Subgroups — #28372484

Order 13952.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812720	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812718	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295575	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#13182028	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295241	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295225	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295219	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295021	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295019	C₂	1	1	yes	maximal, characteristic, complemented
15	#463×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×#13182028	C₂²	1	1	yes	characteristic
18	C₂×#13182028	C₂²	1	1	yes	characteristic
19	#16812720	—	2	4	no	—
20	#16812718	—	2	4	no	—
21	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
22	C₂×#13182019	C₂²	1	1	yes	characteristic
23	C₂×#13182018	C₂²	1	1	yes	characteristic
24	C₂×#13182017	C₂²	1	1	yes	characteristic
25	C₂×#13182017	C₂²	1	1	yes	characteristic
26	C₂×#13182014	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
29	#16812697	C₂²	1	1	yes	characteristic
30	#16812695	C₂²	1	1	yes	characteristic
31	#16812687	C₂²	1	1	yes	characteristic
32	#16812672	—	2	1	no	—
33	#16812672	—	2	1	no	—
34	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
35	#16812664	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
38	#16812659	—	2	1	no	—
39	#16812659	—	2	1	no	—
40	#16812659	—	2	1	no	—
41	#16812657	C₂²	1	1	yes	characteristic
42	#16812655	—	2	1	no	—
43	C₂×C₄×Dic₁₀₉	—	2	1	no	—
44	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
45	C₄⋊#11724784	C₂²	1	1	yes	characteristic
46	#16812628	C₂²	1	1	yes	characteristic
47	#16812628	C₂²	1	1	yes	characteristic
48	#16812624	C₂²	1	1	yes	characteristic
49	#16812623	C₂²	1	1	yes	characteristic
50	#16812622	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	#16812620	C₂²	1	1	yes	characteristic
53	#16812619	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#124×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#103×C₁₀₉	C₂²	1	1	yes	characteristic
60	#102×C₁₀₉	C₂²	1	1	yes	characteristic
61	#13182030	—	2	2	no	—
62	#13182030	—	2	2	no	—
63	#13182028	—	2	2	no	—
64	#13182028	—	2	2	no	—
65	#13182028	—	2	2	no	—
66	#13182028	D₄	1	4	yes	complemented
67	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
68	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
69	C₂×D₄₃₆	—	2	1	no	—
70	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
72	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
73	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
74	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
75	C₂²×Dic₁₀₉	—	2	1	no	—
76	C₂²×Dic₁₀₉	—	2	1	no	—
77	C₂²×Dic₁₀₉	—	2	1	no	—
78	C₂×C₄×D₁₀₉	—	2	1	no	—
79	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
80	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
81	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
82	Dic₄₃₆	—	2	2	no	—
