Subgroups — C₈×#13182028 #28371987

Order 13952.

456 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182028	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295150	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₁₀₉	C₂	1	2	yes	maximal, complemented
6	#20295089	C₂	1	2	yes	maximal, complemented
7	#20295078	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295076	C₂	1	2	yes	maximal, complemented
9	C₂×C₈×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182028	C₄	1	1	yes	characteristic
13	C₂²×#11724784	C₂²	1	2	yes	—
14	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
17	#16812664	C₄	1	1	yes	characteristic
18	D₄×Dic₁₀₉	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
20	#16812659	C₄	1	2	yes	—
21	#16812657	C₄	1	1	yes	characteristic
22	#16812655	C₄	1	2	yes	—
23	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
24	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
25	C₄⋊#11724784	C₂²	1	1	yes	characteristic
26	#16812628	C₂²	1	2	yes	—
27	#16812628	C₂²	1	2	yes	—
28	C₂²⋊#11724784	C₂²	1	2	yes	—
29	#16812611	C₂²	1	2	yes	—
30	C₄×#11724784	C₂²	1	1	yes	characteristic
31	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₀₉	C₂²	1	2	yes	—
34	C₈×Dic₁₀₉	C₂²	1	2	yes	—
35	C₈×Dic₁₀₉	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
37	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#13182028	C₈	1	8	yes	direct product factor
42	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×#11724784	—	2	4	no	—
50	C₂×#11724784	C₂³	1	2	yes	—
51	C₂×#11724784	C₂³	1	1	yes	characteristic
52	C₂×#11724784	C₂³	1	1	yes	characteristic
53	C₈×D₁₀₉	—	2	2	no	—
54	C₈×D₁₀₉	—	2	2	no	—
55	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
56	#13182008	C₂×C₄	1	2	yes	—
57	#13182008	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
59	#13182006	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₀₉	C₂³	1	2	yes	—
66	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
68	C₂×C₈×C₁₀₉	—	2	4	no	—
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
71	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
74	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₁₀₉	C₂×C₈	1	4	yes	complemented
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₁₀₉	C₂×C₈	1	8	yes	complemented
85	C₂×Dic₁₀₉	—	2	4	no	—
86	C₂×Dic₁₀₉	C₂×C₈	1	8	yes	complemented
87	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₀₉	—	2	1	no	—
93	C₄×D₁₀₉	—	2	1	no	—
94	C₄×D₁₀₉	—	2	2	no	—
95	C₄×D₁₀₉	C₂×C₈	1	4	yes	complemented
96	Dic₂₁₈	C₂×C₈	1	4	yes	complemented
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	1	no	—
99	C₈×C₁₀₉	#41	1	2	yes	complemented
100	#11724784	—	2	1	no	—
101	#11724784	—	2	2	no	—
102	#11724784	#41	1	2	yes	complemented
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	C₂²×C₈	1	4	yes	—
106	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₁₀₉	—	2	1	no	—
108	C₂×D₁₀₉	—	2	1	no	—
109	C₂×D₁₀₉	C₂²×C₈	1	2	yes	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	—	2	1	no	—
112	C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
113	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
115	Dic₁₀₉	—	2	1	no	—
116	Dic₁₀₉	—	2	2	no	—
117	Dic₁₀₉	C₂²×C₈	1	4	yes	—
118	Dic₁₀₉	C₂²×C₈	1	2	yes	—
119	Dic₁₀₉	#131	1	2	yes	—
120	C₂×C₁₀₉	—	2	4	no	—
121	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
122	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
123	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
124	D₁₀₉	—	2	2	no	—
125	C₈×#41	—	109	1	no	maximal, Sylow
126	C₁₀₉	C₈×#41	1	1	yes	characteristic, complemented, Sylow
127	C₄×#41	—	109	1	no	—
128	C₈×Q₈	—	109	1	no	—
129	C₈×Q₈	—	109	1	no	—
130	C₈×D₄	—	109	2	no	—
131	C₈×D₄	—	109	1	no	—
132	C₈×D₄	—	109	2	no	—
133	C₈×D₄	D₁₀₉	1	1	yes	characteristic, complemented
134	#431	—	109	1	no	—
135	#431	—	109	2	no	—
136	C₂×C₄×C₈	—	109	2	no	—
137	C₂×C₄×C₈	—	109	1	no	—
138	C₂×#41	—	109	1	no	—
139	C₂²×C₈	—	109	2	no	—
140	C₂²×C₈	—	109	1	no	—
141	C₂²×C₈	—	109	1	no	—
142	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
143	C₄×Q₈	—	109	1	no	—
