Subgroups — #28371854

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812728	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182028	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295131	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295126	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295120	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295118	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295117	C₂	1	1	yes	maximal, characteristic, complemented
10	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295033	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295024	C₂	1	1	yes	maximal, characteristic
14	#20294978	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812728	C₄	1	8	yes	complemented
19	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
20	C₂×#13182019	C₂²	1	1	yes	characteristic
21	C₂×#13182017	C₂²	1	1	yes	characteristic
22	C₂×#13182012	C₂²	1	1	yes	characteristic
23	C₂×#13182012	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₁₈	—	2	1	no	—
26	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
28	#16812664	—	2	1	no	—
29	#16812664	C₂²	1	1	yes	characteristic
30	#16812664	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
33	#16812659	—	2	1	no	—
34	#16812657	C₂²	1	1	yes	characteristic
35	#16812655	—	2	1	no	—
36	C₄⋊C₄×D₁₀₉	—	2	1	no	—
37	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₀₉	—	2	1	no	—
39	C₄²×D₁₀₉	—	2	1	no	—
40	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
41	#16812632	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812630	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	#16812622	C₂²	1	1	yes	characteristic
48	#16812622	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812610	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
56	#107×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	#97×C₁₀₉	C₂²	1	1	yes	characteristic
60	#13182028	—	2	2	no	—
61	#13182028	C₂×C₄	1	4	yes	—
62	C₂×Dic₂₁₈	—	2	1	no	—
63	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
65	Q₈×D₁₀₉	—	2	2	no	—
66	Q₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂²×Dic₁₀₉	—	2	1	no	—
69	C₂²×Dic₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
75	Dic₄₃₆	C₂×C₄	1	4	yes	complemented
76	#13182019	C₂×C₄	1	4	yes	complemented
77	#13182017	C₂×C₄	1	4	yes	complemented
78	#13182012	—	2	2	no	—
79	#13182012	C₂×C₄	1	4	yes	complemented
80	C₂×#11724784	C₂³	1	1	yes	characteristic
81	C₂×#11724784	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
89	#13182008	—	2	1	no	—
90	C₂²⋊Dic₁₀₉	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	—	2	1	no	—
97	C₄×Dic₁₀₉	—	2	1	no	—
98	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
99	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
100	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₁₀₉	—	2	1	no	—
104	SD₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
105	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
111	C₄²×C₁₀₉	—	2	1	no	—
112	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
113	C₂³×C₁₀₉	—	2	1	no	—
114	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×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	—	4	2	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	C₂²⋊D₁₀₉	—	2	4	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	4	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	4	2	no	—
133	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₄×D₁₀₉	—	2	4	no	—
137	C₄×D₁₀₉	—	2	2	no	—
138	C₄×D₁₀₉	—	4	1	no	—
139	C₄×D₁₀₉	—	4	2	no	—
140	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
141	C₄×D₁₀₉	#41	1	2	yes	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	—	2	4	no	—
145	Dic₂₁₈	C₂²×C₄	1	2	yes	—
146	Dic₂₁₈	C₂²×C₄	1	2	yes	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
149	#11724784	—	2	1	no	—
150	#11724784	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₀₉	—	2	2	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	2	1	no	—
157	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
158	C₄×C₁₀₉	—	2	2	no	—
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	2	1	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	—	4	1	no	—
163	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
164	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
165	Dic₁₀₉	—	2	2	no	—
166	Dic₁₀₉	—	2	2	no	—
167	Dic₁₀₉	—	2	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	C₄×D₄	1	2	yes	—
171	Dic₁₀₉	#137	1	2	yes	complemented
172	C₂×C₁₀₉	—	4	2	no	—
173	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
175	C₂×C₁₀₉	#575	1	1	yes	characteristic
176	D₁₀₉	—	2	2	no	—
177	C₄×#137	—	109	1	no	maximal, Sylow
178	C₁₀₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
179	C₂×#137	—	109	1	no	—
180	C₄×#41	—	109	1	no	—
