Subgroups — #28372827

Order 13952.

440 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	#20295582	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295323	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295315	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295314	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295305	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295304	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295048	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295029	C₂	1	1	yes	maximal, characteristic
14	#20295029	C₂	1	1	yes	maximal, characteristic
15	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₉	—	2	1	no	—
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	C₂×#13182012	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic, complemented
21	#16812698	C₂²	1	1	yes	characteristic, complemented
22	#16812691	C₂²	1	1	yes	characteristic, complemented
23	#16812677	—	2	1	no	—
24	#16812677	—	2	1	no	—
25	#16812664	—	2	1	no	—
26	#16812664	—	2	1	no	—
27	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	#16812661	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
31	#16812659	—	2	1	no	—
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812657	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
36	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic, complemented
38	#16812632	C₂²	1	1	yes	characteristic, complemented
39	#16812632	C₂²	1	1	yes	characteristic, complemented
40	#16812631	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	#125×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	#106×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×Dic₂₁₈	—	2	1	no	—
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	D₄×D₁₀₉	—	2	2	no	—
65	D₄×D₁₀₉	Q₈	1	4	yes	complemented
66	C₂²×Dic₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	4	4	no	—
71	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
74	#13182012	—	2	2	no	—
75	#13182012	—	2	2	no	—
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₂×#11724784	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂³×C₁₀₉	—	2	1	no	—
108	C₂²×D₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	—	2	4	no	—
110	C₂²×D₁₀₉	—	4	2	no	—
111	C₂²×D₁₀₉	—	4	2	no	—
112	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
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	—	4	2	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	—	4	2	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	—	2	2	no	—
133	C₄×D₁₀₉	—	4	1	no	—
134	C₄×D₁₀₉	—	4	2	no	—
135	C₄×D₁₀₉	—	4	2	no	—
136	C₄×D₁₀₉	—	4	2	no	—
137	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
138	C₄×D₁₀₉	#41	1	2	yes	—
139	Dic₂₁₈	—	4	2	no	—
140	Dic₂₁₈	—	4	2	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	1	no	—
151	C₂×D₁₀₉	—	2	2	no	—
152	C₂×D₁₀₉	—	4	8	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₂×D₁₀₉	—	4	1	no	—
155	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
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	C₂²×Q₈	1	1	yes	characteristic
162	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
168	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₁₀₉	—	4	2	no	—
170	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₁₀₉	#575	1	1	yes	characteristic
173	D₁₀₉	—	2	2	no	—
174	D₁₀₉	—	4	2	no	—
175	#3077	—	109	1	no	maximal, Sylow
176	C₁₀₉	#3077	1	1	yes	characteristic, complemented, Sylow
177	#533	—	109	1	no	—
178	C₂×C₄×D₄	—	109	1	no	—
179	#478	—	109	1	no	—
180	#478	—	109	1	no	—
181	#478	—	109	1	no	—
182	#478	D₁₀₉	1	1	yes	characteristic, complemented
183	#476	—	109	1	no	—
184	#476	—	109	1	no	—
185	#474	—	109	1	no	—
186	#474	—	109	1	no	—
187	#428	—	109	1	no	—
188	#428	—	109	1	no	—
189	#423	—	109	1	no	—
190	#418	—	109	1	no	—
191	#418	—	109	1	no	—
192	C₂²×D₄	—	109	1	no	—
193	C₂³×C₄	—	218	1	no	—
194	C₂×M₁₆	—	109	1	no	—
195	C₂×M₁₆	—	109	1	no	—
196	C₄⋊Q₈	—	109	1	no	—
197	C₄⋊Q₈	—	109	1	no	—
198	#125	—	109	1	no	—
199	#125	C₂×D₁₀₉	1	1	yes	characteristic, complemented
200	C₂²⋊Q₈	—	218	1	no	—
201	C₂²⋊Q₈	—	218	1	no	—
202	C₄×Q₈	—	218	1	no	—
203	C₄×Q₈	—	218	1	no	—
204	C₄×D₄	—	109	1	no	—
205	C₄×D₄	—	109	1	no	—
206	C₄×D₄	—	109	1	no	—
