Subgroups — #28362750

Order 13952.

448 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182012	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812657	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294789	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294788	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294788	C₂	1	1	yes	maximal, characteristic, complemented
7	#340×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₀₉	—	2	1	no	—
9	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#13182012	—	2	2	no	—
11	C₂×#13182012	—	2	2	no	—
12	C₂×#13182012	C₄	1	4	yes	complemented
13	C₂²×#11724784	C₂²	1	1	yes	characteristic
14	#16812657	—	2	2	no	—
15	#16812657	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#13182008	—	2	1	no	—
19	C₄⋊C₄×D₁₀₉	—	2	2	no	—
20	C₄⋊C₄×D₁₀₉	C₄	1	4	yes	complemented
21	C₂×#13182006	—	2	1	no	—
22	C₂×C₄×Dic₁₀₉	—	2	1	no	—
23	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×Dic₁₀₉	—	2	1	no	—
28	C₂²×Dic₁₀₉	—	2	1	no	—
29	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₀₉	—	2	4	no	—
31	C₂×C₄×D₁₀₉	—	4	2	no	—
32	C₂×C₄×D₁₀₉	—	4	2	no	—
33	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
37	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
38	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
39	#13182012	—	2	8	no	—
40	C₂×#11724784	—	2	1	no	—
41	C₂×#11724784	—	2	1	no	—
42	C₂×#11724784	C₂×C₄	1	2	yes	complemented
43	C₄⋊Dic₁₀₉	—	2	1	no	—
44	C₄⋊Dic₁₀₉	—	2	1	no	—
45	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	complemented
47	#13182008	—	2	4	no	—
48	#13182006	—	2	4	no	—
49	C₄×Dic₁₀₉	—	2	4	no	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	—	2	1	no	—
54	C₂×C₈×C₁₀₉	—	2	1	no	—
55	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
56	C₄⋊C₄×C₁₀₉	—	2	1	no	—
57	C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
60	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×D₁₀₉	—	2	1	no	—
62	C₂²×D₁₀₉	—	2	1	no	—
63	C₂²×D₁₀₉	—	2	1	no	—
64	C₂²×D₁₀₉	—	2	1	no	—
65	C₂²×D₁₀₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×D₁₀₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	4	1	no	—
71	C₂×C₄×C₁₀₉	—	4	1	no	—
72	C₂×C₄×C₁₀₉	—	4	1	no	—
73	C₂×C₄×C₁₀₉	—	4	1	no	—
74	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
80	C₂×Dic₁₀₉	—	2	2	no	—
81	C₂×Dic₁₀₉	—	2	2	no	—
82	C₂×Dic₁₀₉	—	4	1	no	—
83	C₂×Dic₁₀₉	—	4	1	no	—
84	C₂×Dic₁₀₉	—	4	1	no	—
85	C₂×Dic₁₀₉	—	4	1	no	—
86	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
87	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
88	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
89	C₄×D₁₀₉	—	4	8	no	—
90	C₄×D₁₀₉	C₄²	1	4	yes	—
91	C₄×D₁₀₉	C₂²⋊C₄	1	4	yes	—
92	C₄×D₁₀₉	C₄⋊C₄	1	4	yes	—
93	C₄×D₁₀₉	C₄⋊C₄	1	4	yes	—
94	C₈×C₁₀₉	—	2	2	no	—
95	#11724784	—	2	2	no	—
96	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂²×C₁₀₉	#137	1	1	yes	characteristic
102	C₂²×C₁₀₉	#137	1	1	yes	characteristic
103	C₂×D₁₀₉	—	2	2	no	—
104	C₂×D₁₀₉	—	2	2	no	—
105	C₂×D₁₀₉	—	2	2	no	—
106	C₂×D₁₀₉	—	2	2	no	—
107	C₂×D₁₀₉	—	2	2	no	—
108	C₂×D₁₀₉	—	2	2	no	—
109	C₂×D₁₀₉	#95	1	4	yes	—
110	C₄×C₁₀₉	—	4	2	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
113	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	Dic₁₀₉	—	4	2	no	—
117	Dic₁₀₉	—	4	2	no	—
118	Dic₁₀₉	#95	1	4	yes	—
119	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
120	C₂×C₁₀₉	#417	1	1	yes	characteristic
121	C₂×C₁₀₉	#417	1	1	yes	characteristic
122	C₂×C₁₀₉	#418	1	1	yes	characteristic
123	C₂×C₁₀₉	#419	1	1	yes	characteristic
124	C₂×C₁₀₉	#428	1	1	yes	characteristic
125	C₂×C₁₀₉	#428	1	1	yes	characteristic
126	D₁₀₉	—	2	4	no	—
127	#1736	—	109	1	no	maximal, Sylow
128	C₁₀₉	#1736	1	1	yes	characteristic, complemented, Sylow
129	C₂²×M₁₆	—	109	1	no	—
130	C₂×#117	—	109	1	no	—
131	C₂²×C₄⋊C₄	—	109	1	no	—
132	#340	—	109	1	no	—
133	#340	—	109	1	no	—
134	#340	—	109	1	no	—
135	#340	D₁₀₉	1	1	yes	characteristic, complemented
136	C₂³×C₄	—	109	1	no	—
137	C₂³×C₄	—	218	1	no	—
138	C₂×M₁₆	—	109	4	no	—
139	C₂×M₁₆	—	218	2	no	—
140	C₂×M₁₆	—	218	2	no	—
