Subgroups — #28371830

Order 13952.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812719	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182029	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295136	C₂	1	2	yes	maximal, complemented
4	#20295131	C₂	1	2	yes	maximal, complemented
5	#20295126	C₂	1	2	yes	maximal, complemented
6	#20295124	C₂	1	2	yes	maximal, complemented
7	#20295062	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295022	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295020	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812719	C₄	1	8	yes	complemented
14	C₂×#13182019	C₂²	1	1	yes	characteristic
15	C₂×#13182017	C₂²	1	1	yes	characteristic
16	C₂×#13182016	C₂²	1	1	yes	characteristic
17	C₂×#13182014	C₂²	1	1	yes	characteristic
18	C₂²×#11724784	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₁₈	C₂²	1	2	yes	—
21	C₄×D₄₃₆	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₀₉	—	2	2	no	—
23	#16812658	C₂²	1	1	yes	characteristic
24	#16812656	—	2	1	no	—
25	C₄²×D₁₀₉	—	2	1	no	—
26	#16812632	C₂²	1	2	yes	—
27	#16812630	C₂²	1	2	yes	—
28	#16812623	C₂²	1	2	yes	—
29	#16812622	C₂²	1	2	yes	—
30	#16812620	C₂²	1	2	yes	—
31	#16812619	C₂²	1	2	yes	—
32	#16812610	C₂²	1	2	yes	—
33	#16812610	C₂²	1	2	yes	—
34	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
36	C₄×D₄×C₁₀₉	—	2	2	no	—
37	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
38	#117×C₁₀₉	—	2	1	no	—
39	#117×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₉	—	2	1	no	—
41	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
42	#13182029	—	2	2	no	—
43	#13182029	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
45	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
47	C₂×C₄×D₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	2	no	—
49	#13182019	C₂×C₄	1	4	yes	complemented
50	#13182017	C₂×C₄	1	4	yes	complemented
51	#13182016	C₂×C₄	1	4	yes	complemented
52	#13182014	C₂×C₄	1	4	yes	complemented
53	C₂×#11724784	—	2	2	no	—
54	C₂×#11724784	C₂×C₄	1	4	yes	complemented
55	C₂×#11724784	C₂³	1	2	yes	—
56	C₂×#11724784	C₂³	1	1	yes	characteristic
57	C₂×#11724784	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
59	#13182008	—	2	2	no	—
60	C₂²⋊Dic₁₀₉	—	2	1	no	—
61	#13182006	—	2	2	no	—
62	C₄×Dic₁₀₉	—	2	1	no	—
63	C₄×Dic₁₀₉	—	2	2	no	—
64	#41×C₁₀₉	—	2	2	no	—
65	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₀₉	—	2	1	no	—
68	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	—	2	1	no	—
70	C₂²×C₄×C₁₀₉	—	2	2	no	—
71	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
72	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	—	2	2	no	—
74	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₄²×C₁₀₉	—	2	2	no	—
79	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
81	C₄²×C₁₀₉	C₂³	1	2	yes	—
82	C₂³×C₁₀₉	—	2	1	no	—
83	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₀₉	—	2	1	no	—
85	Q₈×C₁₀₉	—	2	1	no	—
86	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₉	—	2	4	no	—
88	D₄×C₁₀₉	—	2	1	no	—
89	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	4	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	—	4	4	no	—
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	#41	1	2	yes	—
100	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₀₉	—	2	4	no	—
102	C₂×Dic₁₀₉	—	2	1	no	—
103	C₂×Dic₁₀₉	—	2	2	no	—
104	D₄₃₆	—	2	1	no	—
105	D₄₃₆	C₂²×C₄	1	2	yes	—
106	C₄×D₁₀₉	—	2	4	no	—
107	C₄×D₁₀₉	—	4	4	no	—
108	Dic₂₁₈	—	2	1	no	—
109	Dic₂₁₈	C₂²×C₄	1	2	yes	—
110	#11724784	—	2	2	no	—
111	#11724784	C₂²×C₄	1	2	yes	—
112	#11724784	C₂²×C₄	1	2	yes	—
113	C₂²×C₁₀₉	—	2	2	no	—
114	C₂²×C₁₀₉	—	2	2	no	—
115	C₂²×C₁₀₉	—	4	2	no	—
116	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
117	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₉	—	2	2	no	—
119	C₂×D₁₀₉	—	4	2	no	—
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	—	2	2	no	—
122	C₄×C₁₀₉	—	2	2	no	—
123	C₄×C₁₀₉	—	4	2	no	—
124	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
125	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
127	Dic₁₀₉	—	2	2	no	—
128	Dic₁₀₉	—	4	2	no	—
129	C₂×C₁₀₉	—	2	2	no	—
130	C₂×C₁₀₉	—	4	2	no	—
131	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₀₉	#575	1	2	yes	—
133	D₁₀₉	—	4	2	no	—
