Subgroups — #28371695

Order 13952.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295494	C₂	1	1	yes	maximal, characteristic
2	#20295087	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295081	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295080	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295077	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295076	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295048	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#20294991	C₂	1	1	yes	maximal, characteristic
12	#20294991	C₂	1	1	yes	maximal, characteristic
13	#20294983	C₂	1	1	yes	maximal, characteristic, complemented
14	#20294980	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	#16812664	C₄	1	1	yes	characteristic
22	#16812664	C₄	1	1	yes	characteristic
23	#16812661	C₄	1	1	yes	characteristic
24	#16812661	C₄	1	1	yes	characteristic
25	#16812659	C₄	1	1	yes	characteristic
26	#16812659	C₄	1	1	yes	characteristic
27	#16812659	C₄	1	1	yes	characteristic
28	#16812659	C₄	1	1	yes	characteristic
29	#16812656	C₂²	1	1	yes	characteristic
30	#16812656	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
33	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊#11724784	C₂²	1	1	yes	characteristic
35	C₄⋊#11724784	C₂²	1	1	yes	characteristic
36	#16812628	C₂²	1	1	yes	characteristic
37	#16812628	C₂²	1	1	yes	characteristic
38	#16812628	C₂²	1	1	yes	characteristic
39	#16812628	C₂²	1	1	yes	characteristic
40	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
41	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812611	C₂²	1	1	yes	characteristic
43	#16812611	C₂²	1	1	yes	characteristic
44	#16812611	C₂²	1	1	yes	characteristic
45	#16812611	C₂²	1	1	yes	characteristic
46	#16812610	C₂²	1	1	yes	characteristic
47	#16812609	C₂²	1	1	yes	characteristic
48	#16812609	C₂²	1	1	yes	characteristic
49	C₄×#11724784	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	#117×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#97×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	2	2	no	—
60	#13182029	—	2	2	no	—
61	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
62	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
73	#13182012	—	2	2	no	—
74	#13182012	—	2	2	no	—
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₂×#11724784	C₂³	1	1	yes	characteristic
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₈×D₁₀₉	—	2	2	no	—
80	C₈×D₁₀₉	—	2	2	no	—
81	#13182008	C₂×C₄	1	1	yes	characteristic
82	#13182008	C₂×C₄	1	1	yes	characteristic
83	#13182008	C₂×C₄	1	1	yes	characteristic
84	#13182008	C₂×C₄	1	1	yes	characteristic
85	#13182006	C₂×C₄	1	1	yes	characteristic
86	#13182006	C₂×C₄	1	1	yes	characteristic
87	#13182006	C₂×C₄	1	1	yes	characteristic
88	#13182006	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
106	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	2	2	no	—
120	C₂²⋊D₁₀₉	—	2	2	no	—
121	C₂×Dic₁₀₉	—	2	2	no	—
122	C₂×Dic₁₀₉	—	4	1	no	—
123	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
129	D₄₃₆	—	2	2	no	—
130	C₄×D₁₀₉	—	2	1	no	—
