Subgroups — #28374874

Order 13952.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295717	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295669	C₂	1	1	yes	maximal, characteristic
3	#20295380	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295355	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295348	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295264	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295252	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295187	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295175	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295091	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295021	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#13182016	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
21	#16812693	—	2	1	no	—
22	#16812691	C₂²	1	1	yes	characteristic, complemented
23	#16812689	—	2	1	no	—
24	#16812684	—	2	1	no	—
25	#16812681	—	2	1	no	—
26	#16812675	C₂²	1	1	yes	characteristic
27	#16812675	C₂²	1	1	yes	characteristic
28	#16812675	C₂²	1	1	yes	characteristic, complemented
29	#16812672	—	2	1	no	—
30	#16812670	C₂²	1	1	yes	characteristic
31	#16812669	—	2	1	no	—
32	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	#16812657	C₂²	1	1	yes	characteristic, complemented
35	#16812656	—	2	1	no	—
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
38	#16812633	C₂²	1	1	yes	characteristic, complemented
39	#16812631	C₂²	1	1	yes	characteristic, complemented
40	#16812630	C₂²	1	1	yes	characteristic, complemented
41	#16812623	C₂²	1	1	yes	characteristic, complemented
42	#16812623	C₂²	1	1	yes	characteristic, complemented
43	#16812623	C₂²	1	1	yes	characteristic, complemented
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	#16812619	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic, complemented
51	C₄×#11724784	C₂²	1	1	yes	characteristic, complemented
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#124×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂³×D₁₀₉	—	2	1	no	—
60	#13182030	—	2	2	no	—
61	#13182030	—	4	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	D₄	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	D₄×D₁₀₉	—	2	2	no	—
68	D₄×D₁₀₉	—	4	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	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
74	#13182016	—	2	2	no	—
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₈×D₁₀₉	—	4	1	no	—
78	C₈×D₁₀₉	—	4	1	no	—
79	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	#13182008	—	2	1	no	—
87	#13182008	—	2	1	no	—
88	#13182008	—	2	1	no	—
89	C₂²⋊Dic₁₀₉	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	#13182006	—	2	1	no	—
95	#13182006	D₄	1	1	yes	characteristic
96	#13182006	D₄	1	1	yes	characteristic
97	C₄×Dic₁₀₉	—	2	1	no	—
98	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₀₉	—	4	1	no	—
102	SD₁₆×C₁₀₉	—	4	1	no	—
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	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₄²×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	C₂²×D₁₀₉	—	4	2	no	—
114	C₂²×D₁₀₉	—	4	2	no	—
115	C₂²×D₁₀₉	—	8	1	no	—
116	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₀₉	—	2	1	no	—
119	Q₈×C₁₀₉	—	2	1	no	—
120	Q₈×C₁₀₉	—	2	1	no	—
121	Q₈×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	2	1	no	—
123	D₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	—	2	1	no	—
127	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₀₉	—	4	2	no	—
131	C₂²⋊D₁₀₉	—	8	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