207	C₄×D₄	—	218	1	no	—
208	C₄×D₄	—	218	1	no	—
209	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
210	#117	—	109	1	no	—
211	#117	—	109	1	no	—
212	C₂×C₄⋊C₄	—	109	1	no	—
213	C₂×C₂²⋊C₄	—	218	1	no	—
214	C₂×C₄²	—	109	1	no	—
215	#107	—	109	1	no	—
216	#107	—	109	1	no	—
217	#107	—	109	1	no	—
218	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
219	#106	—	109	1	no	—
220	#106	—	109	1	no	—
221	#106	—	109	1	no	—
222	#106	C₂×D₁₀₉	1	1	yes	characteristic, complemented
223	C₄⋊C₈	—	109	1	no	—
224	C₄⋊C₈	—	109	1	no	—
225	C₄⋊C₈	—	109	1	no	—
226	C₄⋊C₈	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	#102	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	C₂×D₁₀₉	1	1	yes	characteristic
234	#102	C₂×D₁₀₉	1	1	yes	characteristic
235	C₂⁴	—	218	1	no	—
236	C₂×Q₈	—	218	1	no	—
237	C₂×Q₈	—	218	1	no	—
238	C₂×D₄	—	109	4	no	—
239	C₂×D₄	—	109	1	no	—
240	C₂×D₄	—	218	2	no	—
241	C₂×D₄	—	218	1	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	—	109	1	no	—
246	C₂²×C₄	—	109	1	no	—
247	C₂²×C₄	—	436	4	no	—
248	C₂²×C₄	—	218	1	no	—
249	C₂²×C₄	—	218	1	no	—
250	C₂²×C₄	—	218	1	no	—
251	C₂²×C₄	—	218	1	no	—
252	M₁₆	—	218	2	no	—
253	M₁₆	—	218	2	no	—
254	C₂×C₈	—	109	1	no	—
255	C₂×C₈	—	109	1	no	—
256	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	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	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	218	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	C₄²	—	109	1	no	—
278	C₄²	—	109	1	no	—
279	C₄²	—	109	1	no	—
280	C₄²	—	218	1	no	—
281	C₄²	—	218	1	no	—
282	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	109	1	no	—
285	C₂³	—	436	2	no	—
286	C₂³	—	436	2	no	—
287	C₂³	—	218	4	no	—
288	C₂³	—	218	1	no	—
289	Q₈	—	436	2	no	—
290	Q₈	—	436	2	no	—
291	D₄	—	2	1	no	—
292	D₄	—	109	2	no	—
293	D₄	—	218	4	no	—
294	D₄	—	218	1	no	—
295	D₄	Q₈×D₁₀₉	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	109	1	no	—
301	C₂×C₄	—	109	1	no	—
302	C₂×C₄	—	109	2	no	—
303	C₂×C₄	—	109	2	no	—
304	C₂×C₄	—	109	1	no	—
305	C₂×C₄	—	436	2	no	—
306	C₂×C₄	—	436	1	no	—
307	C₂×C₄	—	436	2	no	—
308	C₂×C₄	—	436	2	no	—
309	C₂×C₄	—	436	2	no	—
310	C₂×C₄	—	218	2	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	—	109	2	no	—
325	C₂²	—	436	8	no	—
326	C₂²	—	436	1	no	—
327	C₂²	—	436	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	—	109	2	no	—
338	C₄	—	109	2	no	—
339	C₄	—	436	1	no	—
340	C₄	—	436	1	no	—
341	C₄	—	436	1	no	—
342	C₄	—	218	1	no	—
343	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
344	C₄	#41×D₁₀₉	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	436	2	no	—
347	C₂	—	218	2	no	—
348	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
349	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
350	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#478×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×D₁₀₉	C₂²
hypercenter	#478	D₁₀₉
hypocenter	C₁₀₉	#3077
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #20295582	C₂	semidirect
3. #20295327	C₂	semidirect
4. #20295326	C₂	semidirect
5. #20295323	C₂	semidirect
6. #20295315	C₂	semidirect
7. #20295314	C₂	semidirect
8. #20295305	C₂	semidirect
9. #20295304	C₂	semidirect
10. #20295067	C₂	semidirect
11. #20295065	C₂	semidirect
12. #20295048	C₂	semidirect
15. #478×C₁₀₉	C₂	semidirect
20. #16812698	C₂²	semidirect
21. #16812698	C₂²	semidirect
22. #16812691	C₂²	semidirect
37. #16812633	C₂²	semidirect
38. #16812632	C₂²	semidirect
39. #16812632	C₂²	semidirect
40. #16812631	C₂²	semidirect
41. #16812630	C₂²	semidirect
42. #16812630	C₂²	semidirect
43. C₄⋊#11724784	C₂²	semidirect
44. #16812628	C₂²	semidirect
45. #16812628	C₂²	semidirect
52. #125×C₁₀₉	C₂²	semidirect
54. #107×C₁₀₉	C₂²	semidirect
55. #106×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
65. D₄×D₁₀₉	Q₈	semidirect
176. C₁₀₉	#3077	semidirect
182. #478	D₁₀₉	semidirect
199. #125	C₂×D₁₀₉	semidirect
218. #107	C₂×D₁₀₉	semidirect
222. #106	C₂×D₁₀₉	semidirect
226. 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
#20295582	2
C₄×D₄×D₁₀₉	2
#20295327	2
#20295326	2
#20295323	2
#20295315	2
#20295314	2
#20295305	2
#20295304	2
#20295067	2
#20295065	2
#20295048	2
#20295029	2
#478×C₁₀₉	2
#3077	109