Subgroups — #28371874

Order 13952.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812731	C₂	1	1	yes	maximal, characteristic
2	#20295508	C₂	1	1	yes	maximal, characteristic
3	#20295502	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295130	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295129	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295128	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295113	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295021	C₂	1	1	yes	maximal, characteristic
13	#103×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#20294983	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812731	C₄	1	8	yes	complemented
19	C₂×#13182019	C₂²	1	1	yes	characteristic
20	C₂×#13182018	C₂²	1	1	yes	characteristic
21	C₂×#13182016	C₂²	1	1	yes	characteristic
22	C₂×#13182012	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁₈	—	2	1	no	—
27	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
28	#16812664	—	2	1	no	—
29	#16812664	C₂²	1	1	yes	characteristic
30	C₄×D₄₃₆	—	2	1	no	—
31	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
32	#16812661	—	2	1	no	—
33	#16812661	C₂²	1	1	yes	characteristic
34	#16812657	—	2	1	no	—
35	#16812657	C₂²	1	1	yes	characteristic
36	#16812656	—	2	1	no	—
37	#16812656	—	2	1	no	—
38	#16812656	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₁₀₉	—	2	1	no	—
40	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
41	#16812633	C₂²	1	1	yes	characteristic
42	#16812632	C₂²	1	1	yes	characteristic
43	#16812630	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812622	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	C₄×#11724784	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#106×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#103×C₁₀₉	C₂²	1	1	yes	characteristic
59	#97×C₁₀₉	C₂²	1	1	yes	characteristic
60	#13182030	—	2	2	no	—
61	#13182030	C₂×C₄	1	4	yes	—
62	C₂×Dic₂₁₈	—	2	1	no	—
63	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
64	Q₈×D₁₀₉	—	2	2	no	—
65	Q₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
66	C₂×D₄₃₆	—	2	1	no	—
67	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	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	#13182019	C₂×C₄	1	4	yes	complemented
75	#13182018	C₂×C₄	1	4	yes	complemented
76	#13182016	C₂×C₄	1	4	yes	complemented
77	#13182012	C₂×C₄	1	4	yes	complemented
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₂×#11724784	C₂³	1	1	yes	characteristic
80	C₈×D₁₀₉	—	2	2	no	—
81	C₄⋊Dic₁₀₉	—	2	1	no	—
82	C₄⋊Dic₁₀₉	—	2	1	no	—
83	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	#13182008	—	2	1	no	—
88	#13182008	D₄	1	1	yes	characteristic
89	#13182008	D₄	1	1	yes	characteristic
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	D₄	1	1	yes	characteristic
94	#13182006	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	—	2	1	no	—
97	C₄×Dic₁₀₉	—	2	1	no	—
98	C₄×Dic₁₀₉	—	2	1	no	—
99	C₄×Dic₁₀₉	—	2	1	no	—
100	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	Q₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
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	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₄²×C₁₀₉	—	2	1	no	—
111	C₄²×C₁₀₉	—	2	1	no	—
112	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