137	D₄₃₆	—	2	2	no	—
138	D₄₃₆	—	2	1	no	—
139	D₄₃₆	—	4	2	no	—
140	D₄₃₆	#41	1	2	yes	—
141	C₄×D₁₀₉	—	2	1	no	—
142	C₄×D₁₀₉	—	2	1	no	—
143	C₄×D₁₀₉	—	2	1	no	—
144	C₄×D₁₀₉	—	2	1	no	—
145	C₄×D₁₀₉	—	4	2	no	—
146	C₄×D₁₀₉	—	4	2	no	—
147	C₄×D₁₀₉	—	8	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	C₈×C₁₀₉	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	#11724784	—	2	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₀₉	—	2	1	no	—
157	C₂×D₁₀₉	—	2	1	no	—
158	C₂×D₁₀₉	—	2	1	no	—
159	C₂×D₁₀₉	—	2	2	no	—
160	C₂×D₁₀₉	—	2	1	no	—
161	C₂×D₁₀₉	—	2	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₂×D₁₀₉	—	8	4	no	—
165	C₂×D₁₀₉	#135	1	2	yes	—
166	C₄×C₁₀₉	—	2	1	no	—
167	C₄×C₁₀₉	—	2	1	no	—
168	C₄×C₁₀₉	—	4	1	no	—
169	C₄×C₁₀₉	—	4	1	no	—
170	C₄×C₁₀₉	—	4	1	no	—
171	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₁₀₉	—	2	1	no	—
174	Dic₁₀₉	—	2	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	—	4	1	no	—
177	Dic₁₀₉	—	4	1	no	—
178	C₂×C₁₀₉	—	8	1	no	—
179	C₂×C₁₀₉	#546	1	1	yes	characteristic
180	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
181	C₂×C₁₀₉	#577	1	1	yes	characteristic
182	D₁₀₉	—	2	2	no	—
183	D₁₀₉	—	4	1	no	—
184	D₁₀₉	—	4	2	no	—
185	#3296	—	109	1	no	maximal, Sylow
186	C₁₀₉	#3296	1	1	yes	characteristic, complemented, Sylow
187	#552	—	109	1	no	—
188	#546	—	109	1	no	—
189	#488	—	109	1	no	—
190	#483	—	109	1	no	—
191	#482	—	109	1	no	—
192	#478	—	109	1	no	—
193	#467	—	109	1	no	—
194	#465	—	109	1	no	—
195	#464	D₁₀₉	1	1	yes	characteristic, complemented
196	#452	—	109	1	no	—
197	#450	—	109	1	no	—
198	C₄×SD₁₆	—	109	1	no	—
199	C₈×D₄	—	109	1	no	—
200	#416	—	109	1	no	—
201	C₂×#102	—	109	1	no	—
202	C₂×#41	—	109	1	no	—
203	C₂²×D₄	—	109	1	no	—
204	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic, complemented
205	C₂×SD₁₆	—	109	1	no	—
206	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
207	C₂²×C₈	—	109	1	no	—
208	C₂²×C₈	—	109	1	no	—
209	#126	—	218	1	no	—
210	#125	—	109	1	no	—
211	#125	—	218	1	no	—
212	#124	C₂×D₁₀₉	1	1	yes	characteristic
213	#123	—	218	1	no	—
214	#123	—	218	1	no	—
215	C₂²⋊Q₈	—	109	1	no	—
216	C₂²⋊Q₈	—	109	1	no	—
217	C₂²⋊Q₈	—	109	1	no	—
218	#121	—	109	1	no	—
219	#121	—	218	1	no	—
220	#120	—	218	1	no	—
221	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
222	C₄×D₄	—	109	1	no	—
223	C₄×D₄	—	109	1	no	—
224	#117	—	109	1	no	—
225	#117	—	218	1	no	—
226	C₂×C₄⋊C₄	—	109	1	no	—
227	C₂×C₂²⋊C₄	—	218	1	no	—
228	#107	—	109	1	no	—
229	#106	—	109	1	no	—
230	#106	—	109	1	no	—
231	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
232	#103	—	109	1	no	—
233	#103	—	109	1	no	—
234	#103	—	109	1	no	—
235	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#102	—	109	1	no	—
241	#102	C₂×D₁₀₉	1	1	yes	characteristic
242	C₂²⋊C₈	—	109	1	no	—
243	C₂²⋊C₈	—	109	1	no	—
244	C₄×C₈	—	109	1	no	—
245	C₂⁴	—	218	1	no	—
246	#41	—	436	1	no	—
247	#41	—	218	2	no	—
248	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂×D₄	—	109	1	no	—
251	C₂×D₄	—	109	1	no	—
252	C₂×D₄	—	436	1	no	—
253	C₂×D₄	—	218	1	no	—
254	C₂×D₄	—	218	1	no	—
255	C₂×D₄	—	218	2	no	—
256	C₂×D₄	—	218	1	no	—
257	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂²×C₄	—	109	1	no	—
259	C₂²×C₄	—	109	1	no	—
260	C₂²×C₄	—	218	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	Q₁₆	—	4	1	no	—
264	SD₁₆	—	4	1	no	—
265	SD₁₆	—	218	2	no	—
266	C₂×C₈	—	109	1	no	—
267	C₂×C₈	—	109	1	no	—
268	C₂×C₈	—	436	1	no	—
269	C₂×C₈	—	436	1	no	—
270	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	109	1	no	—
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	109	1	no	—
