Subgroups — #28375804

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	#20295758	C₂	1	1	yes	maximal, characteristic
2	#20295752	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295389	C₂	1	1	yes	maximal, characteristic
4	#20295387	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295380	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295335	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295314	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295250	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295224	C₂	1	1	yes	maximal, characteristic
11	#20295153	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295133	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182016	C₂²	1	1	yes	characteristic
17	#16812698	—	2	1	no	—
18	#16812698	C₂²	1	1	yes	characteristic
19	#16812695	C₂²	1	1	yes	characteristic
20	#16812691	—	2	1	no	—
21	#16812691	C₂²	1	1	yes	characteristic
22	#16812690	—	2	1	no	—
23	#16812690	C₂²	1	1	yes	characteristic
24	#16812688	C₂²	1	1	yes	characteristic
25	#16812685	—	2	1	no	—
26	#16812683	—	2	1	no	—
27	#16812681	—	2	1	no	—
28	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₁₈	—	2	1	no	—
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	#16812659	—	2	1	no	—
36	#16812633	C₂²	1	1	yes	characteristic
37	#16812631	C₂²	1	1	yes	characteristic
38	#16812630	C₂²	1	1	yes	characteristic
39	#16812628	C₂²	1	1	yes	characteristic
40	#16812628	C₂²	1	1	yes	characteristic
41	#16812623	C₂²	1	1	yes	characteristic
42	#16812623	C₂²	1	1	yes	characteristic
43	#16812623	C₂²	1	1	yes	characteristic
44	#16812621	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812609	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₄×#11724784	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#124×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
61	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	#13182016	—	2	2	no	—
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	—	2	1	no	—
73	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	Q₁₆×C₁₀₉	—	2	2	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	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
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	Q₈×C₁₀₉	—	2	1	no	—
110	Q₈×C₁₀₉	#41	1	2	yes	—
111	D₄×C₁₀₉	—	2	1	no	—
112	D₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	4	2	no	—
120	C₂²⋊D₁₀₉	—	8	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	8	1	no	—
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	2	1	no	—
131	C₄×D₁₀₉	—	8	1	no	—
132	Dic₂₁₈	—	4	1	no	—
133	Dic₂₁₈	—	4	2	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	—	4	1	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	2	no	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	—	2	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	—	2	1	no	—
156	Dic₁₀₉	—	4	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	C₂×C₁₀₉	—	8	1	no	—
160	C₂×C₁₀₉	#548	1	1	yes	characteristic
161	C₂×C₁₀₉	#575	1	1	yes	characteristic
162	C₂×C₁₀₉	#577	1	1	yes	characteristic
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	D₁₀₉	1	1	yes	characteristic, complemented
176	#460	—	109	1	no	—
177	#446	—	109	1	no	—
178	#441	—	109	1	no	—
179	#440	—	109	1	no	—
180	C₄×SD₁₆	—	109	1	no	—
181	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
182	C₂×SD₁₆	—	109	1	no	—
183	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
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	—	218	1	no	—
193	#124	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	1	no	—
209	C₄⋊C₈	—	109	1	no	—
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	C₂×D₁₀₉	1	1	yes	characteristic
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	C₂×D₁₀₉	1	1	yes	characteristic
221	#97	—	109	1	no	—
222	#97	—	109	1	no	—
223	C₄×C₈	—	109	1	no	—
224	C₂×Q₈	—	109	1	no	—
225	C₂×Q₈	—	218	1	no	—
226	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	—	218	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	—	218	1	no	—
233	C₂²×C₄	—	218	1	no	—
234	Q₁₆	—	2	2	no	—
235	SD₁₆	—	2	2	no	—
236	SD₁₆	—	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	—	109	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	—	218	1	no	—
256	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	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	—	2	1	no	—
264	C₄²	—	109	1	no	—
265	C₄²	—	109	1	no	—
266	C₄²	—	218	1	no	—
267	C₄²	—	218	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	—	2	1	no	—
275	Q₈	—	436	1	no	—
276	Q₈	—	436	2	no	—
277	Q₈	#13182028	1	2	yes	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	872	1	no	—
281	D₄	—	436	2	no	—
282	D₄	—	218	1	no	—
283	D₄	—	218	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	872	1	no	—
288	C₂×C₄	—	872	1	no	—
289	C₂×C₄	—	109	1	no	—
290	C₂×C₄	—	109	1	no	—
291	C₂×C₄	—	109	1	no	—
292	C₂×C₄	—	109	1	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	—	4	1	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	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	436	1	no	—
316	C₄	—	436	1	no	—
317	C₄	—	436	1	no	—
318	C₄	—	218	1	no	—
319	C₄	—	218	1	no	—
320	C₄	—	218	1	no	—
321	C₄	—	218	1	no	—
322	C₄	C₂×#13182028	1	1	yes	characteristic
323	C₄	#16812746	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	872	1	no	—
326	C₂	#20295682	1	1	yes	characteristic
327	C₂	#20295823	1	1	yes	characteristic
328	C₂	#20295835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#464×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812695	C₂²
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3371
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. #20295752	C₂	semidirect
4. #20295387	C₂	semidirect
5. #20295380	C₂	semidirect
6. #20295335	C₂	semidirect
7. #20295327	C₂	semidirect
8. #20295314	C₂	semidirect
9. #20295250	C₂	semidirect
11. #20295153	C₂	semidirect
12. #20295133	C₂	semidirect
13. #20295127	C₂	semidirect
14. C₄×#13182016	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
165. C₁₀₉	#3371	semidirect
175. #464	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295758	2
#20295752	2
#20295389	2
#20295387	2
#20295380	2
#20295335	2
#20295327	2
#20295314	2
#20295250	2
#20295224	2
#20295153	2
#20295133	2
#20295127	2
C₄×#13182016	2
#464×C₁₀₉	2
#3371	109