Subgroups — #28371869

Order 13952.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812730	C₂	1	1	yes	maximal, characteristic
2	#20295507	C₂	1	1	yes	maximal, characteristic
3	#20295504	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295130	C₂	1	2	yes	maximal, complemented
5	#20295129	C₂	1	2	yes	maximal, complemented
6	#20295118	C₂	1	2	yes	maximal, complemented
7	#20295111	C₂	1	2	yes	maximal, complemented
8	#20295066	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295023	C₂	1	1	yes	maximal, characteristic
10	#20294981	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812730	C₄	1	8	yes	complemented
14	C₂×#13182019	C₂²	1	2	yes	—
15	C₂×#13182016	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₀₉	—	2	1	no	—
18	Q₈×Dic₁₀₉	—	2	1	no	—
19	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
20	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
21	#16812664	C₂²	1	2	yes	—
22	#16812661	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₁₀₉	—	2	2	no	—
24	#16812658	—	2	2	no	—
25	#16812658	C₂²	1	1	yes	characteristic
26	#16812658	C₂²	1	1	yes	characteristic
27	#16812657	—	2	1	no	—
28	C₄⋊C₄×D₁₀₉	—	2	1	no	—
29	#16812632	C₂²	1	2	yes	—
30	#16812630	C₂²	1	2	yes	—
31	#16812624	C₂²	1	1	yes	characteristic
32	#16812624	C₂²	1	1	yes	characteristic
33	#16812621	C₂²	1	1	yes	characteristic
34	#16812621	C₂²	1	1	yes	characteristic
35	#16812609	C₂²	1	2	yes	—
36	C₈×Dic₁₀₉	C₂²	1	2	yes	—
37	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#103×C₁₀₉	C₂²	1	2	yes	—
41	#103×C₁₀₉	C₂²	1	2	yes	—
42	#13182029	—	2	2	no	—
43	#13182029	C₂×C₄	1	4	yes	—
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	8	yes	complemented
50	#13182016	C₂×C₄	1	8	yes	complemented
51	#13182013	C₂×C₄	1	4	yes	complemented
52	C₂×#11724784	C₂³	1	2	yes	—
53	C₄⋊Dic₁₀₉	—	2	2	no	—
54	C₄⋊Dic₁₀₉	—	2	2	no	—
55	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
56	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
57	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
59	#13182008	—	2	2	no	—
60	C₂²⋊Dic₁₀₉	—	2	1	no	—
61	C₂²⋊Dic₁₀₉	D₄	1	2	yes	—
62	#13182006	—	2	2	no	—
63	C₄×Dic₁₀₉	—	2	1	no	—
64	C₄×Dic₁₀₉	—	2	2	no	—
65	C₄×Dic₁₀₉	—	2	2	no	—
66	C₄×Dic₁₀₉	—	2	2	no	—
67	C₄×Dic₁₀₉	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₀₉	—	2	1	no	—
69	C₂×Q₈×C₁₀₉	—	2	2	no	—
70	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
71	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₀₉	—	2	1	no	—
73	C₂²×C₄×C₁₀₉	—	2	1	no	—
74	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₉	—	2	1	no	—
79	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
80	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₁₀₉	—	2	1	no	—
82	Q₈×C₁₀₉	—	2	4	no	—
83	Q₈×C₁₀₉	—	2	2	no	—
84	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₁₀₉	—	2	4	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	8	1	no	—
89	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	#41	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₁₀₉	—	2	4	no	—
94	C₂×Dic₁₀₉	—	2	1	no	—
95	C₂×Dic₁₀₉	—	2	1	no	—
96	C₂×Dic₁₀₉	—	2	1	no	—
97	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
98	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
99	D₄₃₆	—	2	1	no	—
100	D₄₃₆	C₂²×C₄	1	2	yes	—
101	C₄×D₁₀₉	—	2	4	no	—
102	C₄×D₁₀₉	—	8	2	no	—
103	Dic₂₁₈	—	2	1	no	—
104	Dic₂₁₈	C₂²×C₄	1	2	yes	—
105	C₈×C₁₀₉	—	2	1	no	—
106	C₈×C₁₀₉	—	2	1	no	—
107	#11724784	C₂²×C₄	1	4	yes	—
108	C₂²×C₁₀₉	—	2	1	no	—
109	C₂²×C₁₀₉	—	2	1	no	—
110	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
111	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₁₀₉	—	2	2	no	—
113	C₂×D₁₀₉	—	4	1	no	—
114	C₂×D₁₀₉	—	4	1	no	—
115	C₄×C₁₀₉	—	2	4	no	—
116	C₄×C₁₀₉	—	4	2	no	—
117	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
118	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
120	Dic₁₀₉	—	2	2	no	—
121	Dic₁₀₉	—	2	2	no	—
122	Dic₁₀₉	—	2	2	no	—
123	Dic₁₀₉	—	4	1	no	—
124	Dic₁₀₉	—	4	1	no	—
125	C₂×C₁₀₉	—	2	2	no	—
126	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₁₀₉	#577	1	1	yes	characteristic
128	C₂×C₁₀₉	#578	1	1	yes	characteristic
129	D₁₀₉	—	4	2	no	—
130	#2947	—	109	1	no	maximal, Sylow
