Subgroups — #28373623

Order 13952.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295573	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295563	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295433	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295412	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295389	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295387	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295280	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295278	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295275	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295269	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×D₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13182014	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	—	2	1	no	—
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	C₂×#13182012	C₂²	1	1	yes	characteristic
21	C₂×#13182012	C₂²	1	1	yes	characteristic
22	#16812698	C₂²	1	1	yes	characteristic
23	#16812695	C₂²	1	1	yes	characteristic
24	#16812694	—	2	1	no	—
25	#16812690	C₂²	1	1	yes	characteristic
26	#16812688	C₂²	1	1	yes	characteristic
27	#16812685	—	2	1	no	—
28	#16812684	—	2	1	no	—
29	#16812675	C₂²	1	1	yes	characteristic
30	#16812674	C₂²	1	1	yes	characteristic
31	#16812674	C₂²	1	1	yes	characteristic
32	#16812672	—	2	1	no	—
33	#16812670	C₂²	1	1	yes	characteristic
34	C₄×D₄₃₆	—	2	1	no	—
35	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
36	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
38	#16812656	—	2	1	no	—
39	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812633	C₂²	1	1	yes	characteristic
41	#16812632	C₂²	1	1	yes	characteristic
42	#16812630	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	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812609	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₄×#11724784	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	4	4	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
73	#13182016	—	2	2	no	—
74	D₈₇₂	—	2	2	no	—
75	#13182014	—	2	2	no	—
76	#13182012	—	2	2	no	—
77	#13182012	—	2	2	no	—
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₂×#11724784	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	—	2	1	no	—
81	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	SD₁₆×C₁₀₉	—	2	2	no	—
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	—	2	1	no	—
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	—	2	1	no	—
108	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	1	no	—
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	—	2	1	no	—
116	D₄×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	—	4	2	no	—
121	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₀₉	—	4	2	no	—
125	C₂²⋊D₁₀₉	—	4	2	no	—
126	C₂²⋊D₁₀₉	—	4	2	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	1	no	—
133	D₄₃₆	—	2	1	no	—
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	—	4	2	no	—
137	D₄₃₆	—	4	2	no	—
138	C₄×D₁₀₉	—	2	2	no	—
139	C₄×D₁₀₉	—	2	1	no	—
140	C₄×D₁₀₉	—	2	1	no	—
141	C₄×D₁₀₉	—	4	1	no	—
142	C₄×D₁₀₉	—	4	2	no	—
143	C₄×D₁₀₉	—	4	2	no	—
144	Dic₂₁₈	—	4	2	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	C₈×C₁₀₉	#41	1	2	yes	complemented
147	#11724784	—	2	1	no	—
148	#11724784	#41	1	2	yes	complemented
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	—	4	1	no	—
151	C₂²×C₁₀₉	—	4	1	no	—
152	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₀₉	—	2	1	no	—
154	C₂×D₁₀₉	—	2	1	no	—
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	4	1	no	—
157	C₂×D₁₀₉	—	4	1	no	—
158	C₂×D₁₀₉	—	4	1	no	—
159	C₂×D₁₀₉	—	4	1	no	—
160	C₂×D₁₀₉	—	4	1	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	C₄×C₁₀₉	—	4	1	no	—
166	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
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	C₂×C₁₀₉	—	8	1	no	—
175	C₂×C₁₀₉	#532	1	1	yes	characteristic
176	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₁₀₉	#575	1	1	yes	characteristic
178	D₁₀₉	—	4	1	no	—
179	D₁₀₉	—	8	1	no	—
180	D₁₀₉	—	8	1	no	—
181	#3171	—	109	1	no	maximal, Sylow
182	C₁₀₉	#3171	1	1	yes	characteristic, complemented, Sylow
183	#532	—	109	1	no	—
184	#529	—	109	1	no	—
185	#501	—	109	1	no	—
186	#496	—	109	1	no	—
187	#490	—	109	1	no	—
188	#489	—	109	1	no	—
189	#469	—	109	1	no	—
190	#469	—	109	1	no	—
191	#468	—	109	1	no	—
192	#468	—	109	1	no	—
193	C₄×SD₁₆	—	109	1	no	—
194	C₄×SD₁₆	D₁₀₉	1	1	yes	characteristic, complemented
195	C₄×D₈	—	109	1	no	—
196	C₄×D₈	—	109	1	no	—
197	C₄×M₁₆	—	109	1	no	—
198	C₂×#41	—	218	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
201	C₂×D₈	—	109	1	no	—
