Subgroups — #28371902

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812718	C₂	1	1	yes	maximal, characteristic
2	#20295505	C₂	1	2	yes	maximal, complemented
3	#20295126	C₂	1	2	yes	maximal, complemented
4	#20295123	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295120	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295024	C₂	1	2	yes	maximal, complemented
10	#20294980	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182028	C₂²	1	2	yes	—
13	#16812718	C₄	1	8	yes	complemented
14	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
15	C₂×#13182017	C₂²	1	2	yes	—
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₈	C₂²	1	2	yes	—
19	#16812664	C₂²	1	2	yes	—
20	D₄×Dic₁₀₉	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₁₀₉	—	2	2	no	—
22	#16812659	—	2	2	no	—
23	#16812656	C₂²	1	1	yes	characteristic
24	#16812655	—	2	2	no	—
25	C₂×#13182008	—	2	2	no	—
26	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
27	#16812633	C₂²	1	1	yes	characteristic
28	#16812632	C₂²	1	2	yes	—
29	#16812624	C₂²	1	2	yes	—
30	#16812622	C₂²	1	2	yes	—
31	#16812620	C₂²	1	2	yes	—
32	#16812610	C₂²	1	1	yes	characteristic
33	#16812609	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
37	#107×C₁₀₉	C₂²	1	1	yes	characteristic
38	#102×C₁₀₉	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	#13182028	—	2	4	no	—
41	#13182028	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₂₁₈	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
44	C₂²×Dic₁₀₉	—	2	2	no	—
45	C₂²×Dic₁₀₉	—	2	2	no	—
46	C₂×C₄×D₁₀₉	—	2	2	no	—
47	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
48	Dic₄₃₆	C₂×C₄	1	4	yes	complemented
49	#13182017	C₂×C₄	1	8	yes	complemented
50	#13182012	—	2	2	no	—
51	C₂×#11724784	C₂³	1	1	yes	characteristic
52	C₂×#11724784	C₂³	1	1	yes	characteristic
53	C₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
54	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
55	#13182008	—	2	2	no	—
56	#13182008	—	2	2	no	—
57	#13182008	—	2	2	no	—
58	#13182008	D₄	1	1	yes	characteristic
59	#13182008	D₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₀₉	—	2	2	no	—
61	#13182006	D₄	1	1	yes	characteristic
62	#13182006	D₄	1	1	yes	characteristic
63	C₄×Dic₁₀₉	—	2	2	no	—
64	C₄×Dic₁₀₉	—	2	2	no	—
65	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	D₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂³×C₁₀₉	—	2	2	no	—
75	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
76	D₄×C₁₀₉	—	2	2	no	—
77	D₄×C₁₀₉	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	8	2	no	—
80	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₀₉	—	2	8	no	—
84	C₂×Dic₁₀₉	—	2	2	no	—
85	C₂×Dic₁₀₉	—	2	8	no	—
86	C₂×Dic₁₀₉	—	2	2	no	—
87	C₂×Dic₁₀₉	—	8	2	no	—
88	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₄×D₁₀₉	—	8	2	no	—
92	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
93	C₄×D₁₀₉	#41	1	2	yes	—
94	Dic₂₁₈	—	2	2	no	—
95	Dic₂₁₈	C₂²×C₄	1	4	yes	—
96	C₈×C₁₀₉	—	2	1	no	—
97	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
98	#11724784	—	2	1	no	—
99	#11724784	C₂²×C₄	1	2	yes	—
100	C₂²×C₁₀₉	—	2	4	no	—
101	C₂²×C₁₀₉	—	4	2	no	—
102	C₂²×C₁₀₉	—	4	2	no	—
103	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₀₉	—	2	1	no	—
105	C₂×D₁₀₉	—	2	1	no	—
106	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	2	1	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
112	Dic₁₀₉	—	2	4	no	—
113	Dic₁₀₉	—	2	1	no	—
114	Dic₁₀₉	—	2	1	no	—
115	Dic₁₀₉	—	4	2	no	—
116	Dic₁₀₉	C₄×D₄	1	2	yes	—
117	C₂×C₁₀₉	—	4	4	no	—
118	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₀₉	#576	1	1	yes	characteristic
120	C₂×C₁₀₉	#578	1	1	yes	characteristic
121	D₁₀₉	—	2	2	no	—
122	#2951	—	109	1	no	maximal, Sylow
123	C₁₀₉	#2951	1	1	yes	characteristic, complemented, Sylow
124	C₂×#135	—	109	1	no	—
125	#520	—	109	2	no	—
126	#440	—	109	2	no	—
127	#440	—	109	2	no	—
128	C₄×Q₁₆	—	109	1	no	—
