Subgroups — #28375742

Order 13952.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295732	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295697	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295392	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295389	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295371	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295329	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295315	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295222	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295220	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295220	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295155	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295137	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295136	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295108	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182029	—	2	1	no	—
17	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
18	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812696	C₂²	1	1	yes	characteristic
22	#16812695	C₂²	1	1	yes	characteristic
23	#16812695	C₂²	1	1	yes	characteristic
24	#16812690	C₂²	1	1	yes	characteristic
25	#16812679	—	2	1	no	—
26	#16812677	—	2	1	no	—
27	#16812677	—	2	1	no	—
28	#16812673	—	2	1	no	—
29	#16812672	—	2	1	no	—
30	#16812672	—	2	1	no	—
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
33	#16812661	C₂²	1	1	yes	characteristic
34	#16812661	C₂²	1	1	yes	characteristic
35	#16812661	C₂²	1	1	yes	characteristic
36	#16812661	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₀₉	—	2	1	no	—
38	#16812630	C₂²	1	1	yes	characteristic
39	#16812628	C₂²	1	1	yes	characteristic
40	#16812628	C₂²	1	1	yes	characteristic
41	#16812621	C₂²	1	1	yes	characteristic
42	#16812621	C₂²	1	1	yes	characteristic
43	#16812620	C₂²	1	1	yes	characteristic
44	#16812619	C₂²	1	1	yes	characteristic
45	#16812619	C₂²	1	1	yes	characteristic
46	#16812619	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812610	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182029	—	4	4	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	—	2	1	no	—
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
66	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
69	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
70	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	—	2	1	no	—
73	C₂×C₄×D₁₀₉	—	2	1	no	—
74	C₂×C₄×D₁₀₉	—	2	1	no	—
75	C₂×C₄×D₁₀₉	—	2	1	no	—
76	D₈₇₂	—	2	2	no	—
77	D₈₇₂	—	2	2	no	—
78	#13182014	—	2	2	no	—
79	C₂×#11724784	C₂³	1	1	yes	characteristic
80	C₂×#11724784	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₀₉	—	2	1	no	—
82	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
83	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	D₄	1	1	yes	characteristic
87	#13182008	D₄	1	1	yes	characteristic
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	1	no	—
109	C₂²×D₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	—	2	1	no	—
111	D₄×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	—	4	2	no	—
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	4	2	no	—
120	C₂²⋊D₁₀₉	—	4	2	no	—
121	C₂²⋊D₁₀₉	—	8	1	no	—
122	C₂²⋊D₁₀₉	—	8	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	D₄₃₆	—	2	1	no	—
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	—	2	1	no	—
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	—	4	2	no	—
134	D₄₃₆	—	8	1	no	—
135	D₄₃₆	#41	1	2	yes	complemented
136	C₄×D₁₀₉	—	4	2	no	—
137	C₄×D₁₀₉	—	4	2	no	—
138	C₄×D₁₀₉	—	4	2	no	—
139	C₄×D₁₀₉	—	8	1	no	—
140	C₄×D₁₀₉	—	8	1	no	—
141	Dic₂₁₈	—	2	2	no	—
142	Dic₂₁₈	—	8	1	no	—
143	C₈×C₁₀₉	—	2	1	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	#11724784	—	2	1	no	—
146	#11724784	—	2	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	—	4	1	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₀₉	—	2	2	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₂×D₁₀₉	—	4	1	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	C₂×D₁₀₉	—	4	1	no	—
159	C₂×D₁₀₉	—	4	1	no	—
160	C₄×C₁₀₉	—	2	1	no	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	—	4	1	no	—
163	C₄×C₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
167	Dic₁₀₉	—	2	1	no	—
168	Dic₁₀₉	—	2	1	no	—
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₁₀₉	—	8	1	no	—
173	C₂×C₁₀₉	#548	1	1	yes	characteristic
174	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₁₀₉	#576	1	1	yes	characteristic
176	D₁₀₉	—	4	2	no	—
177	D₁₀₉	—	8	1	no	—
178	D₁₀₉	—	8	1	no	—
179	#3366	—	109	1	no	maximal, Sylow
180	C₁₀₉	#3366	1	1	yes	characteristic, complemented, Sylow
181	#554	—	109	1	no	—
182	#550	—	109	1	no	—
