Subgroups — #28366196

Order 13952.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812679	C₂	1	1	yes	maximal, characteristic
2	#20294930	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294930	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294924	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294922	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294918	C₂	1	1	yes	maximal, characteristic
7	#20294917	C₂	1	1	yes	maximal, characteristic
8	#20294916	C₂	1	1	yes	maximal, characteristic
9	#20294915	C₂	1	1	yes	maximal, characteristic
10	#20294912	C₂	1	1	yes	maximal, characteristic, complemented
11	#20294911	C₂	1	1	yes	maximal, characteristic, complemented
12	#20294909	C₂	1	1	yes	maximal, characteristic, complemented
13	#20294888	C₂	1	1	yes	maximal, characteristic
14	#20294885	C₂	1	1	yes	maximal, characteristic
15	#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812679	C₄	1	8	yes	complemented
19	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×#13182008	C₂²	1	1	yes	characteristic
23	C₂×#13182008	C₂²	1	1	yes	characteristic
24	C₂×#13182008	C₂²	1	1	yes	characteristic
25	C₂×#13182008	C₂²	1	1	yes	characteristic
26	C₂×#13182006	C₂²	1	1	yes	characteristic
27	C₂×#13182006	C₂²	1	1	yes	characteristic
28	C₂×#13182006	C₂²	1	1	yes	characteristic
29	C₂×#13182006	C₂²	1	1	yes	characteristic
30	C₂×#13182006	C₂²	1	1	yes	characteristic
31	C₂×#13182006	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	#16812606	C₂²	1	1	yes	characteristic
36	#16812606	C₂²	1	1	yes	characteristic
37	#16812606	C₂²	1	1	yes	characteristic
38	#16812606	C₂²	1	1	yes	characteristic
39	#16812606	C₂²	1	1	yes	characteristic
40	#16812606	C₂²	1	1	yes	characteristic
41	#16812606	C₂²	1	1	yes	characteristic
42	#16812606	C₂²	1	1	yes	characteristic
43	#16812606	C₂²	1	1	yes	characteristic
44	#16812606	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
48	#95×C₁₀₉	C₂²	1	1	yes	characteristic
49	#95×C₁₀₉	C₂²	1	1	yes	characteristic
50	#95×C₁₀₉	C₂²	1	1	yes	characteristic
51	#95×C₁₀₉	C₂²	1	1	yes	characteristic
52	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁₈	—	4	1	no	—
54	C₂×Dic₂₁₈	—	4	1	no	—
55	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
56	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₉	—	4	1	no	—
64	C₂×C₄×D₁₀₉	—	4	1	no	—
65	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
66	C₄⋊Dic₁₀₉	—	4	1	no	—
67	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
68	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
69	#13182008	—	4	1	no	—
70	#13182008	—	4	1	no	—
71	#13182008	—	4	1	no	—
72	#13182008	—	4	1	no	—
73	#13182006	—	4	1	no	—
74	#13182006	—	4	1	no	—
75	#13182006	—	4	1	no	—
76	#13182006	—	4	1	no	—
77	#13182006	C₂×C₄	1	4	yes	—
78	#13182006	C₂×C₄	1	4	yes	—
79	C₄×Dic₁₀₉	—	4	1	no	—
80	C₄×Dic₁₀₉	—	4	1	no	—
81	C₄×Dic₁₀₉	—	4	1	no	—
82	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₀₉	—	4	1	no	—
90	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
91	C₄²×C₁₀₉	—	4	1	no	—
92	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
93	C₂²×D₁₀₉	—	2	1	no	—
94	C₂²×D₁₀₉	—	2	1	no	—
95	C₂²×D₁₀₉	—	2	1	no	—
