Subgroups — #28371772

Order 13952.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812706	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812705	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295572	C₂	1	1	yes	maximal, characteristic
4	#20295101	C₂	1	2	yes	maximal, complemented
5	#20295100	C₂	1	2	yes	maximal, complemented
6	#20295094	C₂	1	2	yes	maximal, complemented
7	#20295089	C₂	1	2	yes	maximal, complemented
8	C₂×#16812628	C₂	1	1	yes	maximal, characteristic, complemented
9	#20294994	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#408×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812706	—	2	4	no	—
14	#16812705	—	2	4	no	—
15	C₂×#13182013	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	2	yes	—
18	C₂²×#11724784	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	2	yes	—
20	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
21	#16812697	C₄	1	1	yes	characteristic
22	#16812695	C₄	1	1	yes	characteristic
23	#16812688	C₄	1	1	yes	characteristic
24	#16812686	C₄	1	1	yes	characteristic
25	#16812671	C₄	1	2	yes	—
26	#16812671	C₄	1	2	yes	—
27	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	#16812628	C₂²	1	1	yes	characteristic
31	#16812628	C₂²	1	1	yes	characteristic
32	#16812628	C₂²	1	1	yes	characteristic
33	#16812628	C₂²	1	1	yes	characteristic
34	#16812611	C₂²	1	2	yes	—
35	#16812611	C₂²	1	2	yes	—
36	#16812609	C₂²	1	2	yes	—
37	C₈×Dic₁₀₉	C₂²	1	2	yes	—
38	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
42	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
43	#13182029	—	2	2	no	—
44	#13182029	—	2	2	no	—
45	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
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	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
53	#13182013	—	2	4	no	—
54	#13182013	D₄	1	4	yes	complemented
55	#13182012	—	2	4	no	—
56	C₂×#11724784	—	2	2	no	—
57	C₂×#11724784	—	2	4	no	—
58	C₂×#11724784	C₂³	1	1	yes	characteristic
59	C₂×#11724784	C₂³	1	1	yes	characteristic
60	C₂×#11724784	C₂³	1	2	yes	—
61	C₈×D₁₀₉	—	2	4	no	—
62	#13182008	C₂×C₄	1	2	yes	—
63	#13182008	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₀₉	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	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
72	#41×C₁₀₉	—	2	2	no	—
73	#41×C₁₀₉	—	2	2	no	—
74	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
76	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₀₉	—	2	2	no	—
80	C₂×C₈×C₁₀₉	—	2	2	no	—
81	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
84	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
85	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
87	Q₈×C₁₀₉	—	2	2	no	—
88	D₄×C₁₀₉	—	2	4	no	—
89	D₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₀₉	#41	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₀₉	—	2	4	no	—
98	C₂²⋊D₁₀₉	—	2	4	no	—
99	C₂²⋊D₁₀₉	—	4	2	no	—
100	C₂×Dic₁₀₉	—	2	2	no	—
101	C₂×Dic₁₀₉	—	4	1	no	—
102	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
106	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
107	D₄₃₆	—	2	2	no	—
108	C₄×D₁₀₉	—	2	2	no	—
109	C₄×D₁₀₉	—	2	2	no	—
110	Dic₂₁₈	—	2	2	no	—
111	C₈×C₁₀₉	—	2	1	no	—
112	C₈×C₁₀₉	—	2	1	no	—
113	C₈×C₁₀₉	—	2	2	no	—
114	#11724784	—	2	1	no	—
115	#11724784	—	2	1	no	—
116	#11724784	C₂×D₄	1	4	yes	—
117	C₂²×C₁₀₉	—	2	1	no	—
118	C₂²×C₁₀₉	—	2	1	no	—
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	—	2	2	no	—
121	C₂²×C₁₀₉	—	2	2	no	—
122	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
123	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₀₉	—	2	2	no	—
125	C₂×D₁₀₉	—	2	2	no	—
126	C₂×D₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	2	2	no	—
128	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
129	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
131	Dic₁₀₉	—	2	2	no	—
132	Dic₁₀₉	—	2	1	no	—
133	Dic₁₀₉	—	2	1	no	—
134	Dic₁₀₉	—	2	1	no	—
135	Dic₁₀₉	#131	1	2	yes	complemented
136	C₂×C₁₀₉	—	2	2	no	—
137	C₂×C₁₀₉	—	4	2	no	—
138	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
140	C₂×C₁₀₉	#568	1	1	yes	characteristic
141	D₁₀₉	—	4	2	no	—
142	#2935	—	109	1	no	maximal, Sylow
143	C₁₀₉	#2935	1	1	yes	characteristic, complemented, Sylow
144	C₂×#131	—	109	1	no	—
145	C₂×#131	—	109	1	no	—
146	#532	—	109	1	no	—
147	#436	—	109	2	no	—
148	#435	—	109	2	no	—