202	C₂×D₈	—	109	1	no	—
203	C₂×M₁₆	—	109	1	no	—
204	C₂×M₁₆	—	109	1	no	—
205	C₄⋊Q₈	—	109	1	no	—
206	#127	—	109	1	no	—
207	#126	—	218	1	no	—
208	#125	—	109	1	no	—
209	#124	—	109	1	no	—
210	#124	—	218	1	no	—
211	#123	—	218	1	no	—
212	C₂²⋊Q₈	—	109	1	no	—
213	#121	—	109	1	no	—
214	#121	—	109	1	no	—
215	#121	—	109	1	no	—
216	#121	—	218	1	no	—
217	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
218	C₄×D₄	—	109	1	no	—
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	218	1	no	—
221	C₄×D₄	—	218	1	no	—
222	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
223	#117	—	218	1	no	—
224	C₂×C₄²	—	109	1	no	—
225	#107	—	109	1	no	—
226	#107	—	109	1	no	—
227	#106	—	109	1	no	—
228	#106	C₂×D₁₀₉	1	1	yes	characteristic
229	#103	—	109	1	no	—
230	#103	C₂×D₁₀₉	1	1	yes	characteristic
231	#102	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#102	—	109	1	no	—
235	#102	—	109	1	no	—
236	#102	C₂×D₁₀₉	1	1	yes	characteristic
237	#97	—	109	1	no	—
238	#97	—	109	1	no	—
239	C₄×C₈	—	109	1	no	—
240	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
241	#41	—	436	4	no	—
242	C₂×Q₈	—	218	1	no	—
243	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₂×D₄	—	109	1	no	—
245	C₂×D₄	—	109	1	no	—
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	—	218	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	—	218	1	no	—
250	C₂×D₄	—	218	1	no	—
251	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	109	1	no	—
254	C₂²×C₄	—	109	1	no	—
255	C₂²×C₄	—	109	1	no	—
256	C₂²×C₄	—	218	1	no	—
257	C₂²×C₄	—	218	1	no	—
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	218	2	no	—
260	D₈	—	218	2	no	—
261	D₈	—	218	2	no	—
262	M₁₆	—	218	2	no	—
263	M₁₆	—	218	2	no	—
264	C₂×C₈	—	109	1	no	—
265	C₂×C₈	—	109	1	no	—
266	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	109	1	no	—
270	C₄⋊C₄	—	109	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	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	—	2	1	no	—
287	C₄²	—	109	1	no	—
288	C₄²	—	109	1	no	—
289	C₄²	—	109	1	no	—
290	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	109	1	no	—
293	C₂³	—	218	1	no	—
294	C₂³	—	218	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	436	2	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	436	2	no	—
301	D₄	—	436	2	no	—
302	D₄	—	436	2	no	—
303	D₄	—	436	2	no	—
304	D₄	—	436	2	no	—
305	D₄	—	218	1	no	—
306	D₄	—	218	1	no	—
307	D₄	—	218	1	no	—
308	D₄	—	218	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	109	1	no	—
314	C₂×C₄	—	109	1	no	—
315	C₂×C₄	—	109	1	no	—
316	C₂×C₄	—	436	1	no	—
317	C₂×C₄	—	436	2	no	—
318	C₂×C₄	—	436	2	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	2	no	—
322	C₂×C₄	—	218	1	no	—
323	C₂×C₄	—	218	1	no	—
324	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
325	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	109	2	no	—
329	C₈	—	218	1	no	—
330	C₈	#13182029	1	2	yes	complemented
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	436	1	no	—
335	C₂²	—	436	1	no	—
336	C₂²	—	436	1	no	—
337	C₂²	—	436	1	no	—
338	C₂²	—	436	1	no	—
339	C₂²	—	436	1	no	—
340	C₂²	—	218	1	no	—
341	C₂²	—	218	1	no	—
342	C₂²	—	218	1	no	—
343	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	436	1	no	—
349	C₄	—	436	1	no	—
350	C₄	—	218	1	no	—
351	C₄	—	218	1	no	—
352	C₄	—	218	1	no	—
353	C₄	#16812726	1	2	yes	—
354	C₄	#41×D₁₀₉	1	1	yes	characteristic
355	C₄	C₂×#13182029	1	1	yes	characteristic
356	C₂	—	8	1	no	—
357	C₂	—	872	1	no	—
358	C₂	—	872	1	no	—
359	C₂	—	436	1	no	—
360	C₂	#20295573	1	1	yes	characteristic
361	C₂	C₂×#16812726	1	1	yes	characteristic
362	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	C₄×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#3171
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20295573	C₂	semidirect
2. #20295563	C₂	semidirect
3. #20295433	C₂	semidirect
4. #20295412	C₂	semidirect
5. #20295389	C₂	semidirect
6. #20295387	C₂	semidirect
7. #20295280	C₂	semidirect
8. #20295278	C₂	semidirect
9. #20295275	C₂	semidirect
10. #20295269	C₂	semidirect
11. C₄×#13182016	C₂	semidirect
12. C₄×D₈₇₂	C₂	semidirect
13. C₄×#13182014	C₂	semidirect
14. C₄×#13182012	C₂	semidirect
15. C₄×SD₁₆×C₁₀₉	C₂	semidirect
146. C₈×C₁₀₉	#41	semidirect
148. #11724784	#41	semidirect
182. C₁₀₉	#3171	semidirect
194. C₄×SD₁₆	D₁₀₉	semidirect
330. C₈	#13182029	semidirect

Maximal subgroups

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

subgroup	index
#20295573	2
#20295563	2
#20295433	2
#20295412	2
#20295389	2
#20295387	2
#20295280	2
#20295278	2
#20295275	2
#20295269	2
C₄×#13182016	2
C₄×D₈₇₂	2
C₄×#13182014	2
C₄×#13182012	2
C₄×SD₁₆×C₁₀₉	2
#3171	109