Subgroups — #28379539

Order 13952.

532 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295975	C₂	1	2	yes	maximal, complemented
2	#20295975	C₂	1	2	yes	maximal, complemented
3	#20295973	C₂	1	2	yes	maximal
4	#20295971	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295970	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295969	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×#89137	C₂	1	2	yes	maximal
9	#20295962	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×(C₂×#89137)⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#117×D₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×#89137	C₂²	1	2	yes	—
13	C₂×C₄⋊#89137	C₂²	1	2	yes	—
14	C₂×C₄⋊#89137	C₂²	1	2	yes	—
15	C₂×C₂²⋊#89137	C₂²	1	2	yes	—
16	C₂×C₄×#89137	C₂²	1	2	yes	—
17	C₂×C₄×#89137	C₂²	1	2	yes	—
18	(C₂×#89137)⋊C₄	C₂²	1	1	yes	characteristic
19	(C₂×#89137)⋊C₄	C₂²	1	1	yes	characteristic
20	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
21	(C₂×#89137)⋊C₄	C₂²	1	1	yes	characteristic
22	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
23	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
24	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
25	(C₂×#89137)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
27	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
28	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
29	#16812658	C₄	1	1	yes	characteristic, complemented
30	#16812657	C₄	1	1	yes	characteristic, complemented
31	#16812657	C₄	1	1	yes	characteristic, complemented
32	#16812656	C₄	1	1	yes	characteristic, complemented
33	#16812656	C₄	1	1	yes	characteristic, complemented
34	#16812655	C₄	1	1	yes	characteristic, complemented
35	#16812655	C₄	1	1	yes	characteristic, complemented
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
40	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
41	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
42	#117×C₁₀₉	C₄	1	1	yes	characteristic, complemented
43	C₂²×#89137	—	2	4	no	—
44	C₂²×#89137	—	2	4	no	—
45	C₂²×#89137	—	4	2	no	—
46	C₂²×#89137	C₂³	1	2	yes	—
47	C₂²×#89137	C₂³	1	2	yes	—
48	C₂²×#89137	C₂³	1	2	yes	—
49	C₂²×#89137	C₂³	1	2	yes	—
50	C₄⋊#89137	—	2	4	no	—
51	C₄⋊#89137	—	4	2	no	—
52	C₂²⋊#89137	—	2	4	no	—
53	C₄×#89137	—	2	4	no	—
54	C₄×#89137	—	4	2	no	—
55	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₁₀₉	—	2	4	no	—
58	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
67	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
68	#13182008	C₂×C₄	1	1	yes	characteristic, complemented
69	#13182008	C₂×C₄	1	1	yes	characteristic, complemented
70	#13182008	C₂×C₄	1	1	yes	characteristic, complemented
71	#13182008	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂²⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
74	#13182006	C₂×C₄	1	1	yes	characteristic
75	#13182006	C₂×C₄	1	1	yes	characteristic
76	#13182006	C₂×C₄	1	1	yes	characteristic
77	#13182006	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
79	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
80	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
81	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
82	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
83	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
86	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
87	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
89	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
90	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×D₁₀₉	—	2	2	no	—
93	C₂²×D₁₀₉	—	2	1	no	—
94	C₂²×D₁₀₉	—	2	1	no	—
95	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₁₀₉	#41	1	4	yes	—
97	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂×#89137	—	2	2	no	—
99	C₂×#89137	—	2	2	no	—
100	C₂×#89137	—	2	2	no	—
101	C₂×#89137	—	2	2	no	—