141	C₂²×C₈	—	109	1	no	—
142	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
143	#117	—	109	4	no	—
144	#117	—	218	2	no	—
145	C₂×C₄⋊C₄	—	109	1	no	—
146	C₂×C₄⋊C₄	—	109	1	no	—
147	C₂×C₄⋊C₄	—	109	4	no	—
148	C₂×C₄⋊C₄	—	218	2	no	—
149	C₂×C₄⋊C₄	—	218	1	no	—
150	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	—	218	1	no	—
153	C₂×C₄²	—	218	1	no	—
154	C₂⁴	—	109	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	109	1	no	—
158	C₂²×C₄	—	109	2	no	—
159	C₂²×C₄	—	109	2	no	—
160	C₂²×C₄	—	109	2	no	—
161	C₂²×C₄	—	109	2	no	—
162	C₂²×C₄	—	109	2	no	—
163	C₂²×C₄	—	109	2	no	—
164	C₂²×C₄	—	436	2	no	—
165	C₂²×C₄	—	436	2	no	—
166	C₂²×C₄	—	218	4	no	—
167	C₂²×C₄	—	218	1	no	—
168	C₂²×C₄	—	218	1	no	—
169	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
170	M₁₆	—	218	8	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	109	2	no	—
174	C₂×C₈	—	218	1	no	—
175	C₂×C₈	—	218	1	no	—
176	C₂×C₈	C₄×D₁₀₉	1	2	yes	complemented
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	109	2	no	—
180	C₄⋊C₄	—	109	2	no	—
181	C₄⋊C₄	—	218	4	no	—
182	C₄⋊C₄	—	218	1	no	—
183	C₄⋊C₄	—	218	1	no	—
184	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
185	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	complemented
186	C₂²⋊C₄	—	218	4	no	—
187	C₄²	—	218	4	no	—
188	C₂³	—	109	2	no	—
189	C₂³	—	109	2	no	—
190	C₂³	—	109	2	no	—
191	C₂³	—	218	1	no	—
192	C₂³	—	218	1	no	—
193	C₂³	—	218	1	no	—
194	C₂³	—	218	1	no	—
195	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	109	2	no	—
203	C₂×C₄	—	109	2	no	—
204	C₂×C₄	—	109	2	no	—
205	C₂×C₄	—	109	4	no	—
206	C₂×C₄	—	109	4	no	—
207	C₂×C₄	—	109	4	no	—
208	C₂×C₄	—	109	4	no	—
209	C₂×C₄	—	436	8	no	—
210	C₂×C₄	—	436	1	no	—
211	C₂×C₄	—	436	1	no	—
212	C₂×C₄	—	436	1	no	—
213	C₂×C₄	—	436	1	no	—
214	C₂×C₄	—	218	2	no	—
215	C₂×C₄	—	218	2	no	—
216	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
219	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
220	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
221	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	218	2	no	—
224	C₂²	—	109	4	no	—
225	C₂²	—	218	2	no	—
226	C₂²	—	218	2	no	—
227	C₂²	—	218	2	no	—
228	C₂²	—	218	2	no	—
229	C₂²	—	218	2	no	—
230	C₂²	—	218	2	no	—
231	C₂²	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
232	C₂²	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
233	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
234	C₂²	#16812723	1	1	yes	characteristic
235	C₂²	#16812726	1	1	yes	characteristic
236	C₂²	#16812728	1	1	yes	characteristic
237	C₂²	#16812731	1	1	yes	characteristic
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	109	4	no	—
241	C₄	—	436	2	no	—
242	C₄	—	436	2	no	—
243	C₄	C₄²×D₁₀₉	1	1	yes	characteristic
244	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
245	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
246	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
247	C₂	—	218	4	no	—
248	C₂	#95×D₁₀₉	1	1	yes	characteristic
249	C₂	#20295024	1	1	yes	characteristic
250	C₂	#20295026	1	1	yes	characteristic
251	C₂	#20295029	1	1	yes	characteristic
252	C₂	#20295033	1	1	yes	characteristic
253	C₂	#20295065	1	1	yes	characteristic
254	C₂	#20295067	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#340×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#340	D₁₀₉
hypocenter	C₁₀₉	#1736
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
4. #20294789	C₂	semidirect
5. #20294788	C₂	semidirect
6. #20294788	C₂	semidirect
7. #340×C₁₀₉	C₂	semidirect
12. C₂×#13182012	C₄	semidirect
15. #16812657	C₄	semidirect
20. C₄⋊C₄×D₁₀₉	C₄	semidirect
42. C₂×#11724784	C₂×C₄	semidirect
46. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
55. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
59. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
128. C₁₀₉	#1736	semidirect
135. #340	D₁₀₉	semidirect
176. C₂×C₈	C₄×D₁₀₉	semidirect
185. C₄⋊C₄	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13182012	2
C₂×#16812657	2
C₂×C₄⋊C₄×D₁₀₉	2
#20294789	2
#20294788	2
#340×C₁₀₉	2
#1736	109