131	C₄×D₁₀₉	—	2	1	no	—
132	C₄×D₁₀₉	—	2	2	no	—
133	C₄×D₁₀₉	—	2	2	no	—
134	C₄×D₁₀₉	—	4	1	no	—
135	C₄×D₁₀₉	—	4	1	no	—
136	C₄×D₁₀₉	#41	1	2	yes	—
137	C₄×D₁₀₉	#41	1	2	yes	—
138	Dic₂₁₈	—	2	2	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	—	2	1	no	—
142	C₈×C₁₀₉	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	#11724784	—	2	1	no	—
145	#11724784	—	2	1	no	—
146	#11724784	—	2	1	no	—
147	C₂²×C₁₀₉	—	2	1	no	—
148	C₂²×C₁₀₉	—	2	1	no	—
149	C₂²×C₁₀₉	—	2	1	no	—
150	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₁₀₉	—	2	1	no	—
152	C₂×D₁₀₉	—	2	1	no	—
153	C₂×D₁₀₉	—	2	1	no	—
154	C₂×D₁₀₉	—	2	1	no	—
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	#117	1	2	yes	—
157	C₄×C₁₀₉	—	2	1	no	—
158	C₄×C₁₀₉	—	2	1	no	—
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	2	1	no	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	2	1	no	—
167	Dic₁₀₉	—	2	1	no	—
168	Dic₁₀₉	#117	1	2	yes	—
169	Dic₁₀₉	#131	1	2	yes	—
170	C₂×C₁₀₉	—	4	1	no	—
171	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
172	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
173	C₂×C₁₀₉	#568	1	1	yes	characteristic
174	D₁₀₉	—	2	2	no	—
175	D₁₀₉	—	4	1	no	—
176	#2925	—	109	1	no	maximal, Sylow
177	C₁₀₉	#2925	1	1	yes	characteristic, complemented, Sylow
178	C₄×#41	—	109	1	no	—
179	#433	—	109	1	no	—
180	#433	—	109	1	no	—
181	#433	—	109	1	no	—
182	#433	D₁₀₉	1	1	yes	characteristic, complemented
183	#432	—	109	1	no	—
184	#432	—	109	1	no	—
185	#431	—	109	1	no	—
186	#431	—	109	1	no	—
187	#423	—	109	1	no	—
188	C₂×C₄⋊C₈	—	109	1	no	—
189	#408	—	109	1	no	—
190	#408	—	109	1	no	—
191	#405	—	109	1	no	—
192	#405	—	109	1	no	—
193	C₂×#41	—	109	1	no	—
194	C₂×M₁₆	—	109	1	no	—
195	C₂×M₁₆	—	109	1	no	—
196	C₂²×C₈	—	109	1	no	—
197	C₂²×C₈	—	109	1	no	—
198	C₄×Q₈	—	109	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×D₄	—	109	1	no	—
201	C₄×D₄	—	109	1	no	—
202	C₄×D₄	—	109	1	no	—
203	C₄×D₄	—	109	1	no	—
204	C₄×D₄	—	109	1	no	—
205	C₄×D₄	—	109	1	no	—
206	#117	—	109	1	no	—
207	#117	—	109	1	no	—
208	#117	C₂×D₁₀₉	1	1	yes	characteristic
209	C₂×C₄²	—	109	1	no	—
210	C₂×C₄²	—	109	1	no	—
211	C₂×C₄²	—	109	1	no	—
212	C₄⋊C₈	—	109	1	no	—
213	C₄⋊C₈	—	109	1	no	—
214	C₄⋊C₈	—	109	1	no	—
215	C₄⋊C₈	—	109	1	no	—
216	C₄⋊C₈	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
220	C₂²⋊C₈	—	109	1	no	—
221	C₂²⋊C₈	—	109	1	no	—
222	C₂²⋊C₈	—	109	1	no	—
223	C₂²⋊C₈	—	109	1	no	—
224	C₂²⋊C₈	—	109	1	no	—
225	C₂²⋊C₈	—	109	1	no	—
226	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	#97	—	109	1	no	—
229	#97	—	109	1	no	—
230	#97	—	109	1	no	—
231	#97	C₂×D₁₀₉	1	1	yes	characteristic
232	C₄×C₈	—	109	1	no	—
233	C₄×C₈	—	109	1	no	—
234	C₄×C₈	—	109	1	no	—
235	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
236	#41	—	218	2	no	—
237	#41	—	218	2	no	—
238	C₂×Q₈	—	109	1	no	—
239	C₂×D₄	—	109	1	no	—
240	C₂×D₄	—	109	1	no	—
241	C₂×D₄	—	109	1	no	—
242	C₂²×C₄	—	109	1	no	—
243	C₂²×C₄	—	109	1	no	—
244	C₂²×C₄	—	109	1	no	—
245	C₂²×C₄	—	109	1	no	—
246	C₂²×C₄	—	109	1	no	—
247	C₂²×C₄	—	109	1	no	—
248	C₂²×C₄	—	109	1	no	—
249	C₂²×C₄	—	109	1	no	—