83	#13182019	—	2	2	no	—
84	#13182018	—	2	2	no	—
85	#13182017	—	2	2	no	—
86	#13182017	—	2	2	no	—
87	#13182014	—	2	2	no	—
88	C₂×#11724784	C₂³	1	1	yes	characteristic
89	C₂×#11724784	C₂³	1	1	yes	characteristic
90	C₈×D₁₀₉	—	2	2	no	—
91	C₈×D₁₀₉	—	2	2	no	—
92	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
93	#13182008	—	2	1	no	—
94	#13182008	—	2	1	no	—
95	#13182008	—	2	1	no	—
96	#13182008	—	2	1	no	—
97	C₂²⋊Dic₁₀₉	—	2	1	no	—
98	#13182006	—	2	1	no	—
99	#13182006	—	2	1	no	—
100	#13182006	—	2	1	no	—
101	C₄×Dic₁₀₉	—	2	1	no	—
102	C₄×Dic₁₀₉	—	2	1	no	—
103	C₄×Dic₁₀₉	—	2	1	no	—
104	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
105	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
106	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₁₀₉	—	2	1	no	—
111	SD₁₆×C₁₀₉	—	2	2	no	—
112	D₈×C₁₀₉	—	2	2	no	—
113	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
116	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
117	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
118	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
119	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
120	C₂³×C₁₀₉	—	2	1	no	—
121	C₂³×C₁₀₉	—	2	1	no	—
122	C₂²×D₁₀₉	—	2	1	no	—
123	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
124	Q₈×C₁₀₉	—	2	1	no	—
125	Q₈×C₁₀₉	—	2	1	no	—
126	D₄×C₁₀₉	—	2	1	no	—
127	D₄×C₁₀₉	—	2	1	no	—
128	D₄×C₁₀₉	—	2	1	no	—
129	D₄×C₁₀₉	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₁₀₉	—	2	1	no	—
131	C₂×C₄×C₁₀₉	—	2	1	no	—
132	C₂×C₄×C₁₀₉	—	4	2	no	—
133	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
136	C₂²⋊D₁₀₉	—	2	4	no	—
137	C₂²⋊D₁₀₉	—	4	2	no	—
138	C₂²⋊D₁₀₉	—	4	2	no	—
139	C₂²⋊D₁₀₉	—	4	2	no	—
140	C₂×Dic₁₀₉	—	2	1	no	—
141	C₂×Dic₁₀₉	—	2	4	no	—
142	C₂×Dic₁₀₉	—	2	1	no	—
143	C₂×Dic₁₀₉	—	2	1	no	—
144	C₂×Dic₁₀₉	—	4	2	no	—
145	C₂×Dic₁₀₉	—	4	2	no	—
146	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
147	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
148	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
149	D₄₃₆	—	2	1	no	—
150	D₄₃₆	—	2	1	no	—
151	D₄₃₆	—	4	2	no	—
152	C₄×D₁₀₉	—	2	2	no	—
153	C₄×D₁₀₉	—	4	1	no	—
154	C₄×D₁₀₉	—	4	2	no	—
155	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
156	C₄×D₁₀₉	#41	1	2	yes	—
157	Dic₂₁₈	—	2	1	no	—
158	Dic₂₁₈	—	2	1	no	—
159	Dic₂₁₈	—	2	1	no	—
160	Dic₂₁₈	C₂×D₄	1	2	yes	—
161	C₈×C₁₀₉	—	2	1	no	—
162	C₈×C₁₀₉	—	2	1	no	—
163	#11724784	—	2	1	no	—
164	#11724784	—	2	1	no	—
165	C₂²×C₁₀₉	—	2	2	no	—
166	C₂²×C₁₀₉	—	4	1	no	—
167	C₂²×C₁₀₉	—	4	1	no	—
168	C₂²×C₁₀₉	—	4	1	no	—
169	C₂²×C₁₀₉	—	4	1	no	—
170	C₂²×C₁₀₉	—	4	1	no	—
171	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
172	C₂×D₁₀₉	—	2	1	no	—
173	C₂×D₁₀₉	—	2	1	no	—
174	C₂×D₁₀₉	—	4	1	no	—
175	C₂×D₁₀₉	—	4	1	no	—
176	C₂×D₁₀₉	—	4	1	no	—
177	C₂×D₁₀₉	#121	1	2	yes	—
178	C₄×C₁₀₉	—	2	1	no	—
179	C₄×C₁₀₉	—	2	1	no	—
180	C₄×C₁₀₉	—	4	1	no	—
181	C₄×C₁₀₉	—	4	1	no	—
182	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
184	Dic₁₀₉	—	2	2	no	—
185	Dic₁₀₉	—	2	1	no	—
186	Dic₁₀₉	—	4	1	no	—
187	Dic₁₀₉	—	4	1	no	—
