Subgroups — #28371739

Order 13952.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812705	C₂	1	2	yes	maximal, complemented
2	#20295573	C₂	1	1	yes	maximal, characteristic
3	#20295104	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295103	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295101	C₂	1	2	yes	maximal, complemented
6	C₈×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₀₉	C₂	1	2	yes	maximal, complemented
8	#20295048	C₂	1	1	yes	maximal, characteristic, complemented
9	#20294991	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	2	yes	—
13	#16812705	—	2	8	no	—
14	C₂×#13182013	C₂²	1	2	yes	—
15	C₂×#13182012	C₂²	1	2	yes	—
16	C₂×#13182012	C₂²	1	2	yes	—
17	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812698	C₄	1	1	yes	characteristic
21	#16812695	C₄	1	1	yes	characteristic
22	#16812685	C₄	1	2	yes	—
23	#16812674	C₄	1	1	yes	characteristic
24	#16812672	C₄	1	2	yes	—
25	#16812670	C₄	1	1	yes	characteristic
26	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
28	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
29	C₄⋊#11724784	C₂²	1	1	yes	characteristic
30	#16812628	C₂²	1	2	yes	—
31	C₂²⋊#11724784	C₂²	1	2	yes	—
32	#16812611	C₂²	1	2	yes	—
33	#16812611	C₂²	1	2	yes	—
34	C₄×#11724784	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
38	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
41	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	#13182029	—	2	4	no	—
43	#13182029	—	2	4	no	—
44	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
45	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
53	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
54	#13182013	—	2	4	no	—
55	#13182012	—	2	4	no	—
56	#13182012	—	2	4	no	—
57	C₂×#11724784	C₂³	1	1	yes	characteristic
58	C₂×#11724784	C₂³	1	2	yes	—
59	C₂×#11724784	C₂³	1	1	yes	characteristic
60	C₈×D₁₀₉	—	2	2	no	—
61	C₈×D₁₀₉	—	2	4	no	—
62	C₈×D₁₀₉	D₄	1	4	yes	complemented
63	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
64	#13182008	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
66	#13182006	C₂×C₄	1	2	yes	—
67	#13182006	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
73	C₂×C₈×C₁₀₉	—	2	4	no	—
74	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
76	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
79	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₁₀₉	—	2	4	no	—
91	C₂²⋊D₁₀₉	—	2	4	no	—
92	C₂²⋊D₁₀₉	—	2	4	no	—
93	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
95	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
97	D₄₃₆	—	2	4	no	—
98	D₄₃₆	—	2	2	no	—
99	C₄×D₁₀₉	—	2	2	no	—
100	C₄×D₁₀₉	—	2	2	no	—
101	C₄×D₁₀₉	—	2	2	no	—
102	C₄×D₁₀₉	—	2	2	no	—
103	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
104	C₄×D₁₀₉	#41	1	2	yes	—
105	Dic₂₁₈	—	2	4	no	—
106	C₈×C₁₀₉	—	2	2	no	—
107	C₈×C₁₀₉	—	2	1	no	—
108	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
109	#11724784	—	2	2	no	—
110	#11724784	—	2	1	no	—
111	#11724784	C₂×D₄	1	2	yes	—
112	C₂²×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	—	2	2	no	—
114	C₂²×C₁₀₉	—	2	2	no	—
115	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₀₉	—	2	1	no	—
117	C₂×D₁₀₉	—	2	1	no	—
118	C₂×D₁₀₉	—	2	2	no	—
119	C₂×D₁₀₉	—	2	2	no	—
120	C₂×D₁₀₉	—	2	2	no	—
121	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
122	C₄×C₁₀₉	—	2	2	no	—
123	C₄×C₁₀₉	—	2	1	no	—
124	C₄×C₁₀₉	#131	1	2	yes	complemented
125	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
127	Dic₁₀₉	—	2	2	no	—
128	Dic₁₀₉	—	2	1	no	—
129	Dic₁₀₉	C₄×D₄	1	2	yes	—
130	Dic₁₀₉	#131	1	2	yes	complemented
131	C₂×C₁₀₉	—	4	2	no	—
132	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
134	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
135	D₁₀₉	—	2	2	no	—
136	D₁₀₉	—	4	2	no	—
137	#2931	—	109	1	no	maximal, Sylow
138	C₁₀₉	#2931	1	1	yes	characteristic, complemented, Sylow
139	C₂×#131	—	109	2	no	—
140	#532	—	109	1	no	—
141	#436	—	109	1	no	—
142	#436	—	109	1	no	—
143	#436	—	109	2	no	—
144	C₈×D₄	—	109	2	no	—
145	C₈×D₄	—	109	1	no	—
146	C₈×D₄	D₁₀₉	1	1	yes	characteristic, complemented
147	#423	—	109	1	no	—
148	#408	—	109	2	no	—
149	C₂×C₄×C₈	—	109	1	no	—
