Subgroups — #28373176

Order 13952.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295571	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295351	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295347	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295309	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295304	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295224	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295219	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295171	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295138	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295135	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295129	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295080	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	—	2	1	no	—
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	C₂×#13182014	C₂²	1	1	yes	characteristic
19	#16812698	C₂²	1	1	yes	characteristic
20	#16812695	C₂²	1	1	yes	characteristic
21	#16812685	—	2	1	no	—
22	#16812673	C₂²	1	1	yes	characteristic
23	#16812671	—	2	1	no	—
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₀₉	—	2	1	no	—
27	D₄×Dic₁₀₉	—	2	1	no	—
28	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	#16812661	C₂²	1	1	yes	characteristic
32	#16812659	—	2	1	no	—
33	#16812658	—	2	1	no	—
34	#16812658	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₀₉	—	2	1	no	—
36	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	C₂²	1	1	yes	characteristic
39	#16812631	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812623	C₂²	1	1	yes	characteristic
44	#16812621	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	#16812609	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#121×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182028	—	4	4	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂²×Dic₁₀₉	—	2	1	no	—
67	C₂²×Dic₁₀₉	—	2	1	no	—
68	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
69	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	#13182016	—	2	2	no	—
72	#13182014	—	2	2	no	—
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	#13182008	—	2	1	no	—
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
94	#41×C₁₀₉	—	2	2	no	—
95	#41×C₁₀₉	—	2	2	no	—
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	—	2	1	no	—
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₀₉	—	2	1	no	—
102	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	SD₁₆×C₁₀₉	—	2	2	no	—
104	D₈×C₁₀₉	—	2	2	no	—
105	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂³×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	#41	1	2	yes	complemented
115	D₄×C₁₀₉	—	2	4	no	—
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	4	2	no	—
120	D₄×C₁₀₉	#41	1	2	yes	complemented
121	C₂×C₄×C₁₀₉	—	2	4	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₀₉	—	4	2	no	—
129	C₂²⋊D₁₀₉	—	4	2	no	—
130	C₂²⋊D₁₀₉	—	4	2	no	—
131	C₂×Dic₁₀₉	—	2	2	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	—	4	1	no	—
136	C₂×Dic₁₀₉	—	4	2	no	—
137	C₂×Dic₁₀₉	—	4	2	no	—
138	C₂×Dic₁₀₉	—	4	2	no	—
139	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
143	D₄₃₆	—	2	1	no	—
144	D₄₃₆	—	2	1	no	—
145	C₄×D₁₀₉	—	4	2	no	—
146	Dic₂₁₈	—	4	2	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	1	no	—
152	C₂²×C₁₀₉	—	2	2	no	—
153	C₂²×C₁₀₉	—	2	1	no	—
154	C₂²×C₁₀₉	—	2	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	—	4	1	no	—
158	C₂²×C₁₀₉	—	4	1	no	—
159	C₂²×C₁₀₉	—	4	1	no	—
160	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	2	1	no	—
165	C₄×C₁₀₉	—	2	2	no	—
166	C₄×C₁₀₉	—	4	1	no	—
167	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	2	1	no	—
171	Dic₁₀₉	—	2	1	no	—
172	Dic₁₀₉	—	4	1	no	—
173	Dic₁₀₉	—	4	1	no	—
174	Dic₁₀₉	—	4	1	no	—
175	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₁₀₉	—	4	1	no	—
177	C₂×C₁₀₉	—	4	2	no	—
178	C₂×C₁₀₉	—	8	1	no	—
179	C₂×C₁₀₉	#525	1	1	yes	characteristic
180	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₁₀₉	#575	1	1	yes	characteristic
182	D₁₀₉	—	8	1	no	—
183	#3115	—	109	1	no	maximal, Sylow
184	C₁₀₉	#3115	1	1	yes	characteristic, complemented, Sylow
185	#532	—	109	1	no	—
186	C₄×#41	—	109	1	no	—
187	#482	—	109	1	no	—
188	#482	—	109	1	no	—
189	#475	—	109	1	no	—
190	#474	—	109	1	no	—
191	#460	—	109	1	no	—
192	#459	—	109	1	no	—
193	#449	—	109	1	no	—
194	#449	D₁₀₉	1	1	yes	characteristic, complemented
195	#442	—	109	1	no	—
196	#442	—	109	1	no	—
197	#440	—	109	1	no	—
198	#440	—	109	1	no	—