129	C₄×Q₁₆	—	109	1	no	—
130	C₄×D₈	—	109	1	no	—
131	C₄×D₈	D₁₀₉	1	1	yes	characteristic, complemented
132	C₂×#107	—	109	1	no	—
133	#417	—	109	2	no	—
134	#405	—	109	1	no	—
135	C₂×#41	—	109	2	no	—
136	#135	—	109	8	no	—
137	C₂×Q₁₆	—	109	1	no	—
138	C₂×SD₁₆	—	109	2	no	—
139	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
140	C₂×M₁₆	—	109	1	no	—
141	C₂²×C₈	—	109	1	no	—
142	C₄×Q₈	—	109	2	no	—
143	C₄×Q₈	—	109	2	no	—
144	C₄×D₄	—	109	2	no	—
145	C₄×D₄	—	218	2	no	—
146	C₄×D₄	—	218	2	no	—
147	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
148	#117	—	109	1	no	—
149	#117	—	218	2	no	—
150	C₂×C₄⋊C₄	—	109	2	no	—
151	C₂×C₄⋊C₄	—	218	2	no	—
152	#107	—	109	1	no	—
153	#107	—	109	2	no	—
154	#107	C₂×D₁₀₉	1	1	yes	characteristic
155	#103	—	109	2	no	—
156	#103	—	109	2	no	—
157	#102	—	109	2	no	—
158	#102	C₂×D₁₀₉	1	2	yes	—
159	#97	—	109	1	no	—
160	#97	—	109	1	no	—
161	C₄×C₈	—	109	1	no	—
162	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
163	#41	—	109	8	no	—
164	#41	—	218	4	no	—
165	C₂×Q₈	—	109	2	no	—
166	C₂×D₄	—	218	2	no	—
167	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	109	1	no	—
170	C₂²×C₄	—	218	2	no	—
171	C₂²×C₄	—	218	2	no	—
172	C₂²×C₄	—	218	2	no	—
173	Q₁₆	—	109	4	no	—
174	SD₁₆	—	109	8	no	—
175	D₈	C₄×D₁₀₉	1	4	yes	complemented
176	M₁₆	—	218	2	no	—
177	C₂×C₈	—	109	1	no	—
178	C₂×C₈	—	109	4	no	—
179	C₂×C₈	—	109	1	no	—
180	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
182	C₄⋊C₄	—	109	1	no	—
183	C₄⋊C₄	—	109	1	no	—
184	C₄⋊C₄	—	109	2	no	—
185	C₄⋊C₄	—	218	2	no	—
186	C₄⋊C₄	—	218	2	no	—
187	C₄⋊C₄	—	218	2	no	—
188	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	109	1	no	—
191	C₂²⋊C₄	—	109	1	no	—
192	C₂²⋊C₄	—	218	2	no	—
193	C₄²	—	109	1	no	—
194	C₄²	—	218	2	no	—
195	C₄²	—	218	2	no	—
196	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	109	1	no	—
199	Q₈	—	109	4	no	—
200	Q₈	—	218	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	218	8	no	—
203	D₄	C₂×C₄×D₁₀₉	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	872	2	no	—
207	C₂×C₄	—	872	2	no	—
208	C₂×C₄	—	109	1	no	—
209	C₂×C₄	—	109	2	no	—
210	C₂×C₄	—	109	2	no	—
211	C₂×C₄	—	109	1	no	—
212	C₂×C₄	—	109	1	no	—
213	C₂×C₄	—	218	8	no	—
214	C₂×C₄	—	218	2	no	—
215	C₂×C₄	—	218	2	no	—
216	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
217	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	109	2	no	—
221	C₈	—	218	1	no	—
222	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	109	2	no	—
227	C₂²	—	218	1	no	—
228	C₂²	—	218	1	no	—
229	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	109	2	no	—
234	C₄	—	436	2	no	—
235	C₄	—	218	4	no	—
236	C₄	—	218	1	no	—
237	C₄	—	218	1	no	—
238	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
239	C₄	#41×D₁₀₉	1	1	yes	characteristic
240	C₂	—	4	4	no	—
241	C₂	—	218	2	no	—
242	C₂	C₄×D₄×D₁₀₉	1	1	yes	characteristic
243	C₂	#20295827	1	1	yes	characteristic
244	C₂	#20295840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×D₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812718	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×D₈	D₁₀₉
hypocenter	C₁₀₉	#2951
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #20295505	C₂	semidirect
3. #20295126	C₂	semidirect
4. #20295123	C₂	semidirect
5. C₄×Dic₄₃₆	C₂	semidirect
6. #20295120	C₂	semidirect
7. D₈×Dic₁₀₉	C₂	semidirect
8. #107×D₁₀₉	C₂	semidirect
9. #20295024	C₂	semidirect
10. #20294980	C₂	semidirect
11. C₄×D₈×C₁₀₉	C₂	semidirect
13. #16812718	C₄	semidirect
41. #13182028	C₂×C₄	semidirect
48. Dic₄₃₆	C₂×C₄	semidirect
49. #13182017	C₂×C₄	semidirect
53. C₈×D₁₀₉	C₂×C₄	semidirect
68. D₈×C₁₀₉	C₂×C₄	semidirect
123. C₁₀₉	#2951	semidirect
131. C₄×D₈	D₁₀₉	semidirect
175. D₈	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×#16812718	2
#20295505	2
#20295126	2
#20295123	2
C₄×Dic₄₃₆	2
#20295120	2
D₈×Dic₁₀₉	2
#107×D₁₀₉	2
#20295024	2
#20294980	2
C₄×D₈×C₁₀₉	2
#2951	109