Subgroups — #28374906

Order 13952.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295731	C₂	1	1	yes	maximal, characteristic
2	#20295689	C₂	1	1	yes	maximal, characteristic
3	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295363	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295320	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295268	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295265	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295184	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295183	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295120	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295089	C₂	1	1	yes	maximal, characteristic
13	C₂×#16812622	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
20	C₂×#13182019	C₂²	1	1	yes	characteristic
21	C₂²×#11724784	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
23	#16812698	—	2	1	no	—
24	#16812698	C₂²	1	1	yes	characteristic
25	#16812697	C₂²	1	1	yes	characteristic
26	#16812689	—	2	1	no	—
27	#16812679	C₂²	1	1	yes	characteristic
28	#16812679	C₂²	1	1	yes	characteristic
29	#16812678	—	2	1	no	—
30	#16812678	C₂²	1	1	yes	characteristic
31	#16812677	—	2	1	no	—
32	#16812677	—	2	1	no	—
33	#16812676	—	2	1	no	—
34	#16812664	C₂²	1	1	yes	characteristic
35	#16812664	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
37	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812659	—	2	1	no	—
39	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812632	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	#16812622	C₂²	1	1	yes	characteristic
46	#16812622	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	#16812611	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	2	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	—	2	1	no	—
63	C₂×Dic₂₁₈	—	4	1	no	—
64	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
67	Q₈×D₁₀₉	—	2	2	no	—
68	Q₈×D₁₀₉	—	4	1	no	—
69	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
70	C₂²×Dic₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	—	2	1	no	—
73	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
74	Dic₄₃₆	—	2	2	no	—
75	Dic₄₃₆	—	4	1	no	—
76	#13182019	—	4	1	no	—
77	C₂×#11724784	—	4	1	no	—
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₂×#11724784	C₂³	1	1	yes	characteristic
80	C₈×D₁₀₉	—	4	1	no	—
81	C₄⋊Dic₁₀₉	—	2	1	no	—
82	C₄⋊Dic₁₀₉	—	2	1	no	—
83	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	#13182008	—	2	1	no	—
88	#13182008	—	2	1	no	—
89	#13182008	—	2	1	no	—
90	#13182008	—	2	1	no	—
91	#13182008	D₄	1	1	yes	characteristic
92	C₂²⋊Dic₁₀₉	—	2	1	no	—
93	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
94	#13182006	—	2	1	no	—
95	#13182006	—	2	1	no	—
96	#13182006	D₄	1	1	yes	characteristic
97	C₄×Dic₁₀₉	—	2	1	no	—
98	C₄×Dic₁₀₉	—	2	1	no	—
99	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₀₉	—	2	1	no	—
102	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	8	1	no	—
122	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₀₉	Q₁₆	1	4	yes	complemented
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	4	2	no	—
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₄×D₁₀₉	—	2	1	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	4	2	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	2	1	no	—
142	Dic₂₁₈	—	4	2	no	—
143	Dic₂₁₈	—	4	2	no	—
144	Dic₂₁₈	—	8	1	no	—
145	Dic₂₁₈	#41	1	2	yes	—
146	C₈×C₁₀₉	—	2	1	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	#11724784	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	C₂²×C₁₀₉	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	1	no	—
152	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
153	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₀₉	—	2	1	no	—
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	C₂×Q₁₆	1	2	yes	—
157	C₄×C₁₀₉	—	2	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₁₀₉	—	2	2	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	2	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	C₂×Q₁₆	1	2	yes	—
171	C₂×C₁₀₉	—	2	2	no	—
172	C₂×C₁₀₉	#546	1	1	yes	characteristic
173	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂×C₁₀₉	#577	1	1	yes	characteristic
175	D₁₀₉	—	2	2	no	—
176	#3299	—	109	1	no	maximal, Sylow
177	C₁₀₉	#3299	1	1	yes	characteristic, complemented, Sylow
178	#554	—	109	1	no	—
179	D₄×Q₈	—	109	1	no	—
180	#487	—	109	1	no	—
181	#484	—	109	1	no	—
182	#484	D₁₀₉	1	1	yes	characteristic, complemented
