Subgroups — #28371691

Order 13952.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812656	C₂	1	1	yes	maximal, characteristic
2	#20295088	C₂	1	2	yes	maximal, complemented
3	#20295086	C₂	1	2	yes	maximal, complemented
4	#20295080	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295079	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295076	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295051	C₂	1	2	yes	maximal, complemented
8	#20294988	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
10	#20294983	C₂	1	2	yes	maximal, complemented
11	#432×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#13182012	C₂²	1	2	yes	—
14	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
15	#16812656	C₄	1	1	yes	characteristic
16	#16812656	C₄	1	1	yes	characteristic
17	#16812656	C₄	1	1	yes	characteristic
18	#16812656	C₄	1	1	yes	characteristic
19	#16812656	C₂²	1	1	yes	characteristic
20	#16812656	C₂²	1	1	yes	characteristic
21	#16812656	C₂²	1	1	yes	characteristic
22	#16812656	C₂²	1	1	yes	characteristic
23	C₂×#13182008	C₄	1	1	yes	characteristic
24	C₂×#13182008	C₄	1	1	yes	characteristic
25	C₂×#13182006	C₄	1	1	yes	characteristic
26	C₂×#13182006	C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	C₄⋊#11724784	C₂²	1	2	yes	complemented
29	#16812628	C₂²	1	2	yes	complemented
30	#16812628	C₂²	1	2	yes	complemented
31	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
32	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
33	#16812611	C₂²	1	2	yes	—
34	#16812611	C₂²	1	2	yes	—
35	#16812609	C₂²	1	2	yes	complemented
36	C₄×#11724784	C₂²	1	2	yes	complemented
37	C₈×Dic₁₀₉	C₂²	1	2	yes	complemented
38	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	#97×C₁₀₉	C₂²	1	2	yes	complemented
43	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	—	2	4	no	—
48	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
52	#13182012	—	2	4	no	—
53	C₂×#11724784	C₂³	1	2	yes	—
54	C₂×#11724784	C₂³	1	2	yes	—
55	C₈×D₁₀₉	—	2	4	no	—
56	#13182008	C₂×C₄	1	1	yes	characteristic
57	#13182008	C₂×C₄	1	1	yes	characteristic
58	#13182008	C₂×C₄	1	1	yes	characteristic
59	#13182008	C₂×C₄	1	1	yes	characteristic
60	#13182008	C₂×C₄	1	1	yes	characteristic
61	#13182008	C₂×C₄	1	1	yes	characteristic
62	#13182008	C₂×C₄	1	1	yes	characteristic
63	#13182008	C₂×C₄	1	1	yes	characteristic
64	#13182006	C₂×C₄	1	1	yes	characteristic
65	#13182006	C₂×C₄	1	1	yes	characteristic
66	#13182006	C₂×C₄	1	1	yes	characteristic
67	#13182006	C₂×C₄	1	1	yes	characteristic
68	#13182006	C₂×C₄	1	1	yes	characteristic
69	#13182006	C₂×C₄	1	1	yes	characteristic
70	#13182006	C₂×C₄	1	1	yes	characteristic
71	#13182006	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
80	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
81	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₀₉	—	2	2	no	—
87	C₂²×D₁₀₉	—	2	2	no	—
88	C₂²×D₁₀₉	—	4	1	no	—
89	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	4	1	no	—