199	#432	—	109	1	no	—
200	C₂×#41	—	218	1	no	—
201	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
202	C₂×SD₁₆	—	109	1	no	—
203	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
204	C₂×D₈	—	109	1	no	—
205	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
206	C₄⋊Q₈	—	109	1	no	—
207	#127	—	109	1	no	—
208	#124	—	218	1	no	—
209	#121	—	109	1	no	—
210	#121	—	218	1	no	—
211	#121	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	—	109	1	no	—
216	C₄×D₄	—	109	1	no	—
217	C₄×D₄	—	109	1	no	—
218	C₄×D₄	—	218	1	no	—
219	C₄×D₄	—	218	1	no	—
220	C₄×D₄	—	218	1	no	—
221	#117	—	109	1	no	—
222	#117	—	218	1	no	—
223	C₂×C₄²	—	109	1	no	—
224	C₂×C₄²	—	218	1	no	—
225	#107	—	109	1	no	—
226	#107	—	109	1	no	—
227	#106	—	109	1	no	—
228	#106	—	109	1	no	—
229	C₄⋊C₈	—	109	1	no	—
230	C₄⋊C₈	—	109	1	no	—
231	#103	—	109	1	no	—
232	#103	C₂×D₁₀₉	1	1	yes	characteristic
233	#102	—	109	1	no	—
234	#102	—	109	1	no	—
235	#102	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	C₂×D₁₀₉	1	1	yes	characteristic
239	C₂²⋊C₈	—	109	1	no	—
240	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
241	#97	—	109	1	no	—
242	#97	—	109	1	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	436	4	no	—
246	C₂×Q₈	—	218	1	no	—
247	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	109	1	no	—
251	C₂×D₄	—	218	1	no	—
252	C₂×D₄	—	218	1	no	—
253	C₂×D₄	—	218	1	no	—
254	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	109	1	no	—
258	C₂²×C₄	—	109	1	no	—
259	C₂²×C₄	—	218	1	no	—
260	C₂²×C₄	—	218	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	218	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	218	2	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	—	109	1	no	—
270	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₄⋊C₄	—	109	1	no	—
273	C₄⋊C₄	—	109	1	no	—
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	218	1	no	—
276	C₄⋊C₄	—	218	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₂²⋊C₄	—	218	1	no	—
286	C₄²	—	109	1	no	—
287	C₄²	—	109	1	no	—
288	C₄²	—	109	1	no	—
289	C₄²	—	109	1	no	—
290	C₄²	—	218	1	no	—
291	C₄²	—	218	1	no	—
292	C₄²	—	218	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	218	1	no	—
296	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	436	2	no	—
299	Q₈	#13182028	1	2	yes	complemented
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	436	2	no	—
306	D₄	—	436	2	no	—
307	D₄	—	436	2	no	—
308	D₄	—	218	1	no	—
309	D₄	—	218	1	no	—
310	D₄	#13182028	1	2	yes	complemented
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	109	1	no	—
317	C₂×C₄	—	109	1	no	—
318	C₂×C₄	—	109	1	no	—
319	C₂×C₄	—	109	1	no	—
320	C₂×C₄	—	436	2	no	—
321	C₂×C₄	—	436	2	no	—
322	C₂×C₄	—	436	2	no	—
323	C₂×C₄	—	436	2	no	—
324	C₂×C₄	—	436	1	no	—
325	C₂×C₄	—	218	2	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	1	no	—
329	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	218	1	no	—
334	C₈	—	218	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	436	1	no	—
345	C₂²	—	436	1	no	—
346	C₂²	—	436	1	no	—
347	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	109	2	no	—
352	C₄	—	436	1	no	—
353	C₄	—	436	1	no	—
354	C₄	—	436	1	no	—
355	C₄	—	218	1	no	—
356	C₄	—	218	1	no	—
357	C₄	—	218	1	no	—
358	C₄	C₂×#13182028	1	1	yes	characteristic
359	C₄	C₂×#13182028	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	872	1	no	—
364	C₂	#20295532	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
366	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#449×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#449	D₁₀₉
hypocenter	C₁₀₉	#3115
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #20295571	C₂	semidirect
2. #41×Dic₁₀₉	C₂	semidirect
3. #20295351	C₂	semidirect
4. #20295347	C₂	semidirect
5. #20295309	C₂	semidirect
6. #20295304	C₂	semidirect
7. #20295224	C₂	semidirect
8. #20295219	C₂	semidirect
9. #20295171	C₂	semidirect
10. #20295138	C₂	semidirect
11. #20295135	C₂	semidirect
12. #20295129	C₂	semidirect
13. #20295127	C₂	semidirect
14. #20295080	C₂	semidirect
15. #449×C₁₀₉	C₂	semidirect
114. Q₈×C₁₀₉	#41	semidirect
120. D₄×C₁₀₉	#41	semidirect
184. C₁₀₉	#3115	semidirect
194. #449	D₁₀₉	semidirect
299. Q₈	#13182028	semidirect
310. D₄	#13182028	semidirect

Maximal subgroups

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

subgroup	index
#20295571	2
#41×Dic₁₀₉	2
#20295351	2
#20295347	2
#20295309	2
#20295304	2
#20295224	2
#20295219	2
#20295171	2
#20295138	2
#20295135	2
#20295129	2
#20295127	2
#20295080	2
#449×C₁₀₉	2
#3115	109