144	C₄×Q₈	—	109	1	no	—
145	C₄×D₄	—	109	1	no	—
146	C₄×D₄	—	109	2	no	—
147	C₄×D₄	—	109	2	no	—
148	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
149	#117	—	109	1	no	—
150	#117	—	109	2	no	—
151	C₂×C₄²	—	109	2	no	—
152	C₂×C₄²	—	109	1	no	—
153	C₄⋊C₈	—	109	2	no	—
154	C₄⋊C₈	—	109	2	no	—
155	C₄⋊C₈	—	109	1	no	—
156	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
157	C₂²⋊C₈	—	109	2	no	—
158	C₂²⋊C₈	—	109	2	no	—
159	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
160	C₄×C₈	—	109	2	no	—
161	C₄×C₈	—	109	1	no	—
162	C₄×C₈	—	109	2	no	—
163	C₄×C₈	—	109	1	no	—
164	C₄×C₈	—	109	2	no	—
165	C₄×C₈	—	109	1	no	—
166	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
167	#41	—	109	8	no	—
168	C₂×Q₈	—	109	1	no	—
169	C₂×D₄	—	109	2	no	—
170	C₂×D₄	C₄×D₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	—	109	2	no	—
172	C₂²×C₄	—	109	1	no	—
173	C₂²×C₄	—	109	2	no	—
174	C₂²×C₄	—	109	1	no	—
175	C₂²×C₄	—	109	1	no	—
176	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	109	1	no	—
179	C₂×C₈	—	109	2	no	—
180	C₂×C₈	—	109	1	no	—
181	C₂×C₈	—	218	2	no	—
182	C₂×C₈	—	218	4	no	—
183	C₂×C₈	—	218	2	no	—
184	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
185	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
187	C₄⋊C₄	—	109	2	no	—
188	C₄⋊C₄	—	109	2	no	—
189	C₄⋊C₄	—	109	1	no	—
190	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
191	C₂²⋊C₄	—	109	2	no	—
192	C₂²⋊C₄	—	109	2	no	—
193	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
194	C₄²	—	109	2	no	—
195	C₄²	—	109	1	no	—
196	C₄²	—	109	2	no	—
197	C₄²	—	109	1	no	—
198	C₄²	—	109	1	no	—
199	C₄²	—	109	2	no	—
200	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
201	C₂³	—	109	1	no	—
202	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
203	Q₈	—	109	4	no	—
204	D₄	—	109	8	no	—
205	D₄	C₈×D₁₀₉	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	109	8	no	—
209	C₂×C₄	—	109	1	no	—
210	C₂×C₄	—	109	1	no	—
211	C₂×C₄	—	109	2	no	—
212	C₂×C₄	—	109	4	no	—
213	C₂×C₄	—	109	1	no	—
214	C₂×C₄	—	109	2	no	—
215	C₂×C₄	—	218	2	no	—
216	C₂×C₄	—	218	4	no	—
217	C₂×C₄	—	218	1	no	—
218	C₂×C₄	—	218	1	no	—
219	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	109	2	no	—
226	C₈	—	218	2	no	—
227	C₈	—	218	1	no	—
228	C₈	#13182028	1	2	yes	direct product factor
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	109	2	no	—
232	C₂²	—	218	1	no	—
233	C₂²	—	218	1	no	—
234	C₂²	C₂×C₈×D₁₀₉	1	4	yes	—
235	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	109	2	no	—
239	C₄	—	109	4	no	—
240	C₄	—	109	2	no	—
241	C₄	—	218	1	no	—
242	C₄	—	218	2	no	—
243	C₄	C₂×C₈×D₁₀₉	1	2	yes	—
244	C₄	#41×D₁₀₉	1	1	yes	characteristic
245	C₄	C₂×#13182028	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	218	2	no	—
248	C₂	C₄×#13182028	1	1	yes	characteristic
249	C₂	C₂²×C₈×D₁₀₉	1	1	yes	characteristic
250	C₂	#131×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₈×D₄×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#13182028	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₈×D₄	D₁₀₉
hypocenter	C₁₀₉	C₈×#41
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₂₁₈	C₂	semidirect
3. #20295150	C₂	semidirect
4. D₄×#11724784	C₂	semidirect
5. C₈×C₂²⋊D₁₀₉	C₂	semidirect
6. #20295089	C₂	semidirect
7. #20295078	C₂	semidirect
8. #20295076	C₂	semidirect
9. C₂×C₈×Dic₁₀₉	C₂	semidirect
10. C₄×C₈×D₁₀₉	C₂	semidirect
11. C₈×D₄×C₁₀₉	C₂	semidirect
41. #13182028	C₈	direct (N × H)
77. D₄×C₁₀₉	C₂×C₈	semidirect
84. C₂²⋊D₁₀₉	C₂×C₈	semidirect
86. C₂×Dic₁₀₉	C₂×C₈	semidirect
95. C₄×D₁₀₉	C₂×C₈	semidirect
96. Dic₂₁₈	C₂×C₈	semidirect
99. C₈×C₁₀₉	#41	semidirect
102. #11724784	#41	semidirect
126. C₁₀₉	C₈×#41	semidirect
133. C₈×D₄	D₁₀₉	semidirect
205. D₄	C₈×D₁₀₉	semidirect
228. C₈	#13182028	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13182028	2
C₈×Dic₂₁₈	2
#20295150	2
D₄×#11724784	2
C₈×C₂²⋊D₁₀₉	2
#20295089	2
#20295078	2
#20295076	2
C₂×C₈×Dic₁₀₉	2
C₄×C₈×D₁₀₉	2
C₈×D₄×C₁₀₉	2
C₈×#41	109

Subgroups — C8×#13182028 #28371987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13182028 #28371987