Subgroups — #28374504

Order 13952.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295650	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295641	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295315	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295308	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295285	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295278	C₂	1	1	yes	maximal, characteristic
8	#20295256	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295254	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295247	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295241	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295240	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295051	C₂	1	1	yes	maximal, characteristic, complemented
15	#476×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×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	C₂×#13182012	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
22	#16812698	C₂²	1	1	yes	characteristic
23	#16812694	—	2	1	no	—
24	#16812692	—	2	1	no	—
25	#16812688	—	2	1	no	—
26	#16812687	C₂²	1	1	yes	characteristic
27	#16812686	C₂²	1	1	yes	characteristic
28	#16812685	—	2	1	no	—
29	#16812684	—	2	1	no	—
30	#16812682	—	2	1	no	—
31	#16812679	C₂²	1	1	yes	characteristic
32	#16812679	C₂²	1	1	yes	characteristic
33	#16812678	—	2	1	no	—
34	#16812675	—	2	1	no	—
35	#16812674	C₂²	1	1	yes	characteristic
36	#16812670	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
38	#16812664	C₂²	1	1	yes	characteristic
39	#16812661	C₂²	1	1	yes	characteristic
40	#16812656	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812622	C₂²	1	1	yes	characteristic
48	#16812622	C₂²	1	1	yes	characteristic
49	#16812622	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812619	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#106×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×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	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
67	Dic₄₃₆	—	4	1	no	—
68	#13182018	—	4	1	no	—
69	#13182017	—	4	1	no	—
70	#13182014	—	4	1	no	—
71	#13182012	—	4	1	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₈×D₁₀₉	—	4	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	D₄	1	1	yes	characteristic
93	#13182006	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₀₉	—	8	1	no	—
112	D₄×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	8	1	no	—
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₀₉	—	8	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	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	D₄₃₆	—	2	1	no	—
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	8	1	no	—
131	C₄×D₁₀₉	—	2	1	no	—
132	C₄×D₁₀₉	—	2	1	no	—
133	C₄×D₁₀₉	—	8	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	8	1	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	#11724784	—	2	1	no	—
142	#11724784	—	2	1	no	—
143	C₂²×C₁₀₉	—	4	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₀₉	—	2	1	no	—
148	C₂×D₁₀₉	—	2	1	no	—
149	C₂×D₁₀₉	—	2	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	2	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	—	8	1	no	—
167	C₂×C₁₀₉	#537	1	1	yes	characteristic
168	C₂×C₁₀₉	#577	1	1	yes	characteristic
169	C₂×C₁₀₉	#577	1	1	yes	characteristic
170	D₁₀₉	—	4	1	no	—
171	D₁₀₉	—	8	1	no	—
172	#3263	—	109	1	no	maximal, Sylow
173	C₁₀₉	#3263	1	1	yes	characteristic, complemented, Sylow
174	#542	—	109	1	no	—
175	#542	—	109	1	no	—
176	#476	—	109	1	no	—
177	#476	D₁₀₉	1	1	yes	characteristic, complemented
178	#475	—	109	1	no	—
179	#475	—	109	1	no	—
180	#470	—	109	1	no	—
181	#469	—	109	1	no	—
182	#465	—	109	1	no	—
183	#465	—	109	1	no	—
184	#464	—	109	1	no	—
185	#464	—	109	1	no	—
186	#463	—	109	1	no	—
187	#463	—	109	1	no	—
188	#424	—	109	1	no	—
189	C₂×Q₁₆	—	109	1	no	—
190	C₂×SD₁₆	—	109	1	no	—
191	C₂×SD₁₆	—	109	1	no	—
192	C₂×D₈	—	109	1	no	—