150	C₂×#41	—	109	2	no	—
151	#131	—	218	8	no	—
152	C₂×M₁₆	—	109	2	no	—
153	C₂×M₁₆	—	109	2	no	—
154	C₂×M₁₆	—	109	2	no	—
155	C₂²×C₈	—	109	1	no	—
156	C₂²×C₈	—	109	1	no	—
157	C₂²×C₈	—	109	2	no	—
158	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
159	C₄⋊Q₈	—	109	1	no	—
160	#127	—	109	1	no	—
161	#124	—	109	2	no	—
162	#121	—	109	1	no	—
163	#121	—	109	1	no	—
164	#121	—	109	2	no	—
165	C₄×D₄	—	109	2	no	—
166	C₄×D₄	—	109	1	no	—
167	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄²	—	109	1	no	—
169	C₄⋊C₈	—	109	2	no	—
170	C₄⋊C₈	—	109	1	no	—
171	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
172	C₂²⋊C₈	—	109	2	no	—
173	C₂²⋊C₈	—	109	2	no	—
174	C₂²⋊C₈	—	109	2	no	—
175	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
176	C₄×C₈	—	109	1	no	—
177	C₄×C₈	—	109	1	no	—
178	C₄×C₈	—	109	1	no	—
179	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
180	#41	—	218	4	no	—
181	#41	—	218	4	no	—
182	C₂×Q₈	—	109	2	no	—
183	C₂×D₄	—	109	2	no	—
184	C₂×D₄	—	109	2	no	—
185	C₂×D₄	—	109	2	no	—
186	C₂×D₄	—	109	1	no	—
187	C₂×D₄	—	109	2	no	—
188	C₂×D₄	C₄×D₁₀₉	1	1	yes	characteristic
189	C₂²×C₄	—	109	1	no	—
190	C₂²×C₄	—	109	1	no	—
191	C₂²×C₄	—	109	1	no	—
192	C₂²×C₄	—	109	2	no	—
193	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
194	M₁₆	—	218	4	no	—
195	M₁₆	—	218	4	no	—
196	M₁₆	—	218	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	109	1	no	—
199	C₂×C₈	—	109	4	no	—
200	C₂×C₈	—	109	1	no	—
201	C₂×C₈	—	109	2	no	—
202	C₂×C₈	—	218	2	no	—
203	C₂×C₈	—	218	4	no	—
204	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
206	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
207	C₄⋊C₄	—	109	2	no	—
208	C₄⋊C₄	—	109	1	no	—
209	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
210	C₂²⋊C₄	—	109	2	no	—
211	C₂²⋊C₄	—	109	2	no	—
212	C₂²⋊C₄	—	109	2	no	—
213	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
214	C₄²	—	109	1	no	—
215	C₄²	—	109	1	no	—
216	C₄²	—	109	1	no	—
217	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
218	C₂³	—	109	1	no	—
219	C₂³	—	109	2	no	—
220	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
221	Q₈	—	218	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	218	4	no	—
224	D₄	—	218	4	no	—
225	D₄	—	218	4	no	—
226	D₄	—	218	4	no	—
227	D₄	—	218	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	109	1	no	—
231	C₂×C₄	—	109	1	no	—
232	C₂×C₄	—	109	2	no	—
233	C₂×C₄	—	109	2	no	—
234	C₂×C₄	—	109	1	no	—
235	C₂×C₄	—	109	2	no	—
236	C₂×C₄	—	218	2	no	—
237	C₂×C₄	—	218	2	no	—
238	C₂×C₄	—	218	2	no	—
239	C₂×C₄	—	218	2	no	—
240	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	109	2	no	—
247	C₈	—	218	2	no	—
248	C₈	—	218	1	no	—
249	C₈	D₄×D₁₀₉	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	109	2	no	—
254	C₂²	—	218	2	no	—
255	C₂²	—	218	2	no	—
256	C₂²	—	218	2	no	—
257	C₂²	—	218	1	no	—
258	C₂²	—	218	1	no	—
259	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	109	2	no	—
263	C₄	—	109	2	no	—
264	C₄	—	218	2	no	—
265	C₄	—	218	1	no	—
266	C₄	#16812705	1	2	yes	complemented
267	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
268	C₄	#41×D₁₀₉	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	436	2	no	—
271	C₂	—	218	2	no	—
272	C₂	C₄×D₄×D₁₀₉	1	1	yes	characteristic
273	C₂	#131×D₁₀₉	1	1	yes	characteristic
274	C₂	C₂×#16812705	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₈×D₄×C₁₀₉	C₂
prime radical	#20295573	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₈×D₄	D₁₀₉
hypocenter	C₁₀₉	#2931
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16812705	C₂	semidirect
3. #20295104	C₂	semidirect
4. #20295103	C₂	semidirect
5. #20295101	C₂	semidirect
6. C₈×D₄₃₆	C₂	semidirect
7. C₈×C₂²⋊D₁₀₉	C₂	semidirect
8. #20295048	C₂	semidirect
9. #20294991	C₂	semidirect
10. C₄×C₈×D₁₀₉	C₂	semidirect
11. C₈×D₄×C₁₀₉	C₂	semidirect
62. C₈×D₁₀₉	D₄	semidirect
124. C₄×C₁₀₉	#131	semidirect
130. Dic₁₀₉	#131	semidirect
138. C₁₀₉	#2931	semidirect
146. C₈×D₄	D₁₀₉	semidirect
266. C₄	#16812705	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812705	2
#20295573	2
#20295104	2
#20295103	2
#20295101	2
C₈×D₄₃₆	2
C₈×C₂²⋊D₁₀₉	2
#20295048	2
#20294991	2
C₄×C₈×D₁₀₉	2
C₈×D₄×C₁₀₉	2
#2931	109