Subgroups — #28373282

Order 13952.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295637	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295498	C₂	1	1	yes	maximal, characteristic
3	#20295366	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295365	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295228	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295175	C₂	1	1	yes	maximal, characteristic
9	#20295175	C₂	1	1	yes	maximal, characteristic
10	#20295129	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	#20295111	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
15	#461×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	C₂×#13182016	C₂²	1	1	yes	characteristic
19	#16812694	—	2	1	no	—
20	#16812694	—	2	1	no	—
21	#16812691	C₂²	1	1	yes	characteristic, complemented
22	#16812689	—	2	1	no	—
23	#16812689	—	2	1	no	—
24	#16812675	C₂²	1	1	yes	characteristic
25	#16812675	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
27	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
28	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	#16812661	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
32	#16812659	—	2	1	no	—
33	#16812657	C₂²	1	1	yes	characteristic
34	#16812656	C₂²	1	1	yes	characteristic
35	C₂×#13182006	—	2	1	no	—
36	C₂²⋊C₄×D₁₀₉	—	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	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
42	#16812623	C₂²	1	1	yes	characteristic, complemented
43	#16812623	C₂²	1	1	yes	characteristic, complemented
44	#16812621	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812611	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
58	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₀₉	—	2	1	no	—
60	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	D₄×D₁₀₉	—	2	2	no	—
63	D₄×D₁₀₉	—	2	2	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
68	#13182016	—	2	2	no	—
69	#13182016	—	2	2	no	—
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	—	2	1	no	—
73	C₄⋊Dic₁₀₉	—	2	1	no	—
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	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	D₄	1	1	yes	characteristic
82	#13182008	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	D₄	1	1	yes	characteristic
90	#13182006	D₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	—	2	1	no	—
99	SD₁₆×C₁₀₉	—	2	2	no	—
100	SD₁₆×C₁₀₉	—	2	2	no	—
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	—	2	1	no	—
104	C₄⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	4	no	—
111	C₂²×D₁₀₉	—	8	1	no	—
112	C₂²×D₁₀₉	—	8	1	no	—
113	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	#41	1	2	yes	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	8	1	no	—
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₀₉	—	2	4	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	8	1	no	—
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	#41	1	2	yes	—
137	C₄×D₁₀₉	—	2	1	no	—
138	C₄×D₁₀₉	—	2	1	no	—
139	C₄×D₁₀₉	—	8	1	no	—
140	C₄×D₁₀₉	—	8	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	C₂²×C₁₀₉	—	2	2	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₀₉	—	2	1	no	—
150	C₂×D₁₀₉	—	2	2	no	—
151	C₂×D₁₀₉	—	2	1	no	—
152	C₂×D₁₀₉	—	2	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	—	8	4	no	—
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	—	2	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	C₂×C₁₀₉	—	4	2	no	—
170	C₂×C₁₀₉	#525	1	1	yes	characteristic
171	C₂×C₁₀₉	#577	1	1	yes	characteristic
172	C₂×C₁₀₉	#577	1	1	yes	characteristic
173	D₁₀₉	—	4	1	no	—
174	D₁₀₉	—	4	2	no	—
175	#3123	—	109	1	no	maximal, Sylow
176	C₁₀₉	#3123	1	1	yes	characteristic, complemented, Sylow
177	#541	—	109	1	no	—
178	#518	—	109	1	no	—
179	#485	—	109	1	no	—
180	#485	—	109	1	no	—
181	#478	—	109	1	no	—
182	#478	—	109	1	no	—
183	#461	—	109	1	no	—
184	#461	D₁₀₉	1	1	yes	characteristic, complemented
185	#450	—	109	1	no	—
186	#450	—	109	1	no	—
187	#440	—	109	1	no	—
188	#440	—	109	1	no	—
189	C₄×SD₁₆	—	109	1	no	—
190	C₄×SD₁₆	—	109	1	no	—
191	#433	—	109	1	no	—
192	C₂²×D₄	—	109	1	no	—
193	C₂×SD₁₆	—	109	1	no	—
194	C₂×SD₁₆	—	109	1	no	—
195	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
196	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
198	#126	—	218	1	no	—
199	#126	—	218	1	no	—