276	C₄⋊C₄	—	109	1	no	—
277	C₄⋊C₄	—	218	1	no	—
278	C₄⋊C₄	—	218	1	no	—
279	C₄⋊C₄	—	218	1	no	—
280	C₄⋊C₄	—	218	1	no	—
281	C₄⋊C₄	—	218	1	no	—
282	C₄⋊C₄	—	218	1	no	—
283	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	109	1	no	—
287	C₂²⋊C₄	—	109	1	no	—
288	C₂²⋊C₄	—	218	1	no	—
289	C₂²⋊C₄	—	218	1	no	—
290	C₂²⋊C₄	—	218	1	no	—
291	C₂²⋊C₄	—	218	1	no	—
292	C₂²⋊C₄	—	218	1	no	—
293	C₂²⋊C₄	—	218	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	218	1	no	—
296	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
297	C₂³	—	2	1	no	—
298	C₂³	—	872	1	no	—
299	C₂³	—	109	1	no	—
300	C₂³	—	109	1	no	—
301	C₂³	—	436	2	no	—
302	C₂³	—	436	2	no	—
303	C₂³	—	218	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	872	1	no	—
311	D₄	—	109	2	no	—
312	D₄	—	436	2	no	—
313	D₄	—	436	2	no	—
314	D₄	—	218	2	no	—
315	D₄	—	218	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	872	1	no	—
320	C₂×C₄	—	109	1	no	—
321	C₂×C₄	—	109	1	no	—
322	C₂×C₄	—	436	2	no	—
323	C₂×C₄	—	436	2	no	—
324	C₂×C₄	—	218	1	no	—
325	C₂×C₄	—	218	1	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	1	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	—	218	1	no	—
331	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
332	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	218	1	no	—
337	C₈	—	218	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	872	4	no	—
342	C₂²	—	109	2	no	—
343	C₂²	—	436	1	no	—
344	C₂²	—	436	1	no	—
345	C₂²	—	218	1	no	—
346	C₂²	—	218	1	no	—
347	C₂²	—	218	2	no	—
348	C₂²	—	218	1	no	—
349	C₂²	—	218	1	no	—
350	C₂²	—	218	1	no	—
351	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	436	1	no	—
358	C₄	—	436	1	no	—
359	C₄	—	436	1	no	—
360	C₄	—	218	1	no	—
361	C₄	—	218	1	no	—
362	C₄	#41×D₁₀₉	1	1	yes	characteristic
363	C₄	#16812744	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	436	2	no	—
366	C₂	—	436	1	no	—
367	C₂	—	218	2	no	—
368	C₂	#20295673	1	1	yes	characteristic
369	C₂	#135×D₁₀₉	1	1	yes	characteristic
370	C₂	#20295835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#464×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295669	C₂
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3296
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #20295717	C₂	semidirect
3. #20295380	C₂	semidirect
4. #20295355	C₂	semidirect
5. #20295348	C₂	semidirect
6. #20295327	C₂	semidirect
7. #20295264	C₂	semidirect
8. #20295252	C₂	semidirect
9. #20295187	C₂	semidirect
10. #20295175	C₂	semidirect
11. C₄×#13182016	C₂	semidirect
12. #20295091	C₂	semidirect
13. #20295021	C₂	semidirect
14. #102×D₁₀₉	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
20. C₂×C₈×D₁₀₉	C₂²	semidirect
22. #16812691	C₂²	semidirect
28. #16812675	C₂²	semidirect
34. #16812657	C₂²	semidirect
38. #16812633	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. #16812623	C₂²	semidirect
42. #16812623	C₂²	semidirect
43. #16812623	C₂²	semidirect
50. #16812611	C₂²	semidirect
51. C₄×#11724784	C₂²	semidirect
52. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
55. C₄×Q₈×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
57. #103×C₁₀₉	C₂²	semidirect
186. C₁₀₉	#3296	semidirect
195. #464	D₁₀₉	semidirect
204. C₂×Q₁₆	C₂×D₁₀₉	semidirect
221. C₄×Q₈	C₂×D₁₀₉	semidirect
231. C₄⋊C₈	C₂×D₁₀₉	semidirect
235. #103	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295717	2
#20295669	2
#20295380	2
#20295355	2
#20295348	2
#20295327	2
#20295264	2
#20295252	2
#20295187	2
#20295175	2
C₄×#13182016	2
#20295091	2
#20295021	2
#102×D₁₀₉	2
#464×C₁₀₉	2
#3296	109