113	C₂²×D₁₀₉	—	2	1	no	—
114	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
118	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
119	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	D₄₃₆	—	2	4	no	—
133	D₄₃₆	—	2	1	no	—
134	D₄₃₆	C₂²×C₄	1	2	yes	—
135	C₄×D₁₀₉	—	2	4	no	—
136	C₄×D₁₀₉	—	2	4	no	—
137	C₄×D₁₀₉	—	8	1	no	—
138	C₄×D₁₀₉	—	8	1	no	—
139	C₄×D₁₀₉	—	8	1	no	—
140	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
141	C₄×D₁₀₉	#41	1	2	yes	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	—	2	4	no	—
144	Dic₂₁₈	C₂²×C₄	1	2	yes	—
145	C₈×C₁₀₉	—	2	1	no	—
146	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
147	#11724784	—	2	1	no	—
148	#11724784	C₂²×C₄	1	2	yes	—
149	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₀₉	—	2	1	no	—
151	C₂×D₁₀₉	—	2	1	no	—
152	C₂×D₁₀₉	—	2	2	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
156	C₄×C₁₀₉	—	2	2	no	—
157	C₄×C₁₀₉	—	2	2	no	—
158	C₄×C₁₀₉	—	2	1	no	—
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
163	C₄×C₁₀₉	C₂×#41	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	C₄×D₄	1	2	yes	—
170	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×C₁₀₉	#577	1	1	yes	characteristic
172	C₂×C₁₀₉	#578	1	1	yes	characteristic
173	D₁₀₉	—	2	2	no	—
174	D₁₀₉	—	4	2	no	—
175	#2947	—	109	1	no	maximal, Sylow
176	C₁₀₉	#2947	1	1	yes	characteristic, complemented, Sylow
177	C₂×#137	—	109	1	no	—
178	#520	—	109	1	no	—
179	#519	—	109	1	no	—
180	#441	—	109	1	no	—
181	#441	D₁₀₉	1	1	yes	characteristic, complemented
182	#440	—	109	1	no	—
183	#440	—	109	1	no	—
184	C₄×Q₁₆	—	109	1	no	—
185	C₄×Q₁₆	—	109	1	no	—
186	C₄×SD₁₆	—	109	1	no	—
187	C₄×SD₁₆	—	109	1	no	—
188	#428	—	109	1	no	—
189	#416	—	109	1	no	—
190	C₂×#103	—	109	1	no	—
191	#405	—	109	1	no	—
192	C₂×#41	—	109	1	no	—
193	C₂²×Q₈	—	109	1	no	—
194	#137	—	109	8	no	—
195	C₂×Q₁₆	—	109	1	no	—
196	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
197	C₂×SD₁₆	—	109	1	no	—
198	C₂×SD₁₆	—	109	1	no	—
199	C₂×M₁₆	—	109	1	no	—
200	C₂²×C₈	—	109	1	no	—
201	C₄×Q₈	—	109	1	no	—
202	C₄×Q₈	—	109	1	no	—
203	C₄×Q₈	—	109	1	no	—
204	C₄×Q₈	—	109	1	no	—
205	C₄×Q₈	—	218	1	no	—
206	C₄×Q₈	—	218	1	no	—
207	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
208	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
209	C₄×D₄	—	109	1	no	—
210	C₄×D₄	—	109	1	no	—
211	C₄×D₄	—	218	1	no	—
212	C₄×D₄	—	218	1	no	—
213	#117	—	109	1	no	—
214	#117	—	109	1	no	—
215	#117	—	218	1	no	—
216	#117	—	218	1	no	—
217	#117	—	218	1	no	—
218	C₂×C₄⋊C₄	—	109	1	no	—
219	C₂×C₄⋊C₄	—	218	1	no	—
220	#107	—	109	1	no	—
221	#107	—	109	1	no	—
222	#106	—	109	1	no	—
223	#106	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	#103	C₂×D₁₀₉	1	1	yes	characteristic
229	#103	C₂×D₁₀₉	1	1	yes	characteristic
230	#102	—	109	1	no	—
231	#102	—	109	1	no	—
232	#97	—	109	1	no	—
233	#97	C₂×D₁₀₉	1	1	yes	characteristic
234	C₄×C₈	—	109	1	no	—
235	C₄×C₈	—	109	1	no	—
236	#41	—	109	4	no	—
237	#41	—	218	2	no	—
238	C₂×Q₈	—	109	4	no	—
239	C₂×Q₈	—	109	1	no	—
240	C₂×Q₈	—	218	1	no	—
241	C₂×Q₈	—	218	2	no	—
242	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₂×D₄	—	109	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂²×C₄	—	109	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	C₂²×C₄	—	218	1	no	—