200	#125	—	109	1	no	—
201	#125	—	218	1	no	—
202	#125	—	218	1	no	—
203	C₂²⋊Q₈	—	109	1	no	—
204	C₂²⋊Q₈	—	109	1	no	—
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×D₄	—	109	1	no	—
208	C₄×D₄	—	109	1	no	—
209	C₄×D₄	—	109	1	no	—
210	C₄×D₄	—	218	1	no	—
211	C₄×D₄	—	218	1	no	—
212	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
213	#117	—	109	1	no	—
214	#117	—	109	1	no	—
215	C₂×C₂²⋊C₄	—	218	1	no	—
216	C₂×C₂²⋊C₄	—	218	1	no	—
217	#107	—	109	1	no	—
218	#107	—	109	1	no	—
219	#106	—	109	1	no	—
220	#106	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
223	#103	—	109	1	no	—
224	#103	—	109	1	no	—
225	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
226	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	#102	—	109	1	no	—
231	C₂²⋊C₈	—	109	1	no	—
232	C₂²⋊C₈	—	109	1	no	—
233	#97	—	109	1	no	—
234	C₄×C₈	—	109	1	no	—
235	C₂⁴	—	218	1	no	—
236	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂×D₄	—	109	1	no	—
239	C₂×D₄	—	218	1	no	—
240	C₂×D₄	—	218	2	no	—
241	C₂×D₄	—	218	2	no	—
242	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	109	1	no	—
245	C₂²×C₄	—	218	1	no	—
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	218	2	no	—
251	SD₁₆	—	218	2	no	—
252	C₂×C₈	—	109	1	no	—
253	C₂×C₈	—	109	1	no	—
254	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	109	1	no	—
260	C₄⋊C₄	—	109	1	no	—
261	C₄⋊C₄	—	109	1	no	—
262	C₄⋊C₄	—	109	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	—	218	1	no	—
265	C₄⋊C₄	—	218	1	no	—
266	C₄⋊C₄	—	218	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	109	1	no	—
272	C₂²⋊C₄	—	109	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₄²	—	109	1	no	—
280	C₄²	—	218	1	no	—
281	C₄²	—	218	1	no	—
282	C₄²	—	218	1	no	—
283	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	—	872	1	no	—
286	C₂³	—	872	1	no	—
287	C₂³	—	109	1	no	—
288	C₂³	—	218	4	no	—
289	C₂³	—	218	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	109	2	no	—
296	D₄	—	218	4	no	—
297	D₄	—	218	1	no	—
298	D₄	#13182028	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	872	1	no	—
304	C₂×C₄	—	872	1	no	—
305	C₂×C₄	—	872	1	no	—
306	C₂×C₄	—	109	1	no	—
307	C₂×C₄	—	109	1	no	—
308	C₂×C₄	—	109	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	D₄×D₁₀₉	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	218	1	no	—
320	C₈	—	218	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	872	4	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	218	1	no	—
328	C₂²	—	218	2	no	—
329	C₂²	—	218	1	no	—
330	C₂²	—	218	1	no	—
331	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	436	1	no	—
338	C₄	—	436	1	no	—
339	C₄	—	436	1	no	—
340	C₄	—	218	1	no	—
341	C₄	—	218	1	no	—
342	C₄	—	218	1	no	—
343	C₄	#41×D₁₀₉	1	1	yes	characteristic
344	C₄	C₂×#13182028	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	436	1	no	—
347	C₂	—	436	2	no	—
348	C₂	#20295536	1	1	yes	characteristic
349	C₂	#20295832	1	1	yes	characteristic
350	C₂	#20295835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#461×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#461	D₁₀₉
hypocenter	C₁₀₉	#3123
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
1. #20295637	C₂	semidirect
3. #20295366	C₂	semidirect
4. #20295365	C₂	semidirect
5. #20295327	C₂	semidirect
6. #20295326	C₂	semidirect
7. #20295228	C₂	semidirect
10. #20295129	C₂	semidirect
11. #20295127	C₂	semidirect
12. SD₁₆×Dic₁₀₉	C₂	semidirect
13. #20295111	C₂	semidirect
14. #20295086	C₂	semidirect
15. #461×C₁₀₉	C₂	semidirect
21. #16812691	C₂²	semidirect
26. Q₈×Dic₁₀₉	C₂²	semidirect
27. Q₈×Dic₁₀₉	C₂²	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. C₄⋊#11724784	C₂²	semidirect
42. #16812623	C₂²	semidirect
43. #16812623	C₂²	semidirect
50. #16812609	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
54. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
57. #103×C₁₀₉	C₂²	semidirect
58. #103×C₁₀₉	C₂²	semidirect
176. C₁₀₉	#3123	semidirect
184. #461	D₁₀₉	semidirect
197. C₄⋊Q₈	C₂×D₁₀₉	semidirect
222. C₄⋊C₈	C₂×D₁₀₉	semidirect
225. #103	C₂×D₁₀₉	semidirect
226. #103	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295637	2
#20295498	2
#20295366	2
#20295365	2
#20295327	2
#20295326	2
#20295228	2
#20295175	2
#20295129	2
#20295127	2
SD₁₆×Dic₁₀₉	2
#20295111	2
#20295086	2
#461×C₁₀₉	2
#3123	109