102	C₂×#89137	—	2	2	no	—
103	C₂×#89137	—	2	2	no	—
104	C₂×#89137	—	2	2	no	—
105	C₂×#89137	—	2	2	no	—
106	C₂×#89137	—	2	2	no	—
107	C₂×#89137	—	2	2	no	—
108	C₂×#89137	—	2	2	no	—
109	C₂×#89137	—	4	8	no	—
110	C₂×#89137	#41	1	4	yes	—
111	C₂×C₄×C₁₀₉	—	2	2	no	—
112	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
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×Dic₁₀₉	—	2	2	no	—
119	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
125	C₄×D₁₀₉	—	2	2	no	—
126	C₄×D₁₀₉	—	2	2	no	—
127	C₄×D₁₀₉	—	2	2	no	—
128	C₄×D₁₀₉	—	2	2	no	—
129	C₄×D₁₀₉	—	2	2	no	—
130	C₄×D₁₀₉	—	2	2	no	—
131	C₄×D₁₀₉	—	2	1	no	—
132	C₄×D₁₀₉	—	2	1	no	—
133	C₄×D₁₀₉	—	2	1	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₂²×C₁₀₉	—	2	1	no	—
136	C₂²×C₁₀₉	—	2	1	no	—
137	C₂²×C₁₀₉	#117	1	2	yes	—
138	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	—	2	1	no	—
141	C₂×D₁₀₉	—	2	4	no	—
142	C₂×D₁₀₉	—	2	4	no	—
143	C₂×D₁₀₉	#117	1	2	yes	—
144	C₂×D₁₀₉	C₂×#41	1	2	yes	—
145	C₂×D₁₀₉	C₂×#41	1	2	yes	—
146	C₂×D₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×D₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
148	#89137	—	2	4	no	—
149	#89137	—	4	2	no	—
150	#89137	—	4	2	no	—
151	#89137	—	4	2	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	2	1	no	—
159	Dic₁₀₉	—	2	1	no	—
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	2	1	no	—
163	Dic₁₀₉	—	2	1	no	—
164	C₂×C₁₀₉	—	2	2	no	—
165	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
166	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
167	C₂×C₁₀₉	#520	1	1	yes	characteristic
168	D₁₀₉	—	2	2	no	—
169	D₁₀₉	#551	1	2	yes	—
170	D₁₀₉	#552	1	2	yes	—
171	#2343	—	109	1	no	maximal, Sylow
172	C₁₀₉	#2343	1	1	yes	characteristic, complemented, Sylow
173	C₂×#117	—	109	1	no	—
174	#387	—	109	2	no	—
175	#387	—	109	2	no	—
176	#385	—	109	2	no	—
177	#383	—	109	1	no	—
178	#382	—	109	1	no	—
179	#381	—	109	1	no	—
180	C₄×C₄⋊C₄	—	109	1	no	—
181	C₄×C₂²⋊C₄	—	109	2	no	—
182	#376	—	109	1	no	—
183	C₂×#95	—	109	1	no	—
184	C₂³×C₄	—	109	1	no	—
185	C₂³×C₄	—	109	2	no	—
186	#117	—	109	1	no	—
187	#117	—	109	1	no	—
188	#117	—	109	1	no	—
189	#117	—	109	1	no	—
190	#117	—	109	1	no	—
191	#117	—	109	1	no	—
192	#117	—	109	1	no	—
193	#117	#89137	1	1	yes	characteristic, complemented
194	C₂×C₄⋊C₄	—	109	1	no	—
195	C₂×C₄⋊C₄	—	109	1	no	—
196	C₂×C₄⋊C₄	—	109	2	no	—
197	C₂×C₄⋊C₄	—	109	2	no	—
198	C₂×C₂²⋊C₄	—	109	1	no	—
199	C₂×C₂²⋊C₄	—	109	1	no	—
200	C₂×C₂²⋊C₄	—	109	2	no	—
201	C₂×C₄²	—	109	1	no	—
202	C₂×C₄²	—	109	1	no	—
203	C₂×C₄²	—	109	2	no	—
204	C₂×C₄²	—	109	2	no	—
205	#95	—	109	1	no	—
206	#95	—	109	1	no	—
207	#95	—	109	2	no	—
208	#95	—	109	1	no	—
209	#95	—	109	2	no	—
210	#95	—	109	2	no	—
211	#95	—	109	2	no	—
212	#95	—	109	1	no	—
213	#95	—	109	2	no	—
214	#95	—	109	2	no	—
215	C₂⁴	—	109	1	no	—
216	C₂²×C₄	—	109	2	no	—
217	C₂²×C₄	—	109	1	no	—
218	C₂²×C₄	—	109	1	no	—
219	C₂²×C₄	—	109	1	no	—
220	C₂²×C₄	—	109	1	no	—
221	C₂²×C₄	—	109	1	no	—
222	C₂²×C₄	—	109	1	no	—
223	C₂²×C₄	—	109	1	no	—
224	C₂²×C₄	—	109	1	no	—
225	C₂²×C₄	—	109	1	no	—
226	C₂²×C₄	—	109	2	no	—
227	C₂²×C₄	—	109	2	no	—
228	C₂²×C₄	—	109	2	no	—
229	C₂²×C₄	—	436	2	no	—
230	C₂²×C₄	—	218	4	no	—
231	C₂²×C₄	—	218	4	no	—
232	C₂²×C₄	—	218	4	no	—
233	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
234	C₄⋊C₄	—	109	1	no	—
235	C₄⋊C₄	—	109	1	no	—
236	C₄⋊C₄	—	109	1	no	—
237	C₄⋊C₄	—	109	1	no	—
238	C₄⋊C₄	—	109	1	no	—
239	C₄⋊C₄	—	109	1	no	—
240	C₄⋊C₄	—	436	2	no	—
241	C₄⋊C₄	—	218	4	no	—
242	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic, complemented
243	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic, complemented
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	—	218	4	no	—
251	C₂²⋊C₄	C₂×#89137	1	1	yes	characteristic
252	C₂²⋊C₄	C₂×#89137	1	1	yes	characteristic
253	C₄²	—	109	1	no	—
