. #20266147	C₂	semidirect
7. C₄×#13108083	C₂	semidirect
13. C₂×#13108083	C₂²	semidirect
14. C₂²×#11717581	C₂²	semidirect
15. #15692322	C₂²	semidirect
16. C₄⋊#11717581	C₂²	semidirect
17. C₂²⋊#11717581	C₂²	semidirect
18. C₂²⋊#11717581	C₂²	semidirect
19. #15692310	C₂²	semidirect
20. C₄×#11717581	C₂²	semidirect
35. #13108083	D₄	direct (N × H)
74. D₄×C₁₀₁	M₁₆	semidirect
96. #11717581	C₂×D₄	semidirect
106. C₂²×C₁₀₁	C₂×M₁₆	semidirect
109. C₄×C₁₀₁	C₂×M₁₆	semidirect
124. C₁₀₁	D₄×M₁₆	semidirect
198. D₄	#13108083	direct (N × H)
230. C₂²	C₂×#13108083	semidirect
240. C₄	C₂×#13108083	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108083	2
#20266212	2
#20266208	2
D₄×#11717581	2
#20266180	2
#20266147	2
C₄×#13108083	2
C₂×D₄×Dic₁₀₁	2
D₄×M₁₆	101

Subgroups — D4×#13108083 #27187868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#13108083 #27187868