Subgroups — #28376060

Order 13952.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295751	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295745	C₂	1	1	yes	maximal, characteristic
3	#20295417	C₂	1	1	yes	maximal, characteristic
4	#20295404	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295246	C₂	1	1	yes	maximal, characteristic
6	#20295245	C₂	1	1	yes	maximal, characteristic
7	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295233	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295229	C₂	1	1	yes	maximal, characteristic
11	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295133	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
15	#497×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182019	C₂²	1	1	yes	characteristic
19	C₂×#13182019	C₂²	1	1	yes	characteristic
20	C₂×#13182017	C₂²	1	1	yes	characteristic
21	#16812697	C₂²	1	1	yes	characteristic
22	#16812697	C₂²	1	1	yes	characteristic
23	#16812697	C₂²	1	1	yes	characteristic
24	#16812692	—	2	1	no	—
25	#16812692	—	2	1	no	—
26	#16812690	—	2	1	no	—
27	#16812689	—	2	1	no	—
28	#16812689	—	2	1	no	—
29	#16812686	C₂²	1	1	yes	characteristic
30	#16812676	—	2	1	no	—
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
33	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
37	#16812664	—	2	1	no	—
38	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
39	#16812655	—	2	1	no	—
40	#16812631	C₂²	1	1	yes	characteristic
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812628	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812624	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812623	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#124×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	Dic₄₃₆	—	2	2	no	—
64	#13182019	—	4	1	no	—
65	#13182019	—	4	1	no	—
66	#13182019	—	4	1	no	—
67	#13182017	—	4	1	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	—	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	#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	#13182008	D₄	1	1	yes	characteristic
85	#13182008	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	C₂²⋊Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	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	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	Q₁₆×C₁₀₉	—	2	2	no	—
101	Q₁₆×C₁₀₉	—	4	1	no	—
102	SD₁₆×C₁₀₉	—	2	2	no	—
103	SD₁₆×C₁₀₉	—	4	1	no	—
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	4	2	no	—
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
135	Dic₂₁₈	—	2	2	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	#41	1	2	yes	—
142	#11724784	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₀₉	—	2	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	4	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
155	Dic₁₀₉	—	2	1	no	—
156	Dic₁₀₉	—	2	1	no	—
157	Dic₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	#137	1	2	yes	—
163	C₂×C₁₀₉	—	8	1	no	—
164	C₂×C₁₀₉	#550	1	1	yes	characteristic
165	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
166	C₂×C₁₀₉	#578	1	1	yes	characteristic
167	#3398	—	109	1	no	maximal, Sylow
168	C₁₀₉	#3398	1	1	yes	characteristic, complemented, Sylow
169	#557	—	109	1	no	—
170	#556	—	109	1	no	—
171	#497	—	109	1	no	—
172	#497	D₁₀₉	1	1	yes	characteristic
173	#494	—	109	1	no	—
174	#464	—	109	1	no	—
175	#464	—	109	1	no	—
176	#462	—	109	1	no	—
177	#462	—	109	1	no	—
178	#462	—	109	1	no	—
179	#461	—	109	1	no	—
180	#446	—	109	1	no	—
181	#441	—	109	1	no	—
182	#441	—	109	1	no	—
183	C₄×SD₁₆	—	109	1	no	—
184	C₂×Q₁₆	—	109	1	no	—
185	C₂×Q₁₆	—	109	1	no	—
186	C₂×Q₁₆	—	109	1	no	—
187	C₂×Q₁₆	—	109	1	no	—
188	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
189	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	109	1	no	—
191	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
192	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
193	C₄⋊Q₈	—	109	1	no	—
194	C₄⋊Q₈	—	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	#125	—	218	1	no	—
200	#125	—	218	1	no	—
201	#125	—	218	1	no	—
202	#124	—	109	1	no	—
203	#124	C₂×D₁₀₉	1	1	yes	characteristic
204	C₂²⋊Q₈	—	218	1	no	—
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×Q₈	—	109	1	no	—
209	C₄×Q₈	—	109	1	no	—
210	C₄×Q₈	—	109	1	no	—
211	C₄×Q₈	—	218	1	no	—
212	C₄×D₄	—	109	1	no	—
213	#117	—	218	1	no	—
214	#106	—	109	1	no	—
215	C₄⋊C₈	—	109	1	no	—
216	C₄⋊C₈	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	—	109	1	no	—
222	#103	—	109	1	no	—
223	#102	—	109	1	no	—
224	#97	—	109	1	no	—
225	#97	C₂×D₁₀₉	1	1	yes	characteristic
226	C₄×C₈	—	109	1	no	—
227	C₂×Q₈	—	109	1	no	—
228	C₂×Q₈	—	109	1	no	—
229	C₂×Q₈	—	109	1	no	—
230	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂²×C₄	—	218	1	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	4	1	no	—
237	Q₁₆	—	436	1	no	—
238	Q₁₆	—	436	1	no	—
239	Q₁₆	—	436	1	no	—
240	Q₁₆	—	218	2	no	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	4	1	no	—
243	SD₁₆	—	436	1	no	—
244	C₂×C₈	—	109	1	no	—
245	C₂×C₈	—	109	1	no	—
246	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	—	109	1	no	—
253	C₄⋊C₄	—	109	1	no	—
254	C₄⋊C₄	—	109	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	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	—	218	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	—	218	1	no	—
265	C₄⋊C₄	—	218	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₄²	—	109	1	no	—
272	C₄²	—	109	1	no	—
273	C₄²	—	218	1	no	—
274	C₄²	—	218	1	no	—
275	C₄²	—	218	1	no	—
276	C₄²	—	218	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	218	2	no	—
287	Q₈	—	218	1	no	—
288	Q₈	—	218	1	no	—
289	Q₈	—	218	1	no	—
290	Q₈	—	218	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	109	1	no	—
297	C₂×C₄	—	109	1	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	109	1	no	—
300	C₂×C₄	—	436	2	no	—
301	C₂×C₄	—	218	1	no	—
302	C₂×C₄	—	218	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	218	1	no	—
310	C₈	—	218	1	no	—
311	C₈	#13182028	1	2	yes	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	109	2	no	—
322	C₄	—	436	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	436	1	no	—
326	C₄	—	218	1	no	—
327	C₄	—	218	1	no	—
328	C₄	—	218	1	no	—
329	C₄	C₂×#13182028	1	1	yes	characteristic
330	C₄	#16812744	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	#20295696	1	1	yes	characteristic
333	C₂	#137×D₁₀₉	1	1	yes	characteristic
334	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#497×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×C₁₀₉	C₂³
hypercenter	#497	D₁₀₉
hypocenter	C₁₀₉	#3398
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. #20295751	C₂	semidirect
4. #20295404	C₂	semidirect
7. #20295234	C₂	semidirect
8. #20295234	C₂	semidirect
9. #20295233	C₂	semidirect
11. #20295154	C₂	semidirect
12. #20295134	C₂	semidirect
13. #20295133	C₂	semidirect
168. C₁₀₉	#3398	semidirect

Maximal subgroups

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

subgroup	index
#20295751	2
#20295745	2
#20295417	2
#20295404	2
#20295246	2
#20295245	2
#20295234	2
#20295233	2
#20295229	2
#20295154	2
#20295134	2
#20295133	2
SD₁₆×Dic₁₀₉	2
#497×C₁₀₉	2
#3398	109