188	Dic₁₀₉	#121	1	2	yes	—
189	Dic₁₀₉	#135	1	2	yes	complemented
190	C₂×C₁₀₉	—	4	2	no	—
191	C₂×C₁₀₉	—	8	1	no	—
192	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
193	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
194	C₂×C₁₀₉	#575	1	1	yes	characteristic
195	D₁₀₉	—	2	2	no	—
196	D₁₀₉	—	8	1	no	—
197	#3035	—	109	1	no	maximal, Sylow
198	C₁₀₉	#3035	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	109	1	no	—
200	C₂×#135	—	109	1	no	—
201	#532	—	109	1	no	—
202	C₄×#41	—	109	1	no	—
203	#464	—	109	1	no	—
204	#464	—	109	1	no	—
205	#463	—	109	1	no	—
206	#463	D₁₀₉	1	1	yes	characteristic, complemented
207	#462	—	109	1	no	—
208	#461	—	109	1	no	—
209	#460	—	109	1	no	—
210	#459	—	109	1	no	—
211	C₂×C₄⋊C₈	—	109	1	no	—
212	#416	—	109	1	no	—
213	#416	—	109	1	no	—
214	C₂×#41	—	109	1	no	—
215	C₂×#41	—	109	1	no	—
216	C₂×#41	—	109	1	no	—
217	#135	—	218	4	no	—
218	#135	—	218	4	no	—
219	C₂×Q₁₆	—	109	1	no	—
220	C₂×Q₁₆	—	109	1	no	—
221	C₂×SD₁₆	—	109	1	no	—
222	C₂×SD₁₆	—	109	1	no	—
223	C₂×SD₁₆	—	109	1	no	—
224	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
225	C₂×D₈	—	109	1	no	—
226	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₂²×C₈	—	109	1	no	—
228	C₂²×C₈	—	109	1	no	—
229	C₄⋊Q₈	—	109	1	no	—
230	#127	—	109	1	no	—
231	#124	—	109	1	no	—
232	#124	C₂×D₁₀₉	1	1	yes	characteristic
233	#121	—	218	1	no	—
234	#121	—	218	1	no	—
235	C₄×Q₈	—	109	1	no	—
236	C₄×Q₈	—	109	1	no	—
237	C₄×D₄	—	109	1	no	—
238	C₄×D₄	—	218	1	no	—
239	C₄×D₄	—	218	1	no	—
240	C₄×D₄	—	218	1	no	—
241	C₄×D₄	—	218	1	no	—
242	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
243	#117	—	109	1	no	—
244	#117	—	218	1	no	—
245	C₂×C₄²	—	109	1	no	—
246	C₂×C₄²	—	218	1	no	—
247	C₄⋊C₈	—	109	1	no	—
248	C₄⋊C₈	—	109	1	no	—
249	C₄⋊C₈	—	109	1	no	—
250	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
251	#103	—	109	1	no	—
252	#103	—	109	1	no	—
253	#103	—	109	1	no	—
254	#103	C₂×D₁₀₉	1	1	yes	characteristic
255	#102	—	109	1	no	—
256	#102	—	109	1	no	—
257	#102	—	109	1	no	—
258	#102	C₂×D₁₀₉	1	1	yes	characteristic
259	#41	—	109	4	no	—
260	#41	—	218	2	no	—
261	#41	—	218	2	no	—
262	#41	—	218	2	no	—
263	#41	—	218	2	no	—
264	#41	—	218	2	no	—
265	C₂×Q₈	—	109	1	no	—
266	C₂×Q₈	—	109	1	no	—
267	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₂×D₄	—	109	1	no	—
269	C₂×D₄	—	218	1	no	—
270	C₂×D₄	—	218	1	no	—
271	C₂×D₄	—	218	1	no	—
272	C₂×D₄	—	218	1	no	—
273	C₂×D₄	—	218	1	no	—
274	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	109	1	no	—
278	C₂²×C₄	—	109	1	no	—
279	C₂²×C₄	—	109	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	Q₁₆	—	218	2	no	—
285	Q₁₆	—	218	2	no	—
286	SD₁₆	—	2	2	no	—
287	SD₁₆	—	218	2	no	—
288	SD₁₆	—	218	2	no	—
289	SD₁₆	—	218	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	218	2	no	—
292	C₂×C₈	—	109	1	no	—
293	C₂×C₈	—	109	1	no	—
294	C₂×C₈	—	218	2	no	—
295	C₂×C₈	—	218	2	no	—
296	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
298	C₄⋊C₄	—	109	1	no	—
299	C₄⋊C₄	—	218	1	no	—
300	C₄⋊C₄	—	218	1	no	—
301	C₄⋊C₄	—	218	1	no	—
