Subgroups — #28375802

Order 13952.

344 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	#20295761	C₂	1	1	yes	maximal, characteristic
2	#20295754	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295391	C₂	1	1	yes	maximal, characteristic
4	#20295386	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295381	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295317	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295227	C₂	1	1	yes	maximal, characteristic
12	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295128	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	#16812699	C₂²	1	1	yes	characteristic
20	#16812697	—	2	1	no	—
21	#16812696	C₂²	1	1	yes	characteristic
22	#16812691	C₂²	1	1	yes	characteristic
23	#16812691	C₂²	1	1	yes	characteristic
24	#16812689	—	2	1	no	—
25	#16812689	—	2	1	no	—
26	#16812688	—	2	1	no	—
27	#16812687	C₂²	1	1	yes	characteristic
28	#16812687	C₂²	1	1	yes	characteristic
29	#16812684	—	2	1	no	—
30	#16812684	—	2	1	no	—
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
35	#16812664	—	2	1	no	—
36	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
37	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
38	#16812661	—	2	1	no	—
39	#16812633	C₂²	1	1	yes	characteristic
40	#16812631	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812624	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812624	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812621	C₂²	1	1	yes	characteristic
49	#16812621	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#97×C₁₀₉	C₂²	1	1	yes	characteristic
57	#97×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	—	2	1	no	—
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	Dic₄₃₆	—	2	2	no	—
69	#13182018	—	2	2	no	—
70	#13182018	—	2	2	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182006	—	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	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
102	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₀₉	—	2	1	no	—
106	C₂²×D₁₀₉	—	2	1	no	—
107	Q₈×C₁₀₉	—	4	1	no	—
108	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₀₉	—	2	1	no	—
116	C₂×Dic₁₀₉	—	2	1	no	—
117	C₂×Dic₁₀₉	—	2	1	no	—
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	D₄₃₆	—	2	1	no	—
122	D₄₃₆	—	2	1	no	—
123	D₄₃₆	—	2	1	no	—
124	D₄₃₆	—	2	1	no	—
125	D₄₃₆	—	4	2	no	—
126	D₄₃₆	—	8	1	no	—
127	C₄×D₁₀₉	—	8	1	no	—
128	C₄×D₁₀₉	—	8	1	no	—
129	Dic₂₁₈	—	2	1	no	—
130	Dic₂₁₈	—	2	1	no	—
131	Dic₂₁₈	—	2	1	no	—
132	Dic₂₁₈	—	4	2	no	—
133	Dic₂₁₈	#41	1	2	yes	—
134	C₈×C₁₀₉	—	2	1	no	—
135	C₈×C₁₀₉	—	2	1	no	—
136	C₈×C₁₀₉	—	2	1	no	—
137	C₈×C₁₀₉	—	2	1	no	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	4	1	no	—
140	C₂×D₁₀₉	—	4	1	no	—
141	C₂×D₁₀₉	—	4	1	no	—
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	—	2	1	no	—
148	C₄×C₁₀₉	—	2	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	2	1	no	—
151	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
152	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₀₉	—	2	2	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	Dic₁₀₉	—	4	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	C₂×C₁₀₉	#548	1	1	yes	characteristic
160	C₂×C₁₀₉	#575	1	1	yes	characteristic
161	C₂×C₁₀₉	#577	1	1	yes	characteristic
162	D₁₀₉	—	8	1	no	—
163	D₁₀₉	—	8	1	no	—
164	#3371	—	109	1	no	maximal, Sylow
165	C₁₀₉	#3371	1	1	yes	characteristic, complemented, Sylow
166	#559	—	109	1	no	—
167	#557	—	109	1	no	—
168	#490	—	109	1	no	—
169	#489	—	109	1	no	—
170	#488	—	109	1	no	—
171	#479	—	109	1	no	—
172	#478	—	109	1	no	—
173	#476	—	109	1	no	—
174	#464	—	109	1	no	—
175	#464	—	109	1	no	—
176	#460	—	109	1	no	—
177	#446	D₁₀₉	1	1	yes	characteristic, complemented
178	#441	—	109	1	no	—
179	#440	—	109	1	no	—
180	C₄×SD₁₆	—	109	1	no	—
181	C₂×Q₁₆	—	109	1	no	—
182	C₂×SD₁₆	—	109	1	no	—
183	C₂×SD₁₆	—	109	1	no	—
184	C₄⋊Q₈	—	109	1	no	—
185	C₄⋊Q₈	—	218	1	no	—
186	#127	—	109	1	no	—
187	#125	—	109	1	no	—
