Subgroups — #28373217

Order 13952.

406 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	#20295649	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295501	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295364	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295351	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295314	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295304	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295243	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295243	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295160	C₂	1	1	yes	maximal, characteristic
10	#20295135	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295127	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295106	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295084	C₂	1	1	yes	maximal, characteristic, complemented
15	#450×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182014	C₂²	1	1	yes	characteristic
17	C₂×#13182014	C₂²	1	1	yes	characteristic
18	#16812698	C₂²	1	1	yes	characteristic, complemented
19	#16812694	—	2	1	no	—
20	#16812688	C₂²	1	1	yes	characteristic, complemented
21	#16812685	—	2	1	no	—
22	#16812681	—	2	1	no	—
23	#16812679	—	2	1	no	—
24	#16812673	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
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	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	#16812661	C₂²	1	1	yes	characteristic, complemented
33	#16812658	C₂²	1	1	yes	characteristic
34	#16812658	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₀₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₀₉	—	2	1	no	—
37	#16812633	C₂²	1	1	yes	characteristic, complemented
38	#16812632	C₂²	1	1	yes	characteristic, complemented
39	#16812631	C₂²	1	1	yes	characteristic, complemented
40	#16812630	C₂²	1	1	yes	characteristic, complemented
41	#16812628	C₂²	1	1	yes	characteristic, complemented
42	#16812628	C₂²	1	1	yes	characteristic, complemented
43	#16812623	C₂²	1	1	yes	characteristic, complemented
44	#16812623	C₂²	1	1	yes	characteristic, complemented
45	#16812621	C₂²	1	1	yes	characteristic, complemented
46	#16812621	C₂²	1	1	yes	characteristic, complemented
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, complemented
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#102×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂²×Dic₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	#13182014	—	2	2	no	—
68	#13182014	—	2	2	no	—
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	#13182008	—	2	1	no	—
78	C₂²⋊Dic₁₀₉	—	2	1	no	—
79	C₂²⋊Dic₁₀₉	—	2	1	no	—
80	C₂²⋊Dic₁₀₉	—	2	1	no	—
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₂⁴×C₁₀₉	—	2	1	no	—
94	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₀₉	—	2	2	no	—
96	C₂×D₄×C₁₀₉	—	2	2	no	—
97	C₂×D₄×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	SD₁₆×C₁₀₉	—	2	2	no	—
102	SD₁₆×C₁₀₉	—	2	2	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	C₂³×C₁₀₉	—	2	4	no	—
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂³×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	—	8	1	no	—
112	C₂³×C₁₀₉	—	8	1	no	—
113	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	4	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	#41	1	2	yes	—
121	D₄×C₁₀₉	#41	1	2	yes	—
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	—	8	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	8	1	no	—
133	C₂×Dic₁₀₉	—	8	1	no	—
134	C₂×Dic₁₀₉	—	8	1	no	—
135	C₂×Dic₁₀₉	—	8	1	no	—
136	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
140	D₄₃₆	—	2	1	no	—
141	D₄₃₆	—	2	1	no	—
142	C₄×D₁₀₉	—	8	1	no	—
143	Dic₂₁₈	—	8	1	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	#11724784	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	C₂²×C₁₀₉	—	2	1	no	—
149	C₂²×C₁₀₉	—	2	2	no	—
150	C₂²×C₁₀₉	—	2	2	no	—
151	C₂²×C₁₀₉	—	2	1	no	—
152	C₂²×C₁₀₉	—	2	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	—	8	4	no	—
158	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
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	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
167	Dic₁₀₉	—	2	1	no	—
168	Dic₁₀₉	—	2	1	no	—
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	2	1	no	—
171	Dic₁₀₉	—	4	1	no	—
172	Dic₁₀₉	—	4	1	no	—
173	Dic₁₀₉	—	4	1	no	—
174	C₂×C₁₀₉	—	4	1	no	—
175	C₂×C₁₀₉	—	4	2	no	—
176	C₂×C₁₀₉	—	4	2	no	—
177	C₂×C₁₀₉	#525	1	1	yes	characteristic
178	C₂×C₁₀₉	#576	1	1	yes	characteristic
179	C₂×C₁₀₉	#577	1	1	yes	characteristic
180	D₁₀₉	—	8	1	no	—
181	#3118	—	109	1	no	maximal, Sylow
182	C₁₀₉	#3118	1	1	yes	characteristic, complemented, Sylow
183	#542	—	109	1	no	—
184	#518	—	109	1	no	—
185	#485	—	109	1	no	—
186	#482	—	109	1	no	—
187	#476	—	109	1	no	—
188	#474	—	109	1	no	—
189	#463	—	109	1	no	—
190	#463	—	109	1	no	—
191	#450	D₁₀₉	1	1	yes	characteristic, complemented
192	#447	—	109	1	no	—
193	#442	—	109	1	no	—
194	#440	—	109	1	no	—
195	C₄×SD₁₆	—	109	1	no	—
196	C₄×D₈	—	109	1	no	—
197	#433	—	109	1	no	—
198	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
199	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
201	C₂×D₈	—	109	1	no	—
202	C₂×D₈	—	109	1	no	—
203	C₄⋊Q₈	—	109	1	no	—
204	#126	—	218	1	no	—
205	#124	—	109	1	no	—
206	#124	—	218	1	no	—
207	#123	—	218	1	no	—
208	C₂²⋊Q₈	—	218	1	no	—
209	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
210	#121	—	109	1	no	—
211	C₄×Q₈	—	109	1	no	—
212	C₄×D₄	—	109	1	no	—
213	C₄×D₄	—	109	1	no	—
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	—	109	1	no	—
216	C₄×D₄	—	109	1	no	—
217	C₄×D₄	—	218	1	no	—
218	C₄×D₄	—	218	1	no	—
219	#117	—	109	1	no	—
220	#117	—	109	1	no	—
221	C₂×C₂²⋊C₄	—	218	1	no	—
222	C₂×C₂²⋊C₄	—	218	1	no	—
223	#107	—	109	1	no	—
224	#107	—	109	1	no	—
225	#106	—	109	1	no	—
226	#106	—	109	1	no	—
227	C₄⋊C₈	—	109	1	no	—
228	C₄⋊C₈	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	—	109	1	no	—
231	#102	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#102	—	109	1	no	—
235	#102	C₂×D₁₀₉	1	1	yes	characteristic
236	#102	C₂×D₁₀₉	1	1	yes	characteristic
237	C₂²⋊C₈	—	109	1	no	—
238	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
239	#97	—	109	1	no	—
240	C₄×C₈	—	109	1	no	—
241	C₂⁴	—	2	1	no	—
242	C₂×Q₈	—	218	1	no	—
243	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	109	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂²×C₄	—	218	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	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	SD₁₆	—	2	2	no	—
258	SD₁₆	—	2	2	no	—
259	D₈	—	218	2	no	—
260	D₈	—	218	2	no	—
261	C₂×C₈	—	109	1	no	—
262	C₂×C₈	—	109	1	no	—
263	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	—	109	1	no	—
268	C₄⋊C₄	—	109	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	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	109	1	no	—
276	C₂²⋊C₄	—	109	1	no	—
277	C₂²⋊C₄	—	218	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	—	109	1	no	—
287	C₄²	—	109	1	no	—
288	C₄²	—	218	1	no	—
289	C₄²	—	218	1	no	—
290	C₂³	—	2	4	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	8	1	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	218	1	no	—
296	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	872	1	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	872	1	no	—
304	D₄	—	218	1	no	—
305	D₄	—	218	1	no	—
306	D₄	#13182028	1	2	yes	—
307	D₄	#13182028	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	872	1	no	—
313	C₂×C₄	—	872	1	no	—
314	C₂×C₄	—	872	1	no	—
315	C₂×C₄	—	872	1	no	—
316	C₂×C₄	—	872	1	no	—
317	C₂×C₄	—	109	1	no	—
318	C₂×C₄	—	109	1	no	—
319	C₂×C₄	—	109	1	no	—
320	C₂×C₄	—	109	1	no	—
321	C₂×C₄	—	218	1	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	C₂³×D₁₀₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	218	1	no	—
329	C₈	—	218	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	8	4	no	—
340	C₂²	—	436	1	no	—
341	C₂²	—	436	1	no	—
342	C₂²	—	436	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	—	436	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	—	218	1	no	—
354	C₄	C₂×#13182028	1	1	yes	characteristic
355	C₄	C₂×#13182028	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	4	2	no	—
358	C₂	—	4	2	no	—
359	C₂	—	872	1	no	—
360	C₂	#20295532	1	1	yes	characteristic
361	C₂	#20295830	1	1	yes	characteristic
362	C₂	#20295832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#450×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295160	C₂
hypercenter	#450	D₁₀₉
hypocenter	C₁₀₉	#3118
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #450

