Subgroups — #28371968

Order 13952.

348 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	#20295580	C₂	1	1	yes	maximal, characteristic
2	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295156	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295155	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295155	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295153	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295051	C₂	1	1	yes	maximal, characteristic
10	#20295051	C₂	1	1	yes	maximal, characteristic
11	#20295048	C₂	1	1	yes	maximal, characteristic
12	#20294983	C₂	1	1	yes	maximal, characteristic
13	#20294980	C₂	1	1	yes	maximal, characteristic
14	#97×D₁₀₉	C₂	1	1	yes	maximal, characteristic
15	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812698	C₄	1	1	yes	characteristic
21	#16812697	C₄	1	1	yes	characteristic
22	#16812689	C₄	1	1	yes	characteristic
23	#16812689	C₄	1	1	yes	characteristic
24	#16812679	C₄	1	1	yes	characteristic
25	#16812677	C₄	1	1	yes	characteristic
26	#16812677	C₄	1	1	yes	characteristic
27	#16812675	C₄	1	1	yes	characteristic
28	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
31	#16812656	C₂²	1	1	yes	characteristic
32	#16812656	C₂²	1	1	yes	characteristic
33	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊#11724784	C₂²	1	1	yes	characteristic
35	C₄⋊#11724784	C₂²	1	1	yes	characteristic
36	C₄⋊#11724784	C₂²	1	1	yes	characteristic
37	#16812628	C₂²	1	1	yes	characteristic
38	#16812628	C₂²	1	1	yes	characteristic
39	#16812628	C₂²	1	1	yes	characteristic
40	#16812628	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812610	C₂²	1	1	yes	characteristic
44	#16812610	C₂²	1	1	yes	characteristic
45	#16812609	C₂²	1	1	yes	characteristic
46	#16812609	C₂²	1	1	yes	characteristic
47	#16812609	C₂²	1	1	yes	characteristic
48	#16812609	C₂²	1	1	yes	characteristic
49	C₄×#11724784	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	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	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
65	#13182012	—	2	2	no	—
66	#13182012	—	2	2	no	—
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₈×D₁₀₉	—	2	2	no	—
72	C₈×D₁₀₉	Q₈	1	4	yes	complemented
73	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
76	#13182008	C₂×C₄	1	1	yes	characteristic
77	#13182008	C₂×C₄	1	1	yes	characteristic
78	#13182008	C₂×C₄	1	1	yes	characteristic
79	#13182008	C₂×C₄	1	1	yes	characteristic
80	#13182008	C₂×C₄	1	1	yes	characteristic
81	#13182008	C₂×C₄	1	1	yes	characteristic
82	#13182008	C₂×C₄	1	1	yes	characteristic
83	#13182008	C₂×C₄	1	1	yes	characteristic
84	#13182008	C₂×C₄	1	1	yes	characteristic
85	#13182008	C₂×C₄	1	1	yes	characteristic
86	#13182006	C₂×C₄	1	1	yes	characteristic
87	#13182006	C₂×C₄	1	1	yes	characteristic
88	#13182006	C₂×C₄	1	1	yes	characteristic
89	#13182006	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	C₂×C₄	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	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
103	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
107	Q₈×C₁₀₉	—	4	1	no	—
108	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
122	C₄×D₁₀₉	—	4	1	no	—
123	C₄×D₁₀₉	—	4	1	no	—
124	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
125	C₄×D₁₀₉	#41	1	2	yes	—
126	Dic₂₁₈	—	4	1	no	—
127	Dic₂₁₈	—	4	1	no	—
128	Dic₂₁₈	—	4	1	no	—
129	C₈×C₁₀₉	—	2	1	no	—
130	C₈×C₁₀₉	—	2	1	no	—
131	C₈×C₁₀₉	—	2	1	no	—
132	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
133	#11724784	—	2	1	no	—
134	#11724784	—	2	1	no	—
135	#11724784	—	2	1	no	—
136	#11724784	C₂×Q₈	1	2	yes	—
137	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₁₀₉	—	2	1	no	—
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	C₄×Q₈	1	2	yes	—
141	C₄×C₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	2	1	no	—
144	C₄×C₁₀₉	—	2	1	no	—
145	C₄×C₁₀₉	—	2	1	no	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
149	Dic₁₀₉	—	2	1	no	—
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	2	1	no	—
153	Dic₁₀₉	—	2	1	no	—
154	Dic₁₀₉	—	2	1	no	—
155	Dic₁₀₉	C₄×Q₈	1	2	yes	—
156	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂×C₁₀₉	#568	1	1	yes	characteristic
158	C₂×C₁₀₉	#568	1	1	yes	characteristic
159	D₁₀₉	—	2	2	no	—