134	#2943	—	109	1	no	maximal, Sylow
135	C₁₀₉	#2943	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	109	1	no	—
137	C₄×#41	—	109	1	no	—
138	C₄×#41	D₁₀₉	1	1	yes	characteristic, complemented
139	#442	—	109	2	no	—
140	#441	—	109	2	no	—
141	#440	—	109	2	no	—
142	#440	—	109	2	no	—
143	#427	—	109	1	no	—
144	#416	—	109	1	no	—
145	#416	—	109	1	no	—
146	C₂×#97	—	109	1	no	—
147	C₂×#41	—	109	1	no	—
148	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
149	#135	—	109	8	no	—
150	C₂×Q₁₆	—	109	1	no	—
151	C₂×SD₁₆	—	109	1	no	—
152	C₂×SD₁₆	—	109	1	no	—
153	C₂×D₈	—	109	1	no	—
154	C₂²×C₈	—	109	1	no	—
155	C₂²×C₈	—	109	1	no	—
156	C₄×Q₈	—	109	2	no	—
157	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	109	2	no	—
160	C₄×D₄	—	218	2	no	—
161	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	109	1	no	—
164	#117	—	218	1	no	—
165	#117	C₂×D₁₀₉	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	218	1	no	—
168	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
169	#107	—	109	2	no	—
170	#106	—	109	2	no	—
171	#103	—	109	2	no	—
172	#103	—	109	2	no	—
173	#102	—	109	2	no	—
174	#102	—	109	2	no	—
175	#97	—	109	2	no	—
176	#97	—	109	2	no	—
177	#41	—	2	2	no	—
178	#41	—	109	4	no	—
179	#41	—	218	2	no	—
180	#41	C₄×D₁₀₉	1	4	yes	complemented
181	C₂×Q₈	—	109	1	no	—
182	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	109	1	no	—
185	C₂×D₄	—	218	1	no	—
186	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	218	2	no	—
190	C₂²×C₄	—	218	1	no	—
191	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
193	Q₁₆	—	109	4	no	—
194	SD₁₆	—	109	4	no	—
195	SD₁₆	—	109	4	no	—
196	D₈	—	109	4	no	—
197	C₂×C₈	—	109	4	no	—
198	C₂×C₈	—	109	1	no	—
199	C₂×C₈	—	109	1	no	—
200	C₂×C₈	—	109	2	no	—
201	C₂×C₈	—	218	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	109	2	no	—
204	C₄⋊C₄	—	218	2	no	—
205	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	218	1	no	—
209	C₂²⋊C₄	—	218	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	218	2	no	—
213	C₄²	—	218	1	no	—
214	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
216	C₄²	C₂²×D₁₀₉	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	218	1	no	—
219	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	109	2	no	—
222	Q₈	—	218	1	no	—
223	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	109	2	no	—
227	D₄	—	218	4	no	—
228	D₄	—	218	1	no	—
229	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	436	4	no	—
236	C₂×C₄	—	218	2	no	—
237	C₂×C₄	—	218	1	no	—
238	C₂×C₄	—	218	4	no	—
239	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
243	C₂×C₄	#13182029	1	2	yes	—
244	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
245	C₈	—	109	2	no	—
246	C₈	—	109	2	no	—
247	C₈	—	218	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	436	2	no	—
252	C₂²	—	218	2	no	—
253	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	436	2	no	—
260	C₄	—	218	2	no	—
261	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
262	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
263	C₄	C₂×#13182029	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	436	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
268	C₂	#20295824	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×#41×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812719	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	#2943
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#13182029	C₂	semidirect
3. #20295136	C₂	semidirect
4. #20295131	C₂	semidirect
5. #20295126	C₂	semidirect
6. #20295124	C₂	semidirect
7. #20295062	C₂	semidirect
8. #20295022	C₂	semidirect
9. #20295020	C₂	semidirect
10. C₂×#16812610	C₂	semidirect
11. C₄×#41×C₁₀₉	C₂	semidirect
13. #16812719	C₄	semidirect
43. #13182029	C₂×C₄	semidirect
49. #13182019	C₂×C₄	semidirect
50. #13182017	C₂×C₄	semidirect
51. #13182016	C₂×C₄	semidirect
52. #13182014	C₂×C₄	semidirect
54. C₂×#11724784	C₂×C₄	semidirect
65. #41×C₁₀₉	C₂×C₄	semidirect
135. C₁₀₉	#2943	semidirect
138. C₄×#41	D₁₀₉	semidirect
180. #41	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812719	2
C₄×#13182029	2
#20295136	2
#20295131	2
#20295126	2
#20295124	2
#20295062	2
#20295022	2
#20295020	2
C₂×#16812610	2
C₄×#41×C₁₀₉	2
#2943	109