Complements

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

N	H ≅ G/N	type
1. #20295649	C₂	semidirect
2. #20295501	C₂	semidirect
3. #20295364	C₂	semidirect
4. #20295351	C₂	semidirect
5. #20295314	C₂	semidirect
6. #20295304	C₂	semidirect
7. #20295243	C₂	semidirect
8. #20295243	C₂	semidirect
10. #20295135	C₂	semidirect
11. #20295127	C₂	semidirect
12. SD₁₆×Dic₁₀₉	C₂	semidirect
13. #20295106	C₂	semidirect
14. #20295084	C₂	semidirect
15. #450×C₁₀₉	C₂	semidirect
18. #16812698	C₂²	semidirect
20. #16812688	C₂²	semidirect
25. Q₈×Dic₁₀₉	C₂²	semidirect
32. #16812661	C₂²	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. #16812628	C₂²	semidirect
42. #16812628	C₂²	semidirect
43. #16812623	C₂²	semidirect
44. #16812623	C₂²	semidirect
45. #16812621	C₂²	semidirect
46. #16812621	C₂²	semidirect
50. #16812609	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
182. C₁₀₉	#3118	semidirect
191. #450	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295649	2
#20295501	2
#20295364	2
#20295351	2
#20295314	2
#20295304	2
#20295243	2
#20295160	2
#20295135	2
#20295127	2
SD₁₆×Dic₁₀₉	2
#20295106	2
#20295084	2
#450×C₁₀₉	2
#3118	109