Subgroups — #28375773

Order 13952.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295748	C₂	1	1	yes	maximal, characteristic
3	#20295387	C₂	1	1	yes	maximal, characteristic
4	#20295386	C₂	1	1	yes	maximal, characteristic
5	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295315	C₂	1	1	yes	maximal, characteristic
7	#20295311	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295250	C₂	1	1	yes	maximal, characteristic
10	#20295248	C₂	1	1	yes	maximal, characteristic
11	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295130	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295111	C₂	1	1	yes	maximal, characteristic
15	#441×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	#16812699	—	2	1	no	—
19	#16812699	C₂²	1	1	yes	characteristic
20	#16812698	—	2	1	no	—
21	#16812698	—	2	1	no	—
22	#16812698	C₂²	1	1	yes	characteristic
23	#16812698	C₂²	1	1	yes	characteristic
24	#16812697	C₂²	1	1	yes	characteristic
25	#16812694	—	2	1	no	—
26	#16812693	—	2	1	no	—
27	#16812688	C₂²	1	1	yes	characteristic
28	#16812687	C₂²	1	1	yes	characteristic
29	#16812677	—	2	1	no	—
30	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁₈	—	2	1	no	—
33	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
34	#16812664	C₂²	1	1	yes	characteristic
35	#16812661	C₂²	1	1	yes	characteristic
36	#16812656	—	2	1	no	—
37	#16812630	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	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
63	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	#13182019	—	2	2	no	—
67	#13182016	—	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	—	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	D₄	1	1	yes	characteristic
82	#13182008	D₄	1	1	yes	characteristic
83	#13182006	—	2	1	no	—
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	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×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
95	Q₁₆×C₁₀₉	—	2	2	no	—
96	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	C₄²×C₁₀₉	—	2	1	no	—
103	C₄²×C₁₀₉	—	2	1	no	—
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₀₉	—	2	1	no	—
106	Q₈×C₁₀₉	—	2	1	no	—
107	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	2	1	no	—
109	Q₈×C₁₀₉	#41	1	2	yes	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₁₀₉	—	2	1	no	—
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	D₄₃₆	—	2	1	no	—
125	D₄₃₆	—	2	1	no	—
126	C₄×D₁₀₉	—	4	2	no	—
127	Dic₂₁₈	—	2	2	no	—
128	Dic₂₁₈	—	2	1	no	—
129	Dic₂₁₈	—	2	1	no	—
130	Dic₂₁₈	—	8	1	no	—
131	Dic₂₁₈	—	8	1	no	—
132	Dic₂₁₈	—	8	1	no	—
133	C₈×C₁₀₉	—	2	1	no	—
134	C₈×C₁₀₉	—	2	1	no	—
135	#11724784	—	2	1	no	—
136	#11724784	—	2	1	no	—
137	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₀₉	—	4	1	no	—
139	C₂×D₁₀₉	—	4	1	no	—
140	C₂×D₁₀₉	—	4	1	no	—
141	C₄×C₁₀₉	—	2	2	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	2	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₀₉	—	2	1	no	—
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	4	1	no	—
153	Dic₁₀₉	—	4	1	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	#137	1	2	yes	—
156	C₂×C₁₀₉	#548	1	1	yes	characteristic
157	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
158	C₂×C₁₀₉	#577	1	1	yes	characteristic
159	D₁₀₉	—	8	1	no	—
160	#3369	—	109	1	no	maximal, Sylow
161	C₁₀₉	#3369	1	1	yes	characteristic, complemented, Sylow
162	Q₈²	—	109	1	no	—
163	#556	—	109	1	no	—
164	#489	—	109	1	no	—
165	#489	—	109	1	no	—
166	#487	—	109	1	no	—
167	#476	—	109	1	no	—
168	#475	—	109	1	no	—
169	#475	—	109	1	no	—
170	#464	—	109	1	no	—
171	#464	—	109	1	no	—
172	#462	—	109	1	no	—
173	#446	—	109	1	no	—
174	#441	—	109	1	no	—
175	#441	D₁₀₉	1	1	yes	characteristic, complemented
176	C₄×SD₁₆	—	109	1	no	—
177	C₂×Q₁₆	—	109	1	no	—