131	C₁₀₉	#2947	1	1	yes	characteristic, complemented, Sylow
132	C₂×#137	—	109	1	no	—
133	#520	—	109	1	no	—
134	#519	—	109	1	no	—
135	#441	—	109	2	no	—
136	#440	—	109	2	no	—
137	C₄×Q₁₆	—	109	2	no	—
138	C₄×SD₁₆	—	109	2	no	—
139	#428	—	109	1	no	—
140	#416	—	109	1	no	—
141	C₂×#103	D₁₀₉	1	1	yes	characteristic, complemented
142	#405	—	109	1	no	—
143	C₂×#41	—	109	1	no	—
144	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
145	#137	—	109	8	no	—
146	C₂×Q₁₆	—	109	2	no	—
147	C₂×SD₁₆	—	109	2	no	—
148	C₂×M₁₆	—	109	1	no	—
149	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
150	C₄×Q₈	—	109	2	no	—
151	C₄×Q₈	—	109	2	no	—
152	C₄×Q₈	—	109	2	no	—
153	C₄×Q₈	—	218	1	no	—
154	C₄×Q₈	—	218	1	no	—
155	C₄×D₄	—	109	2	no	—
156	C₄×D₄	—	218	2	no	—
157	#117	—	109	1	no	—
158	#117	—	109	1	no	—
159	#117	—	218	2	no	—
160	#117	—	218	1	no	—
161	C₂×C₄⋊C₄	—	218	1	no	—
162	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
163	#107	—	109	2	no	—
164	#106	—	109	2	no	—
165	#103	—	109	1	no	—
166	#103	—	109	1	no	—
167	#103	C₂×D₁₀₉	1	2	yes	—
168	#103	C₂×D₁₀₉	1	2	yes	—
169	#102	—	109	1	no	—
170	#102	—	109	1	no	—
171	#97	—	109	2	no	—
172	C₄×C₈	—	109	2	no	—
173	#41	—	109	4	no	—
174	#41	—	218	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	109	1	no	—
178	C₂×Q₈	C₄×D₁₀₉	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
180	C₂×D₄	—	109	1	no	—
181	C₂×D₄	—	218	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	218	2	no	—
185	C₂²×C₄	—	218	1	no	—
186	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
187	Q₁₆	—	109	8	no	—
188	SD₁₆	—	109	8	no	—
189	M₁₆	—	109	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	109	2	no	—
192	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	109	1	no	—
196	C₄⋊C₄	—	109	1	no	—
197	C₄⋊C₄	—	109	1	no	—
198	C₄⋊C₄	—	109	1	no	—
199	C₄⋊C₄	—	218	2	no	—
200	C₄⋊C₄	—	218	2	no	—
201	C₄⋊C₄	—	218	2	no	—
202	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
203	C₂²⋊C₄	—	109	2	no	—
204	C₂²⋊C₄	—	218	2	no	—
205	C₂²⋊C₄	—	218	1	no	—
206	C₄²	—	109	2	no	—
207	C₄²	—	218	2	no	—
208	C₄²	—	218	2	no	—
209	C₄²	—	218	2	no	—
210	C₄²	—	218	1	no	—
211	C₂³	—	218	1	no	—
212	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	109	2	no	—
216	Q₈	—	218	1	no	—
217	Q₈	C₂×C₄×D₁₀₉	1	4	yes	—
218	D₄	—	109	2	no	—
219	D₄	—	218	4	no	—
220	D₄	—	218	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	872	2	no	—
226	C₂×C₄	—	109	2	no	—
227	C₂×C₄	—	109	2	no	—
228	C₂×C₄	—	218	1	no	—
229	C₂×C₄	—	218	1	no	—
230	C₂×C₄	—	218	1	no	—
231	C₂×C₄	—	218	4	no	—
232	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
233	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
234	C₂×C₄	#13182028	1	2	yes	—
235	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	109	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	436	1	no	—
242	C₂²	—	436	1	no	—
243	C₂²	—	218	2	no	—
244	C₂²	#16812659	1	2	yes	—
245	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	436	1	no	—
249	C₄	—	436	1	no	—
250	C₄	—	218	2	no	—
251	C₄	—	218	2	no	—
252	C₄	—	218	2	no	—
253	C₄	#16812659	1	2	yes	—
254	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
255	C₄	C₂×#13182028	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	436	2	no	—
258	C₂	C₂×#16812659	1	1	yes	characteristic
259	C₂	#20295832	1	1	yes	characteristic
260	C₂	#20295839	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#103×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#103	D₁₀₉
hypocenter	C₁₀₉	#2947
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
3. #20295504	C₂	semidirect
4. #20295130	C₂	semidirect
5. #20295129	C₂	semidirect
6. #20295118	C₂	semidirect
7. #20295111	C₂	semidirect
8. #20295066	C₂	semidirect
10. #20294981	C₂	semidirect
11. C₂×#103×C₁₀₉	C₂	semidirect
13. #16812730	C₄	semidirect
49. #13182019	C₂×C₄	semidirect
50. #13182016	C₂×C₄	semidirect
51. #13182013	C₂×C₄	semidirect
70. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
131. C₁₀₉	#2947	semidirect
141. C₂×#103	D₁₀₉	semidirect
178. C₂×Q₈	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×#16812730	2
#20295507	2
#20295504	2
#20295130	2
#20295129	2
#20295118	2
#20295111	2
#20295066	2
#20295023	2
#20294981	2
C₂×#103×C₁₀₉	2
#2947	109