96	C₂²×D₁₀₉	—	2	1	no	—
97	C₂²×D₁₀₉	—	2	1	no	—
98	C₂²×D₁₀₉	—	2	1	no	—
99	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	2	1	no	—
106	C₂×C₄×C₁₀₉	—	2	1	no	—
107	C₂×C₄×C₁₀₉	—	2	1	no	—
108	C₂×C₄×C₁₀₉	—	2	1	no	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	—	2	1	no	—
136	C₂×Dic₁₀₉	—	2	1	no	—
137	C₂×Dic₁₀₉	—	2	1	no	—
138	C₂×Dic₁₀₉	—	2	1	no	—
139	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
140	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
141	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
142	C₄×D₁₀₉	—	4	4	no	—
143	Dic₂₁₈	—	4	4	no	—
144	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
145	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×D₁₀₉	—	2	2	no	—
152	C₂×D₁₀₉	—	2	2	no	—
153	C₂×D₁₀₉	—	2	2	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	—	4	1	no	—
156	C₂×D₁₀₉	—	4	1	no	—
157	C₂×D₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	—	2	2	no	—
159	C₄×C₁₀₉	—	2	2	no	—
160	C₄×C₁₀₉	—	2	2	no	—
161	C₄×C₁₀₉	—	4	1	no	—
162	C₄×C₁₀₉	—	4	1	no	—
163	C₄×C₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	2	2	no	—
166	Dic₁₀₉	—	2	2	no	—
167	Dic₁₀₉	—	2	2	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	Dic₁₀₉	—	4	1	no	—
172	C₂×C₁₀₉	#518	1	1	yes	characteristic
173	C₂×C₁₀₉	#519	1	1	yes	characteristic
174	C₂×C₁₀₉	#520	1	1	yes	characteristic
175	C₂×C₁₀₉	#562	1	1	yes	characteristic
176	C₂×C₁₀₉	#562	1	1	yes	characteristic
177	C₂×C₁₀₉	#563	1	1	yes	characteristic
178	C₂×C₁₀₉	#563	1	1	yes	characteristic
179	D₁₀₉	—	4	2	no	—
180	#2378	—	109	1	no	maximal, Sylow
181	C₁₀₉	#2378	1	1	yes	characteristic, complemented, Sylow
182	C₂×C₂²⋊Q₈	—	109	1	no	—
183	#391	—	109	1	no	—
184	#391	—	109	1	no	—
185	#389	—	109	1	no	—
186	#389	—	109	1	no	—
187	#388	—	109	1	no	—
188	#388	—	109	1	no	—
189	#388	—	109	1	no	—
190	#388	—	109	1	no	—
191	#387	—	109	1	no	—
192	#387	—	109	1	no	—
193	#387	—	109	1	no	—
194	#387	D₁₀₉	1	1	yes	characteristic, complemented
195	#381	—	109	1	no	—
196	#380	—	109	1	no	—
197	C₂²×Q₈	—	109	1	no	—
198	C₂³×C₄	—	109	1	no	—
199	C₂²⋊Q₈	—	109	8	no	—
200	C₂×C₄⋊C₄	—	109	1	no	—
201	C₂×C₄⋊C₄	—	109	1	no	—
202	C₂×C₄⋊C₄	—	109	1	no	—
203	C₂×C₄⋊C₄	—	109	1	no	—
204	C₂×C₄⋊C₄	—	109	1	no	—
205	C₂×C₄⋊C₄	—	109	1	no	—
206	C₂×C₄⋊C₄	—	109	1	no	—
207	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	109	1	no	—
210	C₂×C₂²⋊C₄	—	109	1	no	—
211	C₂×C₂²⋊C₄	—	109	1	no	—
212	C₂×C₂²⋊C₄	—	109	1	no	—
213	C₂×C₂²⋊C₄	—	109	1	no	—
214	C₂×C₂²⋊C₄	—	109	1	no	—
215	C₂×C₄²	—	109	1	no	—
216	C₂×C₄²	—	109	1	no	—
217	C₂×C₄²	—	109	1	no	—
218	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
219	#95	—	109	1	no	—
220	#95	—	109	1	no	—
221	#95	—	109	1	no	—
222	#95	—	109	1	no	—
223	#95	—	109	1	no	—
224	#95	—	109	1	no	—
225	#95	—	109	1	no	—
226	#95	—	109	1	no	—
227	#95	—	109	1	no	—
228	#95	—	109	1	no	—
229	#95	C₂×D₁₀₉	1	1	yes	characteristic
230	#95	C₂×D₁₀₉	1	1	yes	characteristic
231	#95	C₂×D₁₀₉	1	1	yes	characteristic
232	#95	C₂×D₁₀₉	1	1	yes	characteristic
233	C₂⁴	—	109	1	no	—
234	C₂×Q₈	—	109	4	no	—
235	C₂×Q₈	—	436	1	no	—
236	C₂×Q₈	—	436	1	no	—
237	C₂²×C₄	—	109	1	no	—
238	C₂²×C₄	—	109	1	no	—