188	#125	—	109	1	no	—
189	#125	—	218	1	no	—
190	#125	—	218	1	no	—
191	#124	—	109	1	no	—
192	#124	—	109	1	no	—
193	#124	—	218	1	no	—
194	#123	—	218	1	no	—
195	#123	—	218	1	no	—
196	C₄×Q₈	—	109	1	no	—
197	C₄×Q₈	—	109	1	no	—
198	C₄×Q₈	—	109	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×Q₈	—	218	1	no	—
201	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
202	C₄×D₄	—	109	1	no	—
203	C₄×D₄	—	109	1	no	—
204	C₄×D₄	—	218	1	no	—
205	#107	—	109	1	no	—
206	#106	—	109	1	no	—
207	#106	—	109	1	no	—
208	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
209	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
210	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
211	#103	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#102	—	109	1	no	—
216	#102	—	109	1	no	—
217	#102	—	109	1	no	—
218	#102	—	109	1	no	—
219	#102	—	109	1	no	—
220	#102	—	109	1	no	—
221	#97	C₂×D₁₀₉	1	1	yes	characteristic
222	#97	C₂×D₁₀₉	1	1	yes	characteristic
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	D₄₃₆	1	1	yes	characteristic
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	—	109	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	—	218	1	no	—
232	C₂²×C₄	—	218	1	no	—
233	C₂²×C₄	—	218	1	no	—
234	Q₁₆	—	218	2	no	—
235	SD₁₆	—	218	2	no	—
236	SD₁₆	—	218	2	no	—
237	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₄⋊C₄	—	109	1	no	—
242	C₄⋊C₄	—	109	1	no	—
243	C₄⋊C₄	—	109	1	no	—
244	C₄⋊C₄	—	109	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	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	D₄₃₆	1	1	yes	characteristic
255	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
256	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
257	C₂²⋊C₄	—	218	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	C₄²	—	218	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	—	218	1	no	—
266	C₄²	—	218	1	no	—
267	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂³	—	218	1	no	—
271	C₂³	—	218	1	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	109	2	no	—
274	Q₈	—	436	2	no	—
275	Q₈	—	218	1	no	—
276	Q₈	—	218	1	no	—
277	Q₈	—	218	1	no	—
278	D₄	—	872	1	no	—
279	D₄	—	436	2	no	—
280	D₄	—	218	1	no	—
281	D₄	—	218	1	no	—
282	D₄	—	218	1	no	—
283	D₄	—	218	1	no	—
284	C₂×C₄	—	872	1	no	—
285	C₂×C₄	—	872	1	no	—
286	C₂×C₄	—	218	1	no	—
287	C₂×C₄	—	218	1	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	—	218	1	no	—
292	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄₃₆	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	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	436	1	no	—
304	C₂²	—	436	1	no	—
305	C₂²	—	436	1	no	—
306	C₂²	—	436	1	no	—
307	C₂²	—	436	1	no	—
308	C₂²	—	436	1	no	—
309	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	436	1	no	—
317	C₄	—	436	1	no	—
318	C₄	—	436	1	no	—
319	C₄	—	436	1	no	—
320	C₄	—	436	1	no	—
321	C₄	—	218	2	no	—
322	C₄	#41×D₁₀₉	1	1	yes	characteristic
323	C₄	#16812747	1	1	yes	characteristic
324	C₂	—	872	1	no	—
325	C₂	—	872	1	no	—
326	C₂	#20295688	1	1	yes	characteristic
327	C₂	#20295826	1	1	yes	characteristic
328	C₂	#20295838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#446×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812696	C₂²
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	#3371
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #20295754	C₂	semidirect
4. #20295386	C₂	semidirect
5. #20295381	C₂	semidirect
6. #20295334	C₂	semidirect
7. #20295327	C₂	semidirect
8. #20295317	C₂	semidirect
9. #20295249	C₂	semidirect
10. #20295249	C₂	semidirect
12. #20295134	C₂	semidirect
13. #20295128	C₂	semidirect
14. C₄×#13182018	C₂	semidirect
15. #446×C₁₀₉	C₂	semidirect
165. C₁₀₉	#3371	semidirect
177. #446	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295761	2
#20295754	2
#20295391	2
#20295386	2
#20295381	2
#20295334	2
#20295327	2
#20295317	2
#20295249	2
#20295227	2
#20295134	2
#20295128	2
C₄×#13182018	2
#446×C₁₀₉	2
#3371	109