Subgroups — #28372856

Order 13952.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295653	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295498	C₂	1	1	yes	maximal, characteristic
3	#20295329	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295324	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295314	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295313	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295305	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295063	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295051	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295029	C₂	1	1	yes	maximal, characteristic
14	#102×D₁₀₉	C₂	1	1	yes	maximal, characteristic
15	#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	#16812698	C₂²	1	1	yes	characteristic, complemented
20	#16812697	—	2	1	no	—
21	#16812691	C₂²	1	1	yes	characteristic, complemented
22	#16812690	C₂²	1	1	yes	characteristic, complemented
23	#16812689	—	2	1	no	—
24	#16812677	—	2	1	no	—
25	#16812675	—	2	1	no	—
26	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
27	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	#16812661	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
30	#16812659	—	2	1	no	—
31	#16812657	C₂²	1	1	yes	characteristic
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812656	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₀₉	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	#16812630	C₂²	1	1	yes	characteristic, complemented
42	#16812630	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
44	#16812628	C₂²	1	1	yes	characteristic, complemented
45	#16812628	C₂²	1	1	yes	characteristic, complemented
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812619	C₂²	1	1	yes	characteristic
51	#16812619	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂³×D₁₀₉	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	D₄×D₁₀₉	—	2	2	no	—
64	D₄×D₁₀₉	Q₈	1	4	yes	complemented
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	#13182012	—	2	2	no	—
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₈×D₁₀₉	—	2	2	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	D₄	1	1	yes	characteristic
84	#13182008	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	D₄	1	1	yes	characteristic
91	#13182006	D₄	1	1	yes	characteristic
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	—	2	1	no	—
96	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂³×C₁₀₉	—	2	1	no	—
107	C₂²×D₁₀₉	—	2	1	no	—
108	C₂²×D₁₀₉	—	2	4	no	—
109	C₂²×D₁₀₉	—	8	1	no	—
110	C₂²×D₁₀₉	—	8	1	no	—
111	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₀₉	—	8	1	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
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	—	8	1	no	—
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₀₉	—	2	4	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	8	1	no	—
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	C₂×Q₈	1	2	yes	—
132	C₄×D₁₀₉	—	8	1	no	—
133	C₄×D₁₀₉	—	8	1	no	—
134	C₄×D₁₀₉	—	8	1	no	—
135	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
136	C₄×D₁₀₉	#41	1	2	yes	—
137	Dic₂₁₈	—	8	1	no	—
138	C₈×C₁₀₉	—	2	1	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	#11724784	—	2	1	no	—
141	#11724784	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	2	no	—
143	C₂²×C₁₀₉	—	4	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₀₉	—	2	1	no	—
147	C₂×D₁₀₉	—	2	1	no	—
148	C₂×D₁₀₉	—	2	2	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	8	4	no	—
152	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
159	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₁₀₉	—	4	2	no	—
167	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₁₀₉	#576	1	1	yes	characteristic
169	C₂×C₁₀₉	#577	1	1	yes	characteristic
170	D₁₀₉	—	2	2	no	—
171	D₁₀₉	—	4	2	no	—
172	#3081	—	109	1	no	maximal, Sylow
173	C₁₀₉	#3081	1	1	yes	characteristic, complemented, Sylow
174	#544	—	109	1	no	—
175	#518	—	109	1	no	—
176	#478	—	109	1	no	—
177	#478	—	109	1	no	—
178	#478	—	109	1	no	—
179	#478	—	109	1	no	—
180	#476	—	109	1	no	—
181	#476	—	109	1	no	—
182	#474	—	109	1	no	—
183	#474	D₁₀₉	1	1	yes	characteristic, complemented
184	#428	—	109	1	no	—
185	#427	—	109	1	no	—
186	#424	—	109	1	no	—
187	#418	—	109	1	no	—
188	C₂×#102	—	109	1	no	—
189	C₂²×D₄	—	109	1	no	—
190	C₂×M₁₆	—	109	1	no	—
191	C₂²×C₈	—	109	1	no	—
192	C₄⋊Q₈	—	109	1	no	—
193	C₄⋊Q₈	—	218	1	no	—
194	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
195	#125	—	109	1	no	—
196	#125	—	109	1	no	—