94	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₁₀₉	—	2	2	no	—
101	C₂×Dic₁₀₉	—	2	2	no	—
102	C₂×Dic₁₀₉	—	4	1	no	—
103	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₀₉	—	2	2	no	—
110	C₄×D₁₀₉	—	2	2	no	—
111	C₄×D₁₀₉	—	2	2	no	—
112	C₄×D₁₀₉	#41	1	4	yes	—
113	C₈×C₁₀₉	—	2	2	no	—
114	C₈×C₁₀₉	—	2	2	no	—
115	#11724784	—	2	2	no	—
116	#11724784	—	2	2	no	—
117	C₂²×C₁₀₉	—	2	1	no	—
118	C₂²×C₁₀₉	—	2	1	no	—
119	C₂²×C₁₀₉	#131	1	2	yes	—
120	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₁₀₉	—	2	1	no	—
122	C₂×D₁₀₉	—	2	1	no	—
123	C₂×D₁₀₉	—	2	1	no	—
124	C₂×D₁₀₉	—	2	1	no	—
125	C₂×D₁₀₉	—	2	1	no	—
126	C₂×D₁₀₉	—	4	4	no	—
127	C₂×D₁₀₉	#117	1	2	yes	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	2	1	no	—
131	C₄×C₁₀₉	—	2	1	no	—
132	C₄×C₁₀₉	—	2	1	no	—
133	C₄×C₁₀₉	C₂×#41	1	2	yes	—
134	Dic₁₀₉	—	2	1	no	—
135	Dic₁₀₉	—	2	1	no	—
136	Dic₁₀₉	—	2	1	no	—
137	Dic₁₀₉	—	2	1	no	—
138	Dic₁₀₉	—	2	1	no	—
139	Dic₁₀₉	#117	1	2	yes	—
140	C₂×C₁₀₉	—	2	2	no	—
141	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
142	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
143	C₂×C₁₀₉	#568	1	1	yes	characteristic
144	D₁₀₉	—	2	2	no	—
145	D₁₀₉	—	4	1	no	—
146	#2924	—	109	1	no	maximal, Sylow
147	C₁₀₉	#2924	1	1	yes	characteristic, complemented, Sylow
148	C₂×#117	—	109	1	no	—
149	#433	—	109	2	no	—
150	#433	—	109	2	no	—
151	#432	—	109	1	no	—
152	#432	D₁₀₉	1	1	yes	characteristic, complemented
153	#431	—	109	1	no	—
154	#431	—	109	1	no	—
155	#424	—	109	2	no	—
156	#407	—	109	1	no	—
157	C₂×C₂²⋊C₈	—	109	1	no	—
158	#405	—	109	2	no	—
159	C₂³×C₄	—	109	1	no	—
160	C₂×M₁₆	—	109	2	no	—
161	C₂²×C₈	—	109	2	no	—
162	#117	—	109	1	no	—
163	#117	—	109	1	no	—
164	#117	—	109	1	no	—
165	#117	—	109	1	no	—
166	#117	—	109	1	no	—
167	#117	—	109	1	no	—
168	#117	—	109	1	no	—
169	#117	—	109	1	no	—
170	C₂×C₄⋊C₄	—	109	1	no	—
171	C₂×C₄⋊C₄	—	109	1	no	—
172	C₂×C₂²⋊C₄	—	109	1	no	—
173	C₂×C₂²⋊C₄	—	109	1	no	—
174	C₂×C₄²	—	109	1	no	—
175	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
176	C₄⋊C₈	—	109	2	no	—
177	C₄⋊C₈	—	109	2	no	—
178	C₄⋊C₈	—	109	2	no	—
179	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	complemented
180	C₂²⋊C₈	—	109	1	no	—
181	C₂²⋊C₈	—	109	1	no	—
182	C₂²⋊C₈	—	109	2	no	—
183	C₂²⋊C₈	—	109	2	no	—
184	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
185	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
186	#97	—	109	2	no	—
187	#97	C₂×D₁₀₉	1	2	yes	complemented
188	C₄×C₈	—	109	2	no	—
189	C₄×C₈	—	109	2	no	—
190	C₂⁴	—	109	1	no	—
191	C₂²×C₄	—	109	1	no	—
192	C₂²×C₄	—	109	1	no	—
193	C₂²×C₄	—	109	1	no	—
194	C₂²×C₄	—	109	1	no	—
195	C₂²×C₄	—	109	1	no	—
196	C₂²×C₄	—	109	1	no	—
197	C₂²×C₄	—	109	1	no	—
198	C₂²×C₄	—	218	4	no	—
199	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
202	M₁₆	—	218	4	no	—
203	C₂×C₈	—	109	2	no	—
204	C₂×C₈	—	109	2	no	—
205	C₂×C₈	—	218	4	no	—
206	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
