Subgroups — #28371875

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	#20295506	C₂	1	1	yes	maximal, characteristic
3	#20295131	C₂	1	2	yes	maximal, complemented
4	#20295124	C₂	1	2	yes	maximal, complemented
5	C₄×#13182019	C₂	1	2	yes	maximal, complemented
6	C₄×#13182016	C₂	1	2	yes	maximal, complemented
7	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295020	C₂	1	1	yes	maximal, characteristic
9	C₂×#16812623	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294982	C₂	1	1	yes	maximal, characteristic, complemented
11	#519×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	C₂²×#11724784	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₈	C₂²	1	2	yes	—
19	C₄×D₄₃₆	C₂²	1	2	yes	—
20	#16812659	—	2	2	no	—
21	#16812657	—	2	1	no	—
22	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₀₉	—	2	1	no	—
24	#16812632	C₂²	1	2	yes	—
25	#16812630	C₂²	1	2	yes	—
26	#16812623	C₂²	1	2	yes	—
27	#16812623	C₂²	1	2	yes	—
28	#16812622	C₂²	1	1	yes	characteristic
29	#16812622	C₂²	1	1	yes	characteristic
30	#16812619	C₂²	1	1	yes	characteristic
31	#16812619	C₂²	1	1	yes	characteristic
32	#16812610	C₂²	1	2	yes	—
33	C₄×#11724784	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₉	—	2	1	no	—
36	C₄×Q₈×C₁₀₉	—	2	1	no	—
37	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
38	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
39	#117×C₁₀₉	—	2	2	no	—
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	#117×C₁₀₉	C₂²	1	1	yes	characteristic
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	C₂²×Dic₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	1	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	#13182019	C₂×C₄	1	8	yes	complemented
51	#13182016	C₂×C₄	1	8	yes	complemented
52	#13182013	C₂×C₄	1	4	yes	complemented
53	C₂×#11724784	—	2	2	no	—
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₂×#11724784	C₂³	1	1	yes	characteristic
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₄⋊Dic₁₀₉	—	2	1	no	—
58	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
59	#13182008	—	2	2	no	—
60	#13182006	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	2	no	—
62	C₂×Q₈×C₁₀₉	—	2	1	no	—
63	C₂×Q₈×C₁₀₉	—	2	2	no	—
64	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
65	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₀₉	—	2	1	no	—
67	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	—	2	2	no	—
69	C₄⋊C₄×C₁₀₉	—	2	2	no	—
70	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
76	C₄²×C₁₀₉	—	2	2	no	—
77	C₄²×C₁₀₉	—	2	2	no	—
78	C₄²×C₁₀₉	—	2	1	no	—
79	C₄²×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	1	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	#41	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₁₀₉	—	2	4	no	—
96	C₂×Dic₁₀₉	—	2	1	no	—
97	C₂×Dic₁₀₉	—	2	2	no	—
98	C₂×Dic₁₀₉	—	8	1	no	—
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	#11724784	—	2	1	no	—
106	#11724784	—	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	—	2	2	no	—
117	C₄×C₁₀₉	—	2	2	no	—
118	C₄×C₁₀₉	—	4	1	no	—
119	C₄×C₁₀₉	—	4	1	no	—
120	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
121	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
123	Dic₁₀₉	—	2	2	no	—
124	Dic₁₀₉	—	4	2	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	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
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	—	109	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	—	109	2	no	—
153	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
154	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
155	C₄×D₄	—	109	2	no	—
156	C₄×D₄	—	218	2	no	—
157	#117	—	2	2	no	—
158	#117	—	218	1	no	—
159	#117	C₂×D₁₀₉	1	1	yes	characteristic
160	#117	C₂×D₁₀₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	109	1	no	—
162	C₂×C₄⋊C₄	—	218	1	no	—
163	#107	—	109	2	no	—
164	#106	—	109	2	no	—
165	#103	—	109	2	no	—
166	#103	—	109	2	no	—
167	#103	—	109	1	no	—
168	#103	—	109	1	no	—
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	—	218	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	—	109	2	no	—
191	C₂×C₈	—	109	1	no	—
192	C₂×C₈	—	109	1	no	—
193	C₂×C₈	—	218	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	109	2	no	—
197	C₄⋊C₄	—	218	2	no	—
198	C₄⋊C₄	—	218	1	no	—
199	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	218	2	no	—
205	C₂²⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	218	2	no	—
210	C₄²	C₂²×D₁₀₉	1	2	yes	—
211	C₂³	—	218	1	no	—
212	C₂³	C₂×C₂²⋊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	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	872	2	no	—
226	C₂×C₄	—	872	1	no	—
227	C₂×C₄	—	218	2	no	—
228	C₂×C₄	—	218	1	no	—
229	C₂×C₄	—	218	4	no	—
230	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
234	C₂×C₄	#13182029	1	2	yes	—
235	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
236	C₈	—	109	4	no	—
237	C₈	—	218	1	no	—
238	C₈	—	218	1	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	C₄×C₂²⋊D₁₀₉	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	436	2	no	—
252	C₄	—	218	2	no	—
253	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
254	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
255	C₄	C₂×#13182029	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	436	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
259	C₂	#20295834	1	1	yes	characteristic
260	C₂	#20295841	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#519×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#519	D₁₀₉
hypocenter	C₁₀₉	#2947
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #20295131	C₂	semidirect
4. #20295124	C₂	semidirect
5. C₄×#13182019	C₂	semidirect
6. C₄×#13182016	C₂	semidirect
7. #20295069	C₂	semidirect
9. C₂×#16812623	C₂	semidirect
10. #20294982	C₂	semidirect
11. #519×C₁₀₉	C₂	semidirect
13. #16812730	C₄	semidirect
50. #13182019	C₂×C₄	semidirect
51. #13182016	C₂×C₄	semidirect
52. #13182013	C₂×C₄	semidirect
64. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
131. C₁₀₉	#2947	semidirect
134. #519	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
#20295506	2
#20295131	2
#20295124	2
C₄×#13182019	2
C₄×#13182016	2
#20295069	2
#20295020	2
C₂×#16812623	2
#20294982	2
#519×C₁₀₉	2
#2947	109