183	#490	—	109	1	no	—
184	#490	—	109	1	no	—
185	#486	—	109	1	no	—
186	#478	—	109	1	no	—
187	#478	D₁₀₉	1	1	yes	characteristic, complemented
188	#476	—	109	1	no	—
189	#459	—	109	1	no	—
190	#459	—	109	1	no	—
191	#459	—	109	1	no	—
192	#446	—	109	1	no	—
193	#442	—	109	1	no	—
194	#442	—	109	1	no	—
195	C₄×D₈	—	109	1	no	—
196	C₂×#41	—	218	1	no	—
197	C₂×D₈	—	109	1	no	—
198	C₂×D₈	—	109	1	no	—
199	C₂×D₈	—	109	1	no	—
200	C₄⋊Q₈	—	109	1	no	—
201	#127	—	109	1	no	—
202	#127	—	109	1	no	—
203	#127	—	109	1	no	—
204	#125	—	109	1	no	—
205	#125	C₂×D₁₀₉	1	1	yes	characteristic
206	C₂²⋊Q₈	—	218	1	no	—
207	C₂²⋊Q₈	—	218	1	no	—
208	C₂²⋊Q₈	—	218	1	no	—
209	#121	—	218	1	no	—
210	#121	—	218	1	no	—
211	#121	—	218	1	no	—
212	C₄×Q₈	—	109	1	no	—
213	C₄×D₄	—	109	1	no	—
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	—	109	1	no	—
216	C₄×D₄	—	109	1	no	—
217	C₄×D₄	—	109	1	no	—
218	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	218	1	no	—
220	#107	C₂×D₁₀₉	1	1	yes	characteristic
221	#106	—	109	1	no	—
222	#106	C₂×D₁₀₉	1	1	yes	characteristic
223	C₄⋊C₈	—	109	1	no	—
224	C₄⋊C₈	—	109	1	no	—
225	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
226	#102	—	109	1	no	—
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	—	109	1	no	—
234	#102	C₂×D₁₀₉	1	1	yes	characteristic
235	#102	C₂×D₁₀₉	1	1	yes	characteristic
236	#97	—	109	1	no	—
237	#97	—	109	1	no	—
238	C₄×C₈	—	109	1	no	—
239	#41	—	436	4	no	—
240	C₂×Q₈	—	109	1	no	—
241	C₂×Q₈	—	218	1	no	—
242	C₂×D₄	—	109	1	no	—
243	C₂×D₄	—	109	1	no	—
244	C₂×D₄	—	109	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	—	218	1	no	—
248	C₂×D₄	—	218	1	no	—
249	C₂×D₄	—	218	1	no	—
250	C₂×D₄	—	218	1	no	—
251	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	218	1	no	—
254	C₂²×C₄	—	218	1	no	—
255	C₂²×C₄	—	218	1	no	—
256	C₂²×C₄	—	218	1	no	—
257	C₂²×C₄	—	218	1	no	—
258	D₈	—	218	2	no	—
259	D₈	—	218	2	no	—
260	D₈	—	218	2	no	—
261	C₂×C₈	—	109	1	no	—
262	C₂×C₈	—	109	1	no	—
263	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	109	1	no	—
268	C₄⋊C₄	—	109	1	no	—
269	C₄⋊C₄	—	109	1	no	—
270	C₄⋊C₄	—	109	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₄⋊C₄	—	218	1	no	—
273	C₄⋊C₄	—	218	1	no	—
274	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₄²	—	109	1	no	—
284	C₄²	—	109	1	no	—
285	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	218	1	no	—
288	C₂³	—	218	1	no	—
289	C₂³	—	218	1	no	—
290	Q₈	—	872	1	no	—
291	Q₈	—	218	2	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	872	1	no	—
295	D₄	—	872	1	no	—
296	D₄	—	872	1	no	—
297	D₄	—	109	2	no	—
298	D₄	—	436	2	no	—
299	D₄	—	436	2	no	—
300	D₄	—	436	2	no	—
301	D₄	—	218	1	no	—
302	D₄	—	218	1	no	—
303	D₄	—	218	1	no	—
304	D₄	—	218	1	no	—
305	D₄	—	218	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	872	1	no	—
311	C₂×C₄	—	872	1	no	—
312	C₂×C₄	—	109	1	no	—
313	C₂×C₄	—	109	1	no	—
314	C₂×C₄	—	109	1	no	—
315	C₂×C₄	—	109	1	no	—
316	C₂×C₄	—	436	2	no	—
317	C₂×C₄	—	436	2	no	—
318	C₂×C₄	—	436	2	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
321	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	218	1	no	—
326	C₈	—	218	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	436	1	no	—
331	C₂²	—	436	1	no	—
332	C₂²	—	436	1	no	—
333	C₂²	—	436	1	no	—
334	C₂²	—	436	1	no	—
335	C₂²	—	436	1	no	—
336	C₂²	—	436	1	no	—
337	C₂²	—	436	1	no	—
338	C₂²	—	218	2	no	—
339	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	109	2	no	—
346	C₄	—	436	1	no	—
347	C₄	—	218	1	no	—
348	C₄	—	218	1	no	—
349	C₄	—	218	1	no	—
350	C₄	#41×D₁₀₉	1	1	yes	characteristic
351	C₄	#16812747	1	1	yes	characteristic
352	C₂	—	8	1	no	—
353	C₂	—	872	1	no	—
354	C₂	—	872	1	no	—
355	C₂	—	436	2	no	—
356	C₂	#20295686	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₁₀₉	1	1	yes	characteristic
358	C₂	#20295828	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
hypercenter	#478	D₁₀₉
hypocenter	C₁₀₉	#3366
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. #20295732	C₂	semidirect
2. #20295697	C₂	semidirect
3. #20295392	C₂	semidirect
4. #20295389	C₂	semidirect
5. #20295371	C₂	semidirect
6. #20295329	C₂	semidirect
7. #20295315	C₂	semidirect
8. #20295222	C₂	semidirect
9. #20295220	C₂	semidirect
10. #20295220	C₂	semidirect
11. #20295155	C₂	semidirect
12. #20295137	C₂	semidirect
13. #20295136	C₂	semidirect
14. #20295108	C₂	semidirect
15. #478×C₁₀₉	C₂	semidirect
135. D₄₃₆	#41	semidirect
180. C₁₀₉	#3366	semidirect
187. #478	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295732	2
#20295697	2
#20295392	2
#20295389	2
#20295371	2
#20295329	2
#20295315	2
#20295222	2
#20295220	2
#20295155	2
#20295137	2
#20295136	2
#20295108	2
#478×C₁₀₉	2
#3366	109