Subgroups — #28375682

Order 13952.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295748	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295735	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295382	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295381	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295336	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295315	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295243	C₂	1	1	yes	maximal, characteristic
10	#20295241	C₂	1	1	yes	maximal, characteristic
11	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295117	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295106	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182017	C₂²	1	1	yes	characteristic
17	C₂×#13182014	C₂²	1	1	yes	characteristic
18	#16812698	C₂²	1	1	yes	characteristic
19	#16812697	C₂²	1	1	yes	characteristic
20	#16812694	—	2	1	no	—
21	#16812693	—	2	1	no	—
22	#16812691	C₂²	1	1	yes	characteristic
23	#16812690	C₂²	1	1	yes	characteristic
24	#16812688	C₂²	1	1	yes	characteristic
25	#16812687	C₂²	1	1	yes	characteristic
26	#16812678	—	2	1	no	—
27	#16812677	—	2	1	no	—
28	#16812676	—	2	1	no	—
29	#16812676	—	2	1	no	—
30	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
31	#16812664	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
34	#16812661	C₂²	1	1	yes	characteristic
35	#16812656	—	2	1	no	—
36	C₂×#13182008	—	2	1	no	—
37	#16812632	C₂²	1	1	yes	characteristic
38	#16812630	C₂²	1	1	yes	characteristic
39	C₄⋊#11724784	C₂²	1	1	yes	characteristic
40	#16812628	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812624	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812623	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812622	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812621	C₂²	1	1	yes	characteristic
49	#16812619	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	#13182017	—	2	2	no	—
67	#13182014	—	2	2	no	—
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	#13182008	—	2	1	no	—
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	D₄	1	1	yes	characteristic
82	#13182008	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	—	2	1	no	—
96	SD₁₆×C₁₀₉	—	2	2	no	—
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄²×C₁₀₉	—	2	1	no	—
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂³×C₁₀₉	—	2	1	no	—
106	C₂²×D₁₀₉	—	2	1	no	—
107	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	2	1	no	—
109	D₄×C₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	#41	1	2	yes	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	8	1	no	—
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	4	2	no	—
122	C₂×Dic₁₀₉	—	8	1	no	—
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	2	1	no	—
129	C₄×D₁₀₉	—	4	2	no	—
130	Dic₂₁₈	—	2	2	no	—
131	Dic₂₁₈	—	2	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	Dic₂₁₈	—	8	1	no	—
134	C₈×C₁₀₉	—	2	1	no	—
135	C₈×C₁₀₉	—	2	1	no	—
136	#11724784	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	C₂²×C₁₀₉	—	2	2	no	—
139	C₂²×C₁₀₉	—	4	1	no	—
140	C₂²×C₁₀₉	—	4	1	no	—
141	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₀₉	—	4	1	no	—
143	C₂×D₁₀₉	—	4	1	no	—
144	C₂×D₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	2	1	no	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	—	4	1	no	—
148	C₄×C₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	—	4	1	no	—
150	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₀₉	—	2	1	no	—
153	Dic₁₀₉	—	2	1	no	—
154	Dic₁₀₉	—	2	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	Dic₁₀₉	—	4	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	#135	1	2	yes	—
159	C₂×C₁₀₉	—	4	2	no	—
160	C₂×C₁₀₉	#548	1	1	yes	characteristic
161	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₀₉	#577	1	1	yes	characteristic
163	D₁₀₉	—	8	1	no	—
164	#3360	—	109	1	no	maximal, Sylow
165	C₁₀₉	#3360	1	1	yes	characteristic, complemented, Sylow
166	#556	—	109	1	no	—
167	#554	—	109	1	no	—
168	#488	—	109	1	no	—
169	#488	—	109	1	no	—
170	#487	—	109	1	no	—
171	#479	—	109	1	no	—
172	#479	—	109	1	no	—
173	#476	—	109	1	no	—
174	#463	—	109	1	no	—