160	#2961	—	109	1	no	maximal, Sylow
161	C₁₀₉	#2961	1	1	yes	characteristic, complemented, Sylow
162	#533	—	109	1	no	—
163	#446	—	109	1	no	—
164	#446	—	109	1	no	—
165	#446	—	109	1	no	—
166	#446	—	109	1	no	—
167	#446	—	109	1	no	—
168	#446	—	109	1	no	—
169	#446	—	109	1	no	—
170	#446	D₁₀₉	1	1	yes	characteristic, complemented
171	#424	—	109	1	no	—
172	#424	—	109	1	no	—
173	#423	—	109	1	no	—
174	#405	—	109	1	no	—
175	#405	—	109	1	no	—
176	C₂×#97	—	109	1	no	—
177	C₂×M₁₆	—	109	1	no	—
178	C₂×M₁₆	—	109	1	no	—
179	C₂²×C₈	—	109	1	no	—
180	C₂²×C₈	—	109	1	no	—
181	C₄⋊Q₈	—	109	1	no	—
182	C₄⋊Q₈	—	109	1	no	—
183	#125	—	109	1	no	—
184	#125	—	109	1	no	—
185	C₂²⋊Q₈	—	109	1	no	—
186	C₂²⋊Q₈	—	109	1	no	—
187	C₂²⋊Q₈	—	109	1	no	—
188	C₂²⋊Q₈	—	109	1	no	—
189	C₄×Q₈	—	109	1	no	—
190	C₄×Q₈	—	109	1	no	—
191	C₄×Q₈	—	109	1	no	—
192	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
193	#117	—	109	1	no	—
194	#117	—	109	1	no	—
195	C₂×C₄²	—	109	1	no	—
196	C₄⋊C₈	—	109	1	no	—
197	C₄⋊C₈	—	109	1	no	—
198	C₄⋊C₈	—	109	1	no	—
199	C₄⋊C₈	—	109	1	no	—
200	C₄⋊C₈	—	109	1	no	—
201	C₄⋊C₈	—	109	1	no	—
202	C₄⋊C₈	—	109	1	no	—
203	C₄⋊C₈	—	109	1	no	—
204	C₄⋊C₈	—	109	1	no	—
205	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
207	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
208	#97	—	109	1	no	—
209	#97	—	109	1	no	—
210	#97	—	109	1	no	—
211	#97	—	109	1	no	—
212	#97	—	109	1	no	—
213	#97	—	109	1	no	—
214	#97	C₂×D₁₀₉	1	1	yes	characteristic
215	#97	C₂×D₁₀₉	1	1	yes	characteristic
216	C₄×C₈	—	109	1	no	—
217	C₄×C₈	—	109	1	no	—
218	C₄×C₈	—	109	1	no	—
219	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
220	C₂×Q₈	—	109	1	no	—
221	C₂×Q₈	—	109	1	no	—
222	C₂×Q₈	—	109	1	no	—
223	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
224	C₂²×C₄	—	109	1	no	—
225	C₂²×C₄	—	109	1	no	—
226	C₂²×C₄	—	109	1	no	—
227	M₁₆	—	218	2	no	—
228	M₁₆	—	218	2	no	—
229	C₂×C₈	—	109	1	no	—
230	C₂×C₈	—	109	4	no	—
231	C₂×C₈	—	109	1	no	—
232	C₂×C₈	—	109	1	no	—
233	C₂×C₈	—	109	1	no	—
234	C₂×C₈	—	218	2	no	—
235	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
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	—	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	—	109	1	no	—
249	C₄⋊C₄	—	109	1	no	—
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
255	C₂²⋊C₄	—	109	1	no	—
256	C₂²⋊C₄	—	109	1	no	—
257	C₂²⋊C₄	—	109	1	no	—
258	C₂²⋊C₄	—	109	1	no	—
259	C₄²	—	109	1	no	—
260	C₄²	—	109	1	no	—
261	C₄²	—	109	1	no	—
262	C₄²	—	109	1	no	—
263	C₄²	—	109	1	no	—
264	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₂³	—	109	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	436	1	no	—
270	Q₈	—	436	1	no	—
271	Q₈	—	436	1	no	—
272	C₂×C₄	—	109	1	no	—
273	C₂×C₄	—	109	2	no	—
274	C₂×C₄	—	109	2	no	—
275	C₂×C₄	—	109	1	no	—
276	C₂×C₄	—	109	1	no	—
277	C₂×C₄	—	109	1	no	—
278	C₂×C₄	—	109	1	no	—
279	C₂×C₄	—	109	1	no	—
280	C₂×C₄	—	109	1	no	—
281	C₂×C₄	—	436	1	no	—
282	C₂×C₄	—	436	1	no	—
283	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	109	2	no	—
294	C₈	—	218	1	no	—
295	C₈	—	218	1	no	—
296	C₈	—	218	1	no	—
297	C₈	Q₈×D₁₀₉	1	2	yes	—
298	C₂²	—	109	2	no	—
299	C₂²	—	218	1	no	—
300	C₂²	—	218	1	no	—
301	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	109	2	no	—
309	C₄	—	218	1	no	—
310	C₄	—	218	1	no	—
311	C₄	—	218	1	no	—
312	C₄	—	218	1	no	—
313	C₄	—	218	1	no	—
314	C₄	—	218	1	no	—
315	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
316	C₄	#41×D₁₀₉	1	1	yes	characteristic
317	C₂	—	218	2	no	—
318	C₂	C₄×Q₈×D₁₀₉	1	1	yes	characteristic
319	C₂	#20295791	1	1	yes	characteristic
320	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#446×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	#2961
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #20295157	C₂	semidirect
3. #20295156	C₂	semidirect
4. #20295155	C₂	semidirect
5. #20295155	C₂	semidirect
6. #20295154	C₂	semidirect
7. #20295154	C₂	semidirect
8. #20295153	C₂	semidirect
15. #446×C₁₀₉	C₂	semidirect
72. C₈×D₁₀₉	Q₈	semidirect
161. C₁₀₉	#2961	semidirect
170. #446	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295580	2
#20295157	2
#20295156	2
#20295155	2
#20295154	2
#20295153	2
#20295051	2
#20295048	2
#20294983	2
#20294980	2
#97×D₁₀₉	2
#446×C₁₀₉	2
#2961	109