183	#477	—	109	1	no	—
184	#467	—	109	1	no	—
185	#467	—	109	1	no	—
186	#462	—	109	1	no	—
187	#451	—	109	1	no	—
188	#451	—	109	1	no	—
189	C₄×Q₁₆	—	109	1	no	—
190	C₈×D₄	—	109	1	no	—
191	C₂×#103	—	109	1	no	—
192	C₂×#103	—	109	1	no	—
193	C₂²×Q₈	—	109	1	no	—
194	C₂²×Q₈	—	109	1	no	—
195	C₂×Q₁₆	—	109	1	no	—
196	C₂×Q₁₆	—	109	1	no	—
197	C₂×Q₁₆	—	109	1	no	—
198	C₂²×C₈	—	109	1	no	—
199	C₂²×C₈	—	109	1	no	—
200	C₄⋊Q₈	—	109	1	no	—
201	C₄⋊Q₈	—	109	1	no	—
202	C₄⋊Q₈	—	218	1	no	—
203	#125	—	218	1	no	—
204	C₂²⋊Q₈	—	109	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	C₂²⋊Q₈	—	218	1	no	—
210	C₂²⋊Q₈	—	218	1	no	—
211	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
212	C₄×Q₈	—	109	1	no	—
213	C₄×Q₈	—	109	1	no	—
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	—	218	1	no	—
216	C₄×D₄	—	218	1	no	—
217	C₂×C₄⋊C₄	—	109	1	no	—
218	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
219	#107	—	109	1	no	—
220	#107	C₂×D₁₀₉	1	1	yes	characteristic
221	#107	C₂×D₁₀₉	1	1	yes	characteristic
222	C₄⋊C₈	—	109	1	no	—
223	#103	—	109	1	no	—
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	—	109	1	no	—
228	#103	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	—	109	1	no	—
231	#103	C₂×D₁₀₉	1	1	yes	characteristic
232	#103	C₂×D₁₀₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	109	1	no	—
234	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
235	C₄×C₈	—	109	1	no	—
236	C₂×Q₈	—	109	1	no	—
237	C₂×Q₈	—	109	1	no	—
238	C₂×Q₈	—	109	1	no	—
239	C₂×Q₈	—	436	1	no	—
240	C₂×Q₈	—	436	1	no	—
241	C₂×Q₈	—	218	1	no	—
242	C₂×Q₈	—	218	1	no	—
243	C₂×Q₈	—	218	2	no	—
244	C₂×Q₈	—	218	2	no	—
245	C₂×Q₈	—	218	1	no	—
246	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₂×D₄	—	109	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	109	1	no	—
250	C₂²×C₄	—	218	1	no	—
251	C₂²×C₄	—	218	1	no	—
252	C₂²×C₄	—	218	1	no	—
253	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	Q₁₆	—	436	1	no	—
255	Q₁₆	—	436	1	no	—
256	Q₁₆	—	218	2	no	—
257	C₂×C₈	—	109	1	no	—
258	C₂×C₈	—	109	1	no	—
259	C₂×C₈	—	436	1	no	—
260	C₂×C₈	—	436	1	no	—
261	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	109	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₄⋊C₄	—	218	1	no	—
273	C₄⋊C₄	—	218	1	no	—
274	C₄⋊C₄	—	218	1	no	—
275	C₄⋊C₄	—	218	1	no	—
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	109	1	no	—
281	C₂²⋊C₄	—	109	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₄²	—	109	1	no	—
286	C₄²	—	218	1	no	—
287	C₄²	—	218	1	no	—
288	C₂³	—	109	1	no	—
289	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	872	1	no	—
293	Q₈	—	109	2	no	—
294	Q₈	—	436	2	no	—
295	Q₈	—	436	2	no	—
296	Q₈	—	218	1	no	—
297	Q₈	—	218	1	no	—
298	Q₈	—	218	1	no	—
299	Q₈	—	218	1	no	—
300	Q₈	—	218	1	no	—
301	D₄	—	109	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	872	1	no	—
310	C₂×C₄	—	109	1	no	—
311	C₂×C₄	—	109	1	no	—
312	C₂×C₄	—	109	1	no	—
313	C₂×C₄	—	436	2	no	—
314	C₂×C₄	—	436	2	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	218	1	no	—
326	C₈	—	218	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	109	2	no	—
330	C₂²	—	218	1	no	—
331	C₂²	—	218	1	no	—
332	C₂²	Q₁₆×D₁₀₉	1	2	yes	—
333	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	109	2	no	—
340	C₄	—	436	1	no	—
341	C₄	—	436	1	no	—
342	C₄	—	436	1	no	—
343	C₄	—	218	1	no	—
344	C₄	—	218	1	no	—
345	C₄	—	218	2	no	—
346	C₄	#41×D₁₀₉	1	1	yes	characteristic
347	C₄	#16812745	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	218	2	no	—
350	C₂	#20295668	1	1	yes	characteristic
351	C₂	C₂×Q₁₆×D₁₀₉	1	1	yes	characteristic
352	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	#484×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₀₉	D₄
hypercenter	#484	D₁₀₉
hypocenter	C₁₀₉	#3299
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #20295375	C₂	semidirect
4. #20295363	C₂	semidirect
5. #20295320	C₂	semidirect
6. #20295268	C₂	semidirect
7. #20295265	C₂	semidirect
8. #20295235	C₂	semidirect
9. #20295184	C₂	semidirect
10. #20295183	C₂	semidirect
11. #20295120	C₂	semidirect
13. C₂×#16812622	C₂	semidirect
14. #103×D₁₀₉	C₂	semidirect
15. #484×C₁₀₉	C₂	semidirect
124. C₂²⋊D₁₀₉	Q₁₆	semidirect
177. C₁₀₉	#3299	semidirect
182. #484	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295731	2
#20295689	2
#20295375	2
#20295363	2
#20295320	2
#20295268	2
#20295265	2
#20295235	2
#20295184	2
#20295183	2
#20295120	2
#20295089	2
C₂×#16812622	2
#103×D₁₀₉	2
#484×C₁₀₉	2
#3299	109