254	C₄²	—	109	1	no	—
255	C₄²	—	109	1	no	—
256	C₄²	—	109	1	no	—
257	C₄²	—	109	1	no	—
258	C₄²	—	109	1	no	—
259	C₄²	—	436	2	no	—
260	C₄²	—	218	4	no	—
261	C₄²	C₂×#89137	1	1	yes	characteristic, complemented
262	C₄²	C₂×#89137	1	1	yes	characteristic, complemented
263	C₂³	—	109	1	no	—
264	C₂³	—	109	4	no	—
265	C₂³	—	109	1	no	—
266	C₂³	—	218	1	no	—
267	C₂³	—	218	2	no	—
268	C₂³	—	218	1	no	—
269	C₂³	C₂²×#89137	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	109	4	no	—
272	C₂×C₄	—	109	1	no	—
273	C₂×C₄	—	109	1	no	—
274	C₂×C₄	—	109	1	no	—
275	C₂×C₄	—	109	1	no	—
276	C₂×C₄	—	109	1	no	—
277	C₂×C₄	—	109	1	no	—
278	C₂×C₄	—	436	8	no	—
279	C₂×C₄	—	218	1	no	—
280	C₂×C₄	—	218	2	no	—
281	C₂×C₄	—	218	1	no	—
282	C₂×C₄	—	218	2	no	—
283	C₂×C₄	—	218	2	no	—
284	C₂×C₄	—	218	2	no	—
285	C₂×C₄	—	218	2	no	—
286	C₂×C₄	—	218	2	no	—
287	C₂×C₄	—	218	1	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	2	no	—
290	C₂×C₄	—	218	2	no	—
291	C₂×C₄	—	218	2	no	—
292	C₂×C₄	—	218	2	no	—
293	C₂×C₄	—	218	2	no	—
294	C₂×C₄	—	218	2	no	—
295	C₂×C₄	—	218	2	no	—
296	C₂×C₄	—	218	2	no	—
297	C₂×C₄	—	218	2	no	—
298	C₂×C₄	—	218	2	no	—
299	C₂×C₄	—	218	2	no	—
300	C₂×C₄	—	218	2	no	—
301	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
302	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
303	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
304	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
305	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
306	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	109	2	no	—
310	C₂²	—	109	2	no	—
311	C₂²	—	109	1	no	—
312	C₂²	—	109	1	no	—
313	C₂²	—	109	2	no	—
314	C₂²	—	218	4	no	—
315	C₂²	—	218	4	no	—
316	C₂²	—	218	1	no	—
317	C₂²	—	218	1	no	—
318	C₂²	#16812778	1	2	yes	—
319	C₂²	C₂³×#89137	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	436	2	no	—
327	C₄	—	436	2	no	—
328	C₄	—	436	2	no	—
329	C₄	—	218	4	no	—
330	C₄	—	218	1	no	—
331	C₄	—	218	1	no	—
332	C₄	—	218	1	no	—
333	C₄	—	218	1	no	—
334	C₄	—	218	1	no	—
335	C₄	—	218	1	no	—
336	C₂	—	2	2	no	—
337	C₂	—	109	2	no	—
338	C₂	—	109	2	no	—
339	C₂	—	218	2	no	—
340	C₂	C₂×#16812778	1	1	yes	characteristic
341	C₂	#41×#89137	1	1	yes	characteristic
342	C₂	#20296095	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#117×C₁₀₉	C₄
prime radical	C₂³×D₁₀₉	C₂³
cosocle	C₂²×D₁₀₉	C₂⁴
hypercenter	#117	#89137
hypocenter	C₁₀₉	#2343
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #20295975	C₂	semidirect
2. #20295975	C₂	semidirect
4. #20295971	C₂	semidirect
5. #20295970	C₂	semidirect
7. C₄×C₄⋊#89137	C₂	semidirect
9. #20295962	C₂	semidirect
29. #16812658	C₄	semidirect
30. #16812657	C₄	semidirect
31. #16812657	C₄	semidirect
32. #16812656	C₄	semidirect
33. #16812656	C₄	semidirect
34. #16812655	C₄	semidirect
35. #16812655	C₄	semidirect
42. #117×C₁₀₉	C₄	semidirect
66. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
67. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
68. #13182008	C₂×C₄	semidirect
69. #13182008	C₂×C₄	semidirect
70. #13182008	C₂×C₄	semidirect
71. #13182008	C₂×C₄	semidirect
78. C₄×Dic₁₀₉	C₂×C₄	semidirect
79. C₄×Dic₁₀₉	C₂×C₄	semidirect
80. C₄×Dic₁₀₉	C₂×C₄	semidirect
81. C₄×Dic₁₀₉	C₂×C₄	semidirect
82. C₄×Dic₁₀₉	C₂×C₄	semidirect
83. C₄×Dic₁₀₉	C₂×C₄	semidirect
85. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
86. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
89. C₄²×C₁₀₉	C₂×C₄	semidirect
90. C₄²×C₁₀₉	C₂×C₄	semidirect
172. C₁₀₉	#2343	semidirect
193. #117	#89137	semidirect
242. C₄⋊C₄	C₂×#89137	semidirect
243. C₄⋊C₄	C₂×#89137	semidirect
261. C₄²	C₂×#89137	semidirect
262. C₄²	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
#20295975	2
#20295973	2
#20295971	2
#20295970	2
#20295969	2
C₄×C₄⋊#89137	2
C₂²⋊C₄×#89137	2
#20295962	2
C₂×(C₂×#89137)⋊C₄	2
#117×D₁₀₉	2
#2343	109