250	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
251	M₁₆	—	218	2	no	—
252	M₁₆	—	218	2	no	—
253	C₂×C₈	—	109	1	no	—
254	C₂×C₈	—	109	1	no	—
255	C₂×C₈	—	109	1	no	—
256	C₂×C₈	—	109	1	no	—
257	C₂×C₈	—	218	2	no	—
258	C₂×C₈	—	218	2	no	—
259	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	—	109	1	no	—
264	C₄⋊C₄	—	109	1	no	—
265	C₄⋊C₄	—	109	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
268	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	109	1	no	—
270	C₂²⋊C₄	—	109	1	no	—
271	C₂²⋊C₄	—	109	1	no	—
272	C₂²⋊C₄	—	109	1	no	—
273	C₂²⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
275	C₄²	—	109	1	no	—
276	C₄²	—	109	1	no	—
277	C₄²	—	109	1	no	—
278	C₄²	—	109	1	no	—
279	C₄²	—	109	1	no	—
280	C₄²	—	109	1	no	—
281	C₄²	—	109	1	no	—
282	C₄²	—	109	1	no	—
283	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
284	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
285	C₂³	—	109	1	no	—
286	C₂³	—	109	1	no	—
287	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
288	Q₈	—	218	2	no	—
289	D₄	—	218	2	no	—
290	D₄	—	218	2	no	—
291	D₄	—	218	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	109	1	no	—
295	C₂×C₄	—	109	1	no	—
296	C₂×C₄	—	109	2	no	—
297	C₂×C₄	—	109	2	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	109	1	no	—
300	C₂×C₄	—	109	1	no	—
301	C₂×C₄	—	109	1	no	—
302	C₂×C₄	—	436	1	no	—
303	C₂×C₄	—	436	1	no	—
304	C₂×C₄	—	436	1	no	—
305	C₂×C₄	—	218	2	no	—
306	C₂×C₄	—	218	2	no	—
307	C₂×C₄	—	218	2	no	—
308	C₂×C₄	—	218	1	no	—
309	C₂×C₄	—	218	1	no	—
310	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	218	1	no	—
321	C₈	—	218	1	no	—
322	C₈	—	218	1	no	—
323	C₈	—	218	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	109	2	no	—
328	C₂²	—	218	1	no	—
329	C₂²	—	218	1	no	—
330	C₂²	—	218	1	no	—
331	C₂²	—	218	1	no	—
332	C₂²	—	218	1	no	—
333	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	109	2	no	—
340	C₄	—	109	2	no	—
341	C₄	—	218	1	no	—
342	C₄	—	218	1	no	—
343	C₄	—	218	1	no	—
344	C₄	—	218	1	no	—
345	C₄	#41×D₁₀₉	1	1	yes	characteristic
346	C₄	#41×D₁₀₉	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	436	1	no	—
349	C₂	—	218	2	no	—
350	C₂	#117×D₁₀₉	1	1	yes	characteristic
351	C₂	#131×D₁₀₉	1	1	yes	characteristic
352	C₂	#20295791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#433×C₁₀₉	C₂
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#433	D₁₀₉
hypocenter	C₁₀₉	#2925
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #20295087	C₂	semidirect
3. #20295086	C₂	semidirect
4. #20295086	C₂	semidirect
5. #20295081	C₂	semidirect
6. #20295080	C₂	semidirect
7. #20295077	C₂	semidirect
8. #20295076	C₂	semidirect
9. #20295048	C₂	semidirect
10. C₄⋊C₈×D₁₀₉	C₂	semidirect
13. #20294983	C₂	semidirect
14. #20294980	C₂	semidirect
15. #433×C₁₀₉	C₂	semidirect
177. C₁₀₉	#2925	semidirect
182. #433	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295494	2
#20295087	2
#20295086	2
#20295081	2
#20295080	2
#20295077	2
#20295076	2
#20295048	2
C₄⋊C₈×D₁₀₉	2
#20294991	2
#20294983	2
#20294980	2
#433×C₁₀₉	2
#2925	109