149	#435	—	109	2	no	—
150	C₈×D₄	—	109	2	no	—
151	C₂×C₄⋊C₈	—	109	1	no	—
152	#408	—	109	1	no	—
153	#408	D₁₀₉	1	1	yes	characteristic, complemented
154	C₄×M₁₆	—	109	1	no	—
155	C₂×#41	—	109	1	no	—
156	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
157	#131	—	218	4	no	—
158	#131	—	218	4	no	—
159	C₂×M₁₆	—	109	1	no	—
160	C₂×M₁₆	—	109	2	no	—
161	C₂×M₁₆	—	109	2	no	—
162	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
163	C₂²×C₈	—	109	1	no	—
164	C₂²×C₈	—	109	2	no	—
165	C₂²×C₈	—	109	2	no	—
166	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
167	C₄⋊Q₈	—	109	1	no	—
168	#127	—	109	1	no	—
169	#124	—	109	1	no	—
170	#124	—	109	1	no	—
171	#121	—	109	2	no	—
172	#121	—	109	2	no	—
173	C₄×D₄	—	109	2	no	—
174	C₄×D₄	—	109	2	no	—
175	C₂×C₄²	—	109	1	no	—
176	C₄⋊C₈	—	109	1	no	—
177	C₄⋊C₈	—	109	1	no	—
178	C₄⋊C₈	—	109	1	no	—
179	C₄⋊C₈	—	109	1	no	—
180	C₂²⋊C₈	—	109	2	no	—
181	C₂²⋊C₈	—	109	2	no	—
182	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
183	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
184	#97	—	109	2	no	—
185	C₄×C₈	—	109	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	218	2	no	—
189	#41	—	218	2	no	—
190	C₂×Q₈	—	109	1	no	—
191	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
192	C₂×D₄	—	109	2	no	—
193	C₂×D₄	—	109	2	no	—
194	C₂×D₄	—	109	2	no	—
195	C₂×D₄	—	109	1	no	—
196	C₂×D₄	C₄×D₁₀₉	1	2	yes	—
197	C₂×D₄	C₄×D₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	—	109	1	no	—
199	C₂²×C₄	—	109	1	no	—
200	C₂²×C₄	—	109	2	no	—
201	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	109	4	no	—
205	M₁₆	—	218	4	no	—
206	M₁₆	—	218	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	109	2	no	—
209	C₂×C₈	—	109	1	no	—
210	C₂×C₈	—	109	1	no	—
211	C₂×C₈	—	218	4	no	—
212	C₂×C₈	—	218	4	no	—
213	C₂×C₈	—	218	2	no	—
214	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
215	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
217	C₄⋊C₄	—	109	2	no	—
218	C₄⋊C₄	—	109	2	no	—
219	C₂²⋊C₄	—	109	2	no	—
220	C₂²⋊C₄	—	109	2	no	—
221	C₂²⋊C₄	—	109	2	no	—
222	C₂²⋊C₄	—	109	2	no	—
223	C₄²	—	109	1	no	—
224	C₄²	—	109	1	no	—
225	C₄²	—	109	1	no	—
226	C₄²	—	109	1	no	—
227	C₂³	—	109	2	no	—
228	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
229	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	218	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	436	2	no	—
235	D₄	—	218	4	no	—
236	D₄	—	218	4	no	—
237	D₄	—	218	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	109	1	no	—
241	C₂×C₄	—	109	1	no	—
242	C₂×C₄	—	109	1	no	—
243	C₂×C₄	—	109	1	no	—
244	C₂×C₄	—	109	2	no	—
245	C₂×C₄	—	436	1	no	—
246	C₂×C₄	—	218	2	no	—
247	C₂×C₄	—	218	2	no	—
248	C₂×C₄	—	218	2	no	—
249	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
251	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
252	C₂×C₄	#13182028	1	2	yes	—
253	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	109	4	no	—
258	C₈	—	218	1	no	—
259	C₈	—	218	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	218	2	no	—
266	C₂²	—	218	2	no	—
267	C₂²	—	218	2	no	—
268	C₂²	#16812659	1	2	yes	—
269	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	109	2	no	—
272	C₄	—	218	2	no	—
273	C₄	—	218	1	no	—
274	C₄	—	218	1	no	—
275	C₄	—	218	1	no	—
276	C₄	#16812659	1	2	yes	—
277	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
278	C₄	C₂×#13182028	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	436	2	no	—
282	C₂	C₂×#16812659	1	1	yes	characteristic
283	C₂	#131×D₁₀₉	1	1	yes	characteristic
284	C₂	#20295791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#408×C₁₀₉	C₂
prime radical	#20295572	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#408	D₁₀₉
hypocenter	C₁₀₉	#2935
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#16812706	C₂	semidirect
2. C₂×#16812705	C₂	semidirect
4. #20295101	C₂	semidirect
5. #20295100	C₂	semidirect
6. #20295094	C₂	semidirect
7. #20295089	C₂	semidirect
8. C₂×#16812628	C₂	semidirect
9. #20294994	C₂	semidirect
10. M₁₆×Dic₁₀₉	C₂	semidirect
11. #408×C₁₀₉	C₂	semidirect
54. #13182013	D₄	semidirect
135. Dic₁₀₉	#131	semidirect
143. C₁₀₉	#2935	semidirect
153. #408	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812706	2
C₂×#16812705	2
#20295572	2
#20295101	2
#20295100	2
#20295094	2
#20295089	2
C₂×#16812628	2
#20294994	2
M₁₆×Dic₁₀₉	2
#408×C₁₀₉	2
#2935	109