197	#125	—	218	1	no	—
198	C₂²⋊Q₈	—	218	1	no	—
199	C₂²⋊Q₈	—	218	1	no	—
200	C₄×D₄	—	109	1	no	—
201	C₄×D₄	—	109	1	no	—
202	C₄×D₄	—	109	1	no	—
203	C₄×D₄	—	218	1	no	—
204	C₄×D₄	—	218	1	no	—
205	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
206	#117	—	109	1	no	—
207	#117	—	109	1	no	—
208	#117	—	109	1	no	—
209	C₂×C₄⋊C₄	—	109	1	no	—
210	C₂×C₂²⋊C₄	—	218	1	no	—
211	C₂×C₂²⋊C₄	—	218	1	no	—
212	#107	—	109	1	no	—
213	#107	—	109	1	no	—
214	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
215	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
216	#106	—	109	1	no	—
217	#106	—	109	1	no	—
218	#106	—	109	1	no	—
219	#106	—	109	1	no	—
220	C₄⋊C₈	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	—	109	1	no	—
223	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	#102	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	#102	C₂×D₁₀₉	1	1	yes	characteristic
231	#102	C₂×D₁₀₉	1	1	yes	characteristic
232	C₂⁴	—	218	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	218	1	no	—
235	C₂×D₄	—	109	4	no	—
236	C₂×D₄	—	109	1	no	—
237	C₂×D₄	—	218	2	no	—
238	C₂×D₄	—	218	1	no	—
239	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	109	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	C₂²×C₄	—	218	1	no	—
244	C₂²×C₄	—	218	1	no	—
245	C₂²×C₄	—	218	1	no	—
246	M₁₆	—	218	2	no	—
247	C₂×C₈	—	109	1	no	—
248	C₂×C₈	—	109	1	no	—
249	C₂×C₈	—	218	2	no	—
250	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	109	1	no	—
254	C₄⋊C₄	—	109	1	no	—
255	C₄⋊C₄	—	109	1	no	—
256	C₄⋊C₄	—	109	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	—	218	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	109	1	no	—
269	C₂²⋊C₄	—	109	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₄²	—	109	1	no	—
276	C₄²	—	218	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	872	1	no	—
281	C₂³	—	872	1	no	—
282	C₂³	—	109	1	no	—
283	C₂³	—	218	4	no	—
284	C₂³	—	218	1	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	872	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	109	2	no	—
289	D₄	—	218	4	no	—
290	D₄	—	218	1	no	—
291	D₄	Q₈×D₁₀₉	1	2	yes	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	872	1	no	—
297	C₂×C₄	—	872	1	no	—
298	C₂×C₄	—	872	1	no	—
299	C₂×C₄	—	872	1	no	—
300	C₂×C₄	—	109	1	no	—
301	C₂×C₄	—	109	2	no	—
302	C₂×C₄	—	109	2	no	—
303	C₂×C₄	—	109	1	no	—
304	C₂×C₄	—	109	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	218	1	no	—
314	C₈	—	218	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	872	4	no	—
319	C₂²	—	109	2	no	—
320	C₂²	—	436	1	no	—
321	C₂²	—	436	1	no	—
322	C₂²	—	218	2	no	—
323	C₂²	—	218	1	no	—
324	C₂²	—	218	1	no	—
325	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	109	2	no	—
332	C₄	—	436	1	no	—
333	C₄	—	436	1	no	—
334	C₄	—	436	1	no	—
335	C₄	—	218	1	no	—
336	C₄	—	218	1	no	—
337	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
338	C₄	#41×D₁₀₉	1	1	yes	characteristic
339	C₂	—	4	2	no	—
340	C₂	—	436	2	no	—
341	C₂	—	218	2	no	—
342	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
343	C₂	#20295827	1	1	yes	characteristic
344	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	#474×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#474	D₁₀₉
hypocenter	C₁₀₉	#3081
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #20295653	C₂	semidirect
3. #20295329	C₂	semidirect
4. #20295327	C₂	semidirect
5. #20295326	C₂	semidirect
6. #20295324	C₂	semidirect
7. #20295314	C₂	semidirect
8. #20295313	C₂	semidirect
9. #20295305	C₂	semidirect
10. #20295067	C₂	semidirect
11. #20295063	C₂	semidirect
12. #20295051	C₂	semidirect
15. #474×C₁₀₉	C₂	semidirect
19. #16812698	C₂²	semidirect
21. #16812691	C₂²	semidirect
22. #16812690	C₂²	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812631	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. #16812630	C₂²	semidirect
42. #16812630	C₂²	semidirect
43. C₄⋊#11724784	C₂²	semidirect
44. #16812628	C₂²	semidirect
45. #16812628	C₂²	semidirect
52. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
54. #107×C₁₀₉	C₂²	semidirect
55. #107×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
64. D₄×D₁₀₉	Q₈	semidirect
173. C₁₀₉	#3081	semidirect
183. #474	D₁₀₉	semidirect
194. C₄⋊Q₈	C₂×D₁₀₉	semidirect
214. #107	C₂×D₁₀₉	semidirect
215. #107	C₂×D₁₀₉	semidirect
223. C₄⋊C₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295653	2
#20295498	2
#20295329	2
#20295327	2
#20295326	2
#20295324	2
#20295314	2
#20295313	2
#20295305	2
#20295067	2
#20295063	2
#20295051	2
#20295029	2
#102×D₁₀₉	2
#474×C₁₀₉	2
#3081	109