178	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
179	C₂×SD₁₆	—	109	1	no	—
180	C₄⋊Q₈	—	109	1	no	—
181	C₄⋊Q₈	—	109	1	no	—
182	C₄⋊Q₈	—	109	1	no	—
183	C₄⋊Q₈	—	109	1	no	—
184	C₄⋊Q₈	—	218	1	no	—
185	C₄⋊Q₈	—	218	1	no	—
186	C₄⋊Q₈	—	218	1	no	—
187	#126	—	218	1	no	—
188	#126	—	218	1	no	—
189	#124	—	109	1	no	—
190	#124	—	109	1	no	—
191	C₂²⋊Q₈	—	218	1	no	—
192	C₄×Q₈	—	109	1	no	—
193	C₄×Q₈	—	109	1	no	—
194	C₄×Q₈	—	109	1	no	—
195	C₄×Q₈	—	109	1	no	—
196	C₄×Q₈	—	218	1	no	—
197	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
198	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
199	C₄×D₄	—	109	1	no	—
200	#117	—	218	1	no	—
201	#106	—	109	1	no	—
202	#106	—	109	1	no	—
203	#106	C₂×D₁₀₉	1	1	yes	characteristic
204	C₄⋊C₈	—	109	1	no	—
205	C₄⋊C₈	—	109	1	no	—
206	C₄⋊C₈	—	109	1	no	—
207	#103	—	109	1	no	—
208	#103	—	109	1	no	—
209	#103	—	109	1	no	—
210	#103	—	109	1	no	—
211	#103	—	109	1	no	—
212	#103	C₂×D₁₀₉	1	1	yes	characteristic
213	#103	C₂×D₁₀₉	1	1	yes	characteristic
214	#102	—	109	1	no	—
215	#102	—	109	1	no	—
216	#102	—	109	1	no	—
217	#97	—	109	1	no	—
218	#97	C₂×D₁₀₉	1	1	yes	characteristic
219	C₄×C₈	—	109	1	no	—
220	C₂×Q₈	—	109	1	no	—
221	C₂×Q₈	—	109	1	no	—
222	C₂×Q₈	—	218	1	no	—
223	C₂×Q₈	—	218	1	no	—
224	C₂×Q₈	—	218	1	no	—
225	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₂×D₄	—	109	1	no	—
228	C₂²×C₄	—	218	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	218	2	no	—
231	SD₁₆	—	218	2	no	—
232	C₂×C₈	—	109	1	no	—
233	C₂×C₈	—	109	1	no	—
234	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	109	1	no	—
239	C₄⋊C₄	—	109	1	no	—
240	C₄⋊C₄	—	109	1	no	—
241	C₄⋊C₄	—	109	1	no	—
242	C₄⋊C₄	—	109	1	no	—
243	C₄⋊C₄	—	218	1	no	—
244	C₄⋊C₄	—	218	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	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₂²⋊C₄	—	218	1	no	—
254	C₂²⋊C₄	—	218	1	no	—
255	C₂²⋊C₄	—	218	1	no	—
256	C₂²⋊C₄	—	218	1	no	—
257	C₂²⋊C₄	—	218	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	109	1	no	—
261	C₄²	—	109	1	no	—
262	C₄²	—	218	1	no	—
263	C₄²	—	218	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂³	—	218	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	872	1	no	—
271	Q₈	—	872	1	no	—
272	Q₈	—	872	1	no	—
273	Q₈	—	218	2	no	—
274	Q₈	—	218	1	no	—
275	Q₈	—	218	1	no	—
276	Q₈	#13182029	1	2	yes	—
277	D₄	—	218	1	no	—
278	D₄	—	218	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	109	1	no	—
284	C₂×C₄	—	109	1	no	—
285	C₂×C₄	—	109	1	no	—
286	C₂×C₄	—	109	1	no	—
287	C₂×C₄	—	436	2	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	218	1	no	—
297	C₈	—	218	1	no	—
298	C₂²	—	436	1	no	—
299	C₂²	—	436	1	no	—
300	C₂²	—	436	1	no	—
301	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	109	2	no	—
309	C₄	—	436	1	no	—
310	C₄	—	436	1	no	—
311	C₄	—	436	1	no	—
312	C₄	—	218	1	no	—
313	C₄	—	218	1	no	—
314	C₄	—	218	1	no	—
315	C₄	C₂×#13182029	1	1	yes	characteristic
316	C₄	#16812746	1	1	yes	characteristic
317	C₂	—	872	1	no	—
318	C₂	#20295685	1	1	yes	characteristic
319	C₂	#137×D₁₀₉	1	1	yes	characteristic
320	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	#441×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄₃₆	C₂³
hypercenter	#441	D₁₀₉
hypocenter	C₁₀₉	#3369
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₁₈	C₂	semidirect
5. #20295375	C₂	semidirect
7. #20295311	C₂	semidirect
8. #20295310	C₂	semidirect
11. #20295234	C₂	semidirect
12. #20295154	C₂	semidirect
13. #20295130	C₂	semidirect
15. #441×C₁₀₉	C₂	semidirect
161. C₁₀₉	#3369	semidirect
175. #441	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₁₈	2
#20295748	2
#20295387	2
#20295386	2
#20295375	2
#20295315	2
#20295311	2
#20295310	2
#20295250	2
#20295248	2
#20295234	2
#20295154	2
#20295130	2
#20295111	2
#441×C₁₀₉	2
#3369	109