239	C₂²×C₄	—	109	1	no	—
240	C₂²×C₄	—	109	4	no	—
241	C₂²×C₄	—	109	1	no	—
242	C₂²×C₄	—	109	1	no	—
243	C₂²×C₄	—	109	1	no	—
244	C₂²×C₄	—	109	1	no	—
245	C₂²×C₄	—	436	1	no	—
246	C₂²×C₄	—	436	1	no	—
247	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	109	4	no	—
256	C₄⋊C₄	—	109	4	no	—
257	C₄⋊C₄	—	436	1	no	—
258	C₄⋊C₄	—	436	1	no	—
259	C₄⋊C₄	—	436	1	no	—
260	C₄⋊C₄	—	436	1	no	—
261	C₄⋊C₄	—	436	1	no	—
262	C₄⋊C₄	C₄×D₁₀₉	1	4	yes	—
263	C₂²⋊C₄	—	109	4	no	—
264	C₂²⋊C₄	—	109	4	no	—
265	C₂²⋊C₄	—	436	1	no	—
266	C₂²⋊C₄	—	436	1	no	—
267	C₂²⋊C₄	—	436	1	no	—
268	C₂²⋊C₄	—	436	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	436	1	no	—
271	C₄²	—	436	1	no	—
272	C₄²	—	436	1	no	—
273	C₂³	—	109	2	no	—
274	C₂³	—	218	1	no	—
275	C₂³	—	218	1	no	—
276	C₂³	—	218	1	no	—
277	C₂³	—	218	1	no	—
278	C₂³	—	218	1	no	—
279	C₂³	—	218	1	no	—
280	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
281	Q₈	—	436	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	109	2	no	—
301	C₂×C₄	—	109	2	no	—
302	C₂×C₄	—	109	2	no	—
303	C₂×C₄	—	436	4	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	—	218	1	no	—
309	C₂×C₄	—	218	1	no	—
310	C₂×C₄	—	218	1	no	—
311	C₂×C₄	—	218	1	no	—
312	C₂×C₄	—	218	1	no	—
313	C₂×C₄	—	218	1	no	—
314	C₂×C₄	—	218	1	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	1	no	—
322	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
323	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
324	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	436	1	no	—
328	C₂²	—	436	1	no	—
329	C₂²	—	218	2	no	—
330	C₂²	—	218	2	no	—
331	C₂²	—	218	2	no	—
332	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
333	C₂²	#16812744	1	1	yes	characteristic
334	C₂²	#16812745	1	1	yes	characteristic
335	C₂²	#16812745	1	1	yes	characteristic
336	C₂²	#16812746	1	1	yes	characteristic
337	C₂²	#16812746	1	1	yes	characteristic
338	C₂²	#16812747	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	436	1	no	—
347	C₄	—	436	1	no	—
348	C₄	—	436	1	no	—
349	C₄	—	436	1	no	—
350	C₄	—	218	2	no	—
351	C₄	—	218	2	no	—
352	C₄	—	218	2	no	—
353	C₂	—	436	2	no	—
354	C₂	#20295500	1	1	yes	characteristic
355	C₂	#20295502	1	1	yes	characteristic
356	C₂	#20295505	1	1	yes	characteristic
357	C₂	#20295772	1	1	yes	characteristic
358	C₂	#20295773	1	1	yes	characteristic
359	C₂	#20295778	1	1	yes	characteristic
360	C₂	#20295780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator	C₂²×C₁₀₉	C₂³×C₄
Fitting	#387×C₁₀₉	C₂
socle, cosocle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	#387	D₁₀₉
hypocenter	C₁₀₉	#2378
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
2. #20294930	C₂	semidirect
3. #20294930	C₂	semidirect
4. #20294924	C₂	semidirect
5. #20294922	C₂	semidirect
10. #20294912	C₂	semidirect
11. #20294911	C₂	semidirect
12. #20294909	C₂	semidirect
15. #387×C₁₀₉	C₂	semidirect
18. #16812679	C₄	semidirect
181. C₁₀₉	#2378	semidirect
194. #387	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812679	2
#20294930	2
#20294924	2
#20294922	2
#20294918	2
#20294917	2
#20294916	2
#20294915	2
#20294912	2
#20294911	2
#20294909	2
#20294888	2
#20294885	2
#387×C₁₀₉	2
#2378	109