175	#463	—	109	1	no	—
176	#461	—	109	1	no	—
177	C₈×Q₈	—	109	1	no	—
178	C₄×SD₁₆	—	109	1	no	—
179	C₄×SD₁₆	D₁₀₉	1	1	yes	characteristic, complemented
180	C₄×D₈	—	109	1	no	—
181	C₂×SD₁₆	—	109	1	no	—
182	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
183	C₂×D₈	—	109	1	no	—
184	C₄⋊Q₈	—	109	1	no	—
185	C₄⋊Q₈	—	109	1	no	—
186	#126	—	218	1	no	—
187	#126	—	218	1	no	—
188	#125	—	109	1	no	—
189	#125	—	109	1	no	—
190	#124	—	109	1	no	—
191	#124	—	109	1	no	—
192	C₂²⋊Q₈	—	218	1	no	—
193	C₂²⋊Q₈	—	218	1	no	—
194	C₂²⋊Q₈	—	218	1	no	—
195	C₂²⋊Q₈	—	218	1	no	—
196	C₄×Q₈	—	109	1	no	—
197	C₄×Q₈	—	109	1	no	—
198	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
199	C₄×D₄	—	109	1	no	—
200	C₄×D₄	—	109	1	no	—
201	C₄×D₄	—	109	1	no	—
202	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
203	#117	—	218	1	no	—
204	C₂×C₄⋊C₄	—	218	1	no	—
205	#107	—	109	1	no	—
206	#106	—	109	1	no	—
207	#106	C₂×D₁₀₉	1	1	yes	characteristic
208	C₄⋊C₈	—	109	1	no	—
209	C₄⋊C₈	—	109	1	no	—
210	C₄⋊C₈	—	109	1	no	—
211	#103	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	C₂×D₁₀₉	1	1	yes	characteristic
217	#102	—	109	1	no	—
218	#102	—	109	1	no	—
219	#102	—	109	1	no	—
220	#102	C₂×D₁₀₉	1	1	yes	characteristic
221	C₄×C₈	—	109	1	no	—
222	C₄×C₈	—	109	1	no	—
223	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
224	C₂×Q₈	—	109	1	no	—
225	C₂×Q₈	—	109	1	no	—
226	C₂×Q₈	—	218	1	no	—
227	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	218	1	no	—
232	C₂²×C₄	—	218	1	no	—
233	C₂²×C₄	—	218	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	218	2	no	—
236	D₈	—	218	2	no	—
237	C₂×C₈	—	109	1	no	—
238	C₂×C₈	—	109	1	no	—
239	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	109	1	no	—
243	C₄⋊C₄	—	109	1	no	—
244	C₄⋊C₄	—	109	1	no	—
245	C₄⋊C₄	—	109	1	no	—
246	C₄⋊C₄	—	109	1	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	—	218	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	218	1	no	—
259	C₂²⋊C₄	—	218	1	no	—
260	C₂²⋊C₄	—	218	1	no	—
261	C₂²⋊C₄	—	218	1	no	—
262	C₂²⋊C₄	—	218	1	no	—
263	C₂²⋊C₄	—	218	1	no	—
264	C₂²⋊C₄	—	218	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	109	1	no	—
267	C₄²	—	109	1	no	—
268	C₄²	—	218	1	no	—
269	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	218	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	872	1	no	—
275	Q₈	—	218	2	no	—
276	Q₈	—	218	1	no	—
277	Q₈	—	218	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	218	1	no	—
280	D₄	—	218	1	no	—
281	D₄	#13182029	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	872	1	no	—
287	C₂×C₄	—	109	1	no	—
288	C₂×C₄	—	109	1	no	—
289	C₂×C₄	—	109	1	no	—
290	C₂×C₄	—	109	1	no	—
291	C₂×C₄	—	436	2	no	—
292	C₂×C₄	—	436	2	no	—
293	C₂×C₄	—	218	1	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	218	1	no	—
302	C₈	—	218	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	436	1	no	—
307	C₂²	—	436	1	no	—
308	C₂²	—	436	1	no	—
309	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	109	2	no	—
316	C₄	—	436	1	no	—
317	C₄	—	436	1	no	—
318	C₄	—	436	1	no	—
319	C₄	—	218	1	no	—
320	C₄	—	218	1	no	—
321	C₄	—	218	1	no	—
322	C₄	C₂×#13182029	1	1	yes	characteristic
323	C₄	#16812746	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	872	1	no	—
326	C₂	#20295685	1	1	yes	characteristic
327	C₂	#135×D₁₀₉	1	1	yes	characteristic
328	C₂	#20295832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182014	C₂²
hypercenter	C₄×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#3360
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20295748	C₂	semidirect
2. #20295735	C₂	semidirect
3. #20295382	C₂	semidirect
4. #20295381	C₂	semidirect
5. #20295375	C₂	semidirect
6. #20295336	C₂	semidirect
7. #20295333	C₂	semidirect
8. #20295315	C₂	semidirect
11. #20295229	C₂	semidirect
12. C₈×Dic₂₁₈	C₂	semidirect
13. #20295117	C₂	semidirect
15. C₄×SD₁₆×C₁₀₉	C₂	semidirect
165. C₁₀₉	#3360	semidirect
179. C₄×SD₁₆	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295748	2
#20295735	2
#20295382	2
#20295381	2
#20295375	2
#20295336	2
#20295333	2
#20295315	2
#20295243	2
#20295241	2
#20295229	2
C₈×Dic₂₁₈	2
#20295117	2
#20295106	2
C₄×SD₁₆×C₁₀₉	2
#3360	109