207	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
208	C₄⋊C₄	—	109	1	no	—
209	C₄⋊C₄	—	109	1	no	—
210	C₄⋊C₄	—	109	1	no	—
211	C₄⋊C₄	—	109	1	no	—
212	C₄⋊C₄	—	109	1	no	—
213	C₄⋊C₄	—	109	1	no	—
214	C₄⋊C₄	—	109	1	no	—
215	C₄⋊C₄	—	109	1	no	—
216	C₂²⋊C₄	—	109	1	no	—
217	C₂²⋊C₄	—	109	1	no	—
218	C₂²⋊C₄	—	109	1	no	—
219	C₂²⋊C₄	—	109	1	no	—
220	C₂²⋊C₄	—	109	1	no	—
221	C₂²⋊C₄	—	109	1	no	—
222	C₂²⋊C₄	—	109	1	no	—
223	C₂²⋊C₄	—	109	1	no	—
224	C₄²	—	109	1	no	—
225	C₄²	—	109	1	no	—
226	C₄²	—	109	1	no	—
227	C₄²	—	109	1	no	—
228	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
229	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
230	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
231	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂³	—	109	1	no	—
233	C₂³	—	109	1	no	—
234	C₂³	—	436	1	no	—
235	C₂³	—	218	2	no	—
236	C₂³	—	218	2	no	—
237	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	109	1	no	—
242	C₂×C₄	—	109	4	no	—
243	C₂×C₄	—	109	1	no	—
244	C₂×C₄	—	109	1	no	—
245	C₂×C₄	—	109	1	no	—
246	C₂×C₄	—	109	1	no	—
247	C₂×C₄	—	109	1	no	—
248	C₂×C₄	—	436	1	no	—
249	C₂×C₄	—	218	2	no	—
250	C₂×C₄	—	218	2	no	—
251	C₂×C₄	—	218	2	no	—
252	C₂×C₄	—	218	2	no	—
253	C₂×C₄	—	218	2	no	—
254	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	218	2	no	—
263	C₈	—	218	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	109	2	no	—
267	C₂²	—	436	4	no	—
268	C₂²	—	218	1	no	—
269	C₂²	—	218	1	no	—
270	C₂²	—	218	1	no	—
271	C₂²	—	218	1	no	—
272	C₂²	—	218	1	no	—
273	C₂²	#16812707	1	2	yes	—
274	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	109	2	no	—
281	C₄	—	218	1	no	—
282	C₄	—	218	1	no	—
283	C₄	—	218	1	no	—
284	C₄	—	218	1	no	—
285	C₄	—	218	1	no	—
286	C₄	#41×D₁₀₉	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	436	1	no	—
289	C₂	—	218	2	no	—
290	C₂	#117×D₁₀₉	1	1	yes	characteristic
291	C₂	C₂×#16812707	1	1	yes	characteristic
292	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#432×C₁₀₉	C₂
prime radical	C₂³×D₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#432	D₁₀₉
hypocenter	C₁₀₉	#2924
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #20295088	C₂	semidirect
3. #20295086	C₂	semidirect
4. #20295080	C₂	semidirect
5. #20295079	C₂	semidirect
6. #20295076	C₂	semidirect
7. #20295051	C₂	semidirect
10. #20294983	C₂	semidirect
11. #432×C₁₀₉	C₂	semidirect
28. C₄⋊#11724784	C₂²	semidirect
29. #16812628	C₂²	semidirect
30. #16812628	C₂²	semidirect
35. #16812609	C₂²	semidirect
36. C₄×#11724784	C₂²	semidirect
37. C₈×Dic₁₀₉	C₂²	semidirect
39. C₄⋊C₈×C₁₀₉	C₂²	semidirect
42. #97×C₁₀₉	C₂²	semidirect
147. C₁₀₉	#2924	semidirect
152. #432	D₁₀₉	semidirect
179. C₄⋊C₈	C₂×D₁₀₉	semidirect
187. #97	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812656	2
#20295088	2
#20295086	2
#20295080	2
#20295079	2
#20295076	2
#20295051	2
#20294988	2
C₂²⋊C₈×D₁₀₉	2
#20294983	2
#432×C₁₀₉	2
#2924	109