193	C₂×M₁₆	—	109	1	no	—
194	C₂²×C₈	—	109	1	no	—
195	C₄⋊Q₈	—	109	1	no	—
196	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
197	#126	—	218	1	no	—
198	#126	—	218	1	no	—
199	#124	—	109	1	no	—
200	#124	—	109	1	no	—
201	#124	—	218	1	no	—
202	#124	—	218	1	no	—
203	#123	—	218	1	no	—
204	#123	—	218	1	no	—
205	C₂²⋊Q₈	—	109	1	no	—
206	C₂²⋊Q₈	—	109	1	no	—
207	C₂²⋊Q₈	—	218	1	no	—
208	C₂²⋊Q₈	—	218	1	no	—
209	#121	—	109	1	no	—
210	#121	—	109	1	no	—
211	C₄×Q₈	—	109	1	no	—
212	C₄×Q₈	—	109	1	no	—
213	C₄×D₄	—	109	1	no	—
214	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
215	#117	—	109	1	no	—
216	#106	—	109	1	no	—
217	#106	—	109	1	no	—
218	#106	C₂×D₁₀₉	1	1	yes	characteristic
219	#106	C₂×D₁₀₉	1	1	yes	characteristic
220	C₄⋊C₈	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	—	109	1	no	—
223	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	#102	C₂×D₁₀₉	1	1	yes	characteristic
231	#102	C₂×D₁₀₉	1	1	yes	characteristic
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	109	1	no	—
234	C₂×Q₈	—	109	1	no	—
235	C₂×Q₈	—	218	1	no	—
236	C₂×D₄	—	109	1	no	—
237	C₂×D₄	—	218	1	no	—
238	C₂×D₄	—	218	1	no	—
239	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	109	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	Q₁₆	—	436	1	no	—
244	SD₁₆	—	436	1	no	—
245	SD₁₆	—	436	1	no	—
246	D₈	—	436	1	no	—
247	M₁₆	—	436	1	no	—
248	C₂×C₈	—	109	1	no	—
249	C₂×C₈	—	109	1	no	—
250	C₂×C₈	—	436	1	no	—
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	109	1	no	—
255	C₄⋊C₄	—	109	1	no	—
256	C₄⋊C₄	—	109	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	109	1	no	—
267	C₂²⋊C₄	—	109	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₄²	—	109	1	no	—
278	C₄²	—	218	1	no	—
279	C₄²	—	218	1	no	—
280	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	109	1	no	—
283	C₂³	—	218	1	no	—
284	Q₈	—	8	1	no	—
285	Q₈	—	872	1	no	—
286	Q₈	—	218	1	no	—
287	Q₈	—	218	1	no	—
288	Q₈	—	218	1	no	—
289	Q₈	—	218	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	872	1	no	—
293	D₄	—	872	1	no	—
294	D₄	—	218	1	no	—
295	D₄	—	218	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	872	1	no	—
301	C₂×C₄	—	109	1	no	—
302	C₂×C₄	—	109	1	no	—
303	C₂×C₄	—	109	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	—	218	1	no	—
309	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
310	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	218	1	no	—
315	C₈	—	218	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	436	1	no	—
320	C₂²	—	436	1	no	—
321	C₂²	—	436	1	no	—
322	C₂²	—	218	1	no	—
323	C₂²	—	218	1	no	—
324	C₂²	—	218	1	no	—
325	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	436	1	no	—
332	C₄	—	436	1	no	—
333	C₄	—	436	1	no	—
334	C₄	—	218	1	no	—
335	C₄	—	218	1	no	—
336	C₄	—	218	1	no	—
337	C₄	#16812745	1	1	yes	characteristic
338	C₄	#16812747	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	872	1	no	—
341	C₂	—	436	1	no	—
342	C₂	#20295606	1	1	yes	characteristic
343	C₂	#20295832	1	1	yes	characteristic
344	C₂	#20295836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#476×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295278	C₂
hypercenter	#476	D₁₀₉
hypocenter	C₁₀₉	#3263
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #20295650	C₂	semidirect
2. #20295641	C₂	semidirect
3. #20295315	C₂	semidirect
4. #20295310	C₂	semidirect
5. #20295308	C₂	semidirect
6. #20295285	C₂	semidirect
8. #20295256	C₂	semidirect
9. #20295254	C₂	semidirect
10. #20295249	C₂	semidirect
11. #20295247	C₂	semidirect
12. #20295241	C₂	semidirect
13. #20295240	C₂	semidirect
14. #20295051	C₂	semidirect
15. #476×C₁₀₉	C₂	semidirect
173. C₁₀₉	#3263	semidirect
177. #476	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295650	2
#20295641	2
#20295315	2
#20295310	2
#20295308	2
#20295285	2
#20295278	2
#20295256	2
#20295254	2
#20295249	2
#20295247	2
#20295241	2
#20295240	2
#20295051	2
#476×C₁₀₉	2
#3263	109