302	C₄⋊C₄	—	218	1	no	—
303	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	2	1	no	—
306	C₂²⋊C₄	—	2	1	no	—
307	C₂²⋊C₄	—	218	1	no	—
308	C₂²⋊C₄	—	218	1	no	—
309	C₂²⋊C₄	—	218	1	no	—
310	C₂²⋊C₄	—	218	1	no	—
311	C₄²	—	109	1	no	—
312	C₄²	—	109	1	no	—
313	C₄²	—	109	1	no	—
314	C₄²	—	218	1	no	—
315	C₄²	—	218	1	no	—
316	C₄²	—	218	1	no	—
317	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
318	C₂³	—	2	1	no	—
319	C₂³	—	2	1	no	—
320	C₂³	—	109	1	no	—
321	C₂³	—	218	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	109	2	no	—
325	Q₈	—	218	1	no	—
326	Q₈	—	218	1	no	—
327	Q₈	—	218	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	436	2	no	—
332	D₄	—	436	2	no	—
333	D₄	—	436	2	no	—
334	D₄	—	436	2	no	—
335	D₄	—	218	4	no	—
336	D₄	—	218	1	no	—
337	D₄	—	218	1	no	—
338	D₄	D₄×D₁₀₉	1	2	yes	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	109	1	no	—
343	C₂×C₄	—	109	1	no	—
344	C₂×C₄	—	109	2	no	—
345	C₂×C₄	—	109	2	no	—
346	C₂×C₄	—	109	1	no	—
347	C₂×C₄	—	436	1	no	—
348	C₂×C₄	—	436	2	no	—
349	C₂×C₄	—	436	2	no	—
350	C₂×C₄	—	436	2	no	—
351	C₂×C₄	—	218	2	no	—
352	C₂×C₄	—	218	4	no	—
353	C₂×C₄	—	218	1	no	—
354	C₂×C₄	—	218	1	no	—
355	C₂×C₄	—	218	1	no	—
356	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
357	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
358	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	218	1	no	—
362	C₈	—	218	1	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	109	2	no	—
370	C₂²	—	436	1	no	—
371	C₂²	—	436	1	no	—
372	C₂²	—	436	1	no	—
373	C₂²	—	218	1	no	—
374	C₂²	—	218	1	no	—
375	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	1	no	—
380	C₄	—	109	2	no	—
381	C₄	—	109	2	no	—
382	C₄	—	436	1	no	—
383	C₄	—	436	1	no	—
384	C₄	—	218	2	no	—
385	C₄	—	218	1	no	—
386	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
387	C₄	#41×D₁₀₉	1	1	yes	characteristic
388	C₂	—	4	2	no	—
389	C₂	—	8	1	no	—
390	C₂	—	872	1	no	—
391	C₂	—	218	2	no	—
392	C₂	#121×D₁₀₉	1	1	yes	characteristic
393	C₂	#135×D₁₀₉	1	1	yes	characteristic
394	C₂	#20295822	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#463×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#463	D₁₀₉
hypocenter	C₁₀₉	#3035
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. C₂×#16812720	C₂	semidirect
2. C₂×#16812718	C₂	semidirect
3. #20295575	C₂	semidirect
4. C₄×#13182028	C₂	semidirect
5. #20295249	C₂	semidirect
6. #20295245	C₂	semidirect
7. #20295241	C₂	semidirect
8. #20295235	C₂	semidirect
9. #20295229	C₂	semidirect
10. #20295225	C₂	semidirect
11. #20295219	C₂	semidirect
12. C₄⋊C₈×D₁₀₉	C₂	semidirect
13. #20295021	C₂	semidirect
14. #20295019	C₂	semidirect
15. #463×C₁₀₉	C₂	semidirect
66. #13182028	D₄	semidirect
189. Dic₁₀₉	#135	semidirect
198. C₁₀₉	#3035	semidirect
206. #463	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812720	2
C₂×#16812718	2
#20295575	2
C₄×#13182028	2
#20295249	2
#20295245	2
#20295241	2
#20295235	2
#20295229	2
#20295225	2
#20295219	2
C₄⋊C₈×D₁₀₉	2
#20295021	2
#20295019	2
#463×C₁₀₉	2
#3035	109