249	C₂²×C₄	—	218	1	no	—
250	C₂²×C₄	—	218	1	no	—
251	C₂²×C₄	—	218	1	no	—
252	Q₁₆	—	109	4	no	—
253	Q₁₆	C₄×D₁₀₉	1	4	yes	complemented
254	SD₁₆	—	109	4	no	—
255	SD₁₆	—	109	4	no	—
256	M₁₆	—	109	4	no	—
257	C₂×C₈	—	109	1	no	—
258	C₂×C₈	—	109	1	no	—
259	C₂×C₈	—	218	2	no	—
260	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	109	1	no	—
265	C₄⋊C₄	—	109	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	—	109	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	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	109	1	no	—
276	C₂²⋊C₄	—	109	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	109	1	no	—
283	C₄²	—	218	1	no	—
284	C₄²	—	218	1	no	—
285	C₄²	—	218	1	no	—
286	C₄²	—	218	1	no	—
287	C₄²	—	218	1	no	—
288	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
289	C₂³	—	109	1	no	—
290	C₂³	—	218	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	109	2	no	—
294	Q₈	—	218	4	no	—
295	Q₈	—	218	1	no	—
296	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
297	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
298	D₄	—	109	2	no	—
299	D₄	—	218	4	no	—
300	D₄	—	218	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	872	1	no	—
306	C₂×C₄	—	872	1	no	—
307	C₂×C₄	—	872	1	no	—
308	C₂×C₄	—	109	1	no	—
309	C₂×C₄	—	109	2	no	—
310	C₂×C₄	—	109	2	no	—
311	C₂×C₄	—	109	1	no	—
312	C₂×C₄	—	109	1	no	—
313	C₂×C₄	—	218	4	no	—
314	C₂×C₄	—	218	4	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	109	2	no	—
323	C₈	—	218	1	no	—
324	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
325	C₂²	—	109	2	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	436	1	no	—
328	C₂²	—	218	2	no	—
329	C₂²	—	218	1	no	—
330	C₂²	—	218	1	no	—
331	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	109	2	no	—
339	C₄	—	436	1	no	—
340	C₄	—	436	1	no	—
341	C₄	—	218	2	no	—
342	C₄	—	218	1	no	—
343	C₄	—	218	1	no	—
344	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
345	C₄	#41×D₁₀₉	1	1	yes	characteristic
346	C₂	—	436	2	no	—
347	C₂	—	218	2	no	—
348	C₂	C₄×D₄×D₁₀₉	1	1	yes	characteristic
349	C₂	#20295835	1	1	yes	characteristic
350	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#441×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182030	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#441	D₁₀₉
hypocenter	C₁₀₉	#2947
LCS	G ▷ 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
3. #20295502	C₂	semidirect
4. #20295130	C₂	semidirect
5. #20295129	C₂	semidirect
6. #20295128	C₂	semidirect
7. Q₁₆×Dic₁₀₉	C₂	semidirect
8. C₄×#13182019	C₂	semidirect
9. C₄×#13182016	C₂	semidirect
10. #20295113	C₂	semidirect
11. #20295065	C₂	semidirect
13. #103×D₁₀₉	C₂	semidirect
14. #20294983	C₂	semidirect
15. #441×C₁₀₉	C₂	semidirect
18. #16812731	C₄	semidirect
65. Q₈×D₁₀₉	C₂×C₄	semidirect
74. #13182019	C₂×C₄	semidirect
75. #13182018	C₂×C₄	semidirect
76. #13182016	C₂×C₄	semidirect
77. #13182012	C₂×C₄	semidirect
103. Q₁₆×C₁₀₉	C₂×C₄	semidirect
176. C₁₀₉	#2947	semidirect
181. #441	D₁₀₉	semidirect
253. Q₁₆	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812731	2
#20295508	2
#20295502	2
#20295130	2
#20295129	2
#20295128	2
Q₁₆×Dic₁₀₉	2
C₄×#13182019	2
C₄×#13182016	2
#20295113	2
#20295065	2
#20295021	2
#103×D₁₀₉	2
#20294983	2
#441×C₁₀₉	2
#2947	109