181	C₂×C₄×Q₈	—	109	1	no	—
182	#441	—	109	1	no	—
183	#441	—	109	1	no	—
184	#440	—	109	1	no	—
185	#440	D₁₀₉	1	1	yes	characteristic, complemented
186	C₄×Q₁₆	—	109	1	no	—
187	C₄×Q₁₆	—	109	1	no	—
188	C₄×SD₁₆	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	#428	—	109	1	no	—
191	#419	—	109	1	no	—
192	#417	—	109	1	no	—
193	C₄×M₁₆	—	109	1	no	—
194	C₂×#41	—	109	1	no	—
195	C₂²×Q₈	—	109	1	no	—
196	#137	—	109	8	no	—
197	C₂×Q₁₆	—	109	1	no	—
198	C₂×Q₁₆	—	109	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
201	C₂×M₁₆	—	109	1	no	—
202	C₂×M₁₆	—	109	1	no	—
203	C₄×Q₈	—	109	1	no	—
204	C₄×Q₈	—	109	1	no	—
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×Q₈	—	218	1	no	—
209	C₄×Q₈	—	218	1	no	—
210	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
211	C₄×D₄	—	109	1	no	—
212	C₄×D₄	—	218	1	no	—
213	C₄×D₄	—	218	1	no	—
214	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
215	#117	—	109	1	no	—
216	#117	—	218	1	no	—
217	C₂×C₄⋊C₄	—	109	1	no	—
218	C₂×C₄⋊C₄	—	218	1	no	—
219	C₂×C₄²	—	109	1	no	—
220	C₂×C₄²	—	218	1	no	—
221	C₂×C₄²	—	218	1	no	—
222	#107	—	109	1	no	—
223	#107	C₂×D₁₀₉	1	1	yes	characteristic
224	#106	—	109	1	no	—
225	#106	—	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	#102	—	109	1	no	—
233	#102	C₂×D₁₀₉	1	1	yes	characteristic
234	#97	—	109	1	no	—
235	#97	C₂×D₁₀₉	1	1	yes	characteristic
236	C₄×C₈	—	109	1	no	—
237	C₄×C₈	—	109	1	no	—
238	#41	—	109	4	no	—
239	#41	—	218	2	no	—
240	C₂×Q₈	—	109	4	no	—
241	C₂×Q₈	—	109	1	no	—
242	C₂×Q₈	—	109	1	no	—
243	C₂×Q₈	—	218	1	no	—
244	C₂×Q₈	—	218	2	no	—
245	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	109	1	no	—
250	C₂²×C₄	—	109	1	no	—
251	C₂²×C₄	—	109	1	no	—
252	C₂²×C₄	—	218	1	no	—
253	C₂²×C₄	—	218	1	no	—
254	C₂²×C₄	—	218	1	no	—
255	C₂²×C₄	—	218	1	no	—
256	Q₁₆	—	109	4	no	—
257	Q₁₆	—	109	4	no	—
258	SD₁₆	—	109	4	no	—
259	SD₁₆	C₄×D₁₀₉	1	4	yes	complemented
260	M₁₆	—	109	4	no	—
261	M₁₆	—	218	2	no	—
262	C₂×C₈	—	109	1	no	—
263	C₂×C₈	—	109	1	no	—
264	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	109	1	no	—
268	C₄⋊C₄	—	109	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	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	109	1	no	—
282	C₄²	—	109	1	no	—
283	C₄²	—	109	1	no	—
284	C₄²	—	218	1	no	—
285	C₄²	—	218	1	no	—
286	C₄²	—	218	1	no	—
287	C₄²	—	218	1	no	—
288	C₄²	—	218	1	no	—
289	C₄²	—	218	1	no	—
290	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	109	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	109	2	no	—
295	Q₈	—	109	2	no	—
296	Q₈	—	218	4	no	—
297	Q₈	—	218	1	no	—
298	Q₈	—	218	1	no	—
299	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
300	D₄	—	2	1	no	—
301	D₄	—	218	4	no	—
302	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	109	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	—	436	1	no	—
313	C₂×C₄	—	436	2	no	—
314	C₂×C₄	—	436	2	no	—
315	C₂×C₄	—	218	2	no	—
316	C₂×C₄	—	218	4	no	—
317	C₂×C₄	—	218	4	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	1	no	—
322	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	109	2	no	—
327	C₈	—	218	1	no	—
328	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	109	2	no	—
333	C₂²	—	218	1	no	—
334	C₂²	—	218	1	no	—
335	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	109	2	no	—
342	C₄	—	109	2	no	—
343	C₄	—	436	1	no	—
344	C₄	—	436	1	no	—
345	C₄	—	218	2	no	—
346	C₄	—	218	2	no	—
347	C₄	—	218	1	no	—
348	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
349	C₄	#41×D₁₀₉	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	218	2	no	—
352	C₂	C₄×D₄×D₁₀₉	1	1	yes	characteristic
353	C₂	#137×D₁₀₉	1	1	yes	characteristic
354	C₂	#20295822	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#440×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182028	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#440	D₁₀₉
hypocenter	C₁₀₉	C₄×#137
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₁₀₉	C₂	semidirect
4. #20295134	C₂	semidirect
5. #20295131	C₂	semidirect
6. #20295126	C₂	semidirect
7. #20295120	C₂	semidirect
8. #20295118	C₂	semidirect
9. #20295117	C₂	semidirect
10. SD₁₆×Dic₁₀₉	C₂	semidirect
11. #20295067	C₂	semidirect
12. #20295033	C₂	semidirect
14. #20294978	C₂	semidirect
15. #440×C₁₀₉	C₂	semidirect
18. #16812728	C₄	semidirect
66. Q₈×D₁₀₉	C₂×C₄	semidirect
75. Dic₄₃₆	C₂×C₄	semidirect
76. #13182019	C₂×C₄	semidirect
77. #13182017	C₂×C₄	semidirect
79. #13182012	C₂×C₄	semidirect
104. SD₁₆×C₁₀₉	C₂×C₄	semidirect
171. Dic₁₀₉	#137	semidirect
178. C₁₀₉	C₄×#137	semidirect
185. #440	D₁₀₉	semidirect
259. SD₁₆	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×#16812728	2
C₄×#13182028	2
C₄×Q₈×D₁₀₉	2
#20295134	2
#20295131	2
#20295126	2
#20295120	2
#20295118	2
#20295117	2
SD₁₆×Dic₁₀₉	2
#20295067	2
#20295033	2
#20295024	2
#20294978	2
#440×C₁₀₉	2
C₄×#137	109