Subgroups — #28373236

Order 13952.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295632	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295505	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295346	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295341	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295330	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295227	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295221	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295172	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295172	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295135	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295125	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295113	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295106	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	C₂×#13182014	C₂²	1	1	yes	characteristic
21	#16812696	C₂²	1	1	yes	characteristic
22	#16812684	—	2	1	no	—
23	#16812684	—	2	1	no	—
24	#16812674	—	2	1	no	—
25	#16812674	—	2	1	no	—
26	#16812674	C₂²	1	1	yes	characteristic
27	#16812674	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
29	#16812664	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	#16812661	C₂²	1	1	yes	characteristic
32	#16812661	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
34	#16812659	—	2	1	no	—
35	#16812656	C₂²	1	1	yes	characteristic
36	#16812655	—	2	1	no	—
37	C₂×#13182008	—	2	1	no	—
38	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812622	C₂²	1	1	yes	characteristic
43	#16812622	C₂²	1	1	yes	characteristic
44	#16812621	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812619	C₂²	1	1	yes	characteristic
47	#16812619	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
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	#13182028	—	2	2	no	—
60	#13182028	—	2	2	no	—
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	—	2	1	no	—
64	C₂×D₄₃₆	—	2	1	no	—
65	C₂×D₄₃₆	—	2	1	no	—
66	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
67	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
69	C₂²×Dic₁₀₉	—	2	1	no	—
70	C₂²×Dic₁₀₉	—	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	C₂³	1	1	yes	characteristic
75	#13182018	—	2	2	no	—
76	#13182018	—	2	2	no	—
77	#13182014	—	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	C₂³	1	1	yes	characteristic
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	D₄	1	1	yes	characteristic
86	#13182008	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	D₄	1	1	yes	characteristic
93	#13182006	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
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	1	no	—
111	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
112	D₄×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	#41	1	2	yes	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	8	1	no	—
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₀₉	—	2	4	no	—
122	C₂×Dic₁₀₉	—	2	4	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	8	1	no	—
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	—	2	1	no	—
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	—	8	1	no	—
134	D₄₃₆	—	8	1	no	—
135	D₄₃₆	—	8	1	no	—
136	D₄₃₆	—	8	1	no	—
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	C₄×D₁₀₉	—	8	1	no	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	#41	1	2	yes	—
144	C₈×C₁₀₉	—	2	1	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	#11724784	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	C₂²×C₁₀₉	—	2	2	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	—	4	1	no	—
151	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₀₉	—	2	1	no	—
153	C₂×D₁₀₉	—	2	1	no	—
154	C₂×D₁₀₉	—	2	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	C₂×D₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	C₄×C₁₀₉	—	4	1	no	—
166	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
168	Dic₁₀₉	—	2	2	no	—
169	Dic₁₀₉	—	2	1	no	—
170	Dic₁₀₉	—	2	1	no	—
171	Dic₁₀₉	—	2	1	no	—
172	Dic₁₀₉	—	4	1	no	—
173	C₂×C₁₀₉	—	4	2	no	—
174	C₂×C₁₀₉	#525	1	1	yes	characteristic
175	C₂×C₁₀₉	#576	1	1	yes	characteristic
176	C₂×C₁₀₉	#577	1	1	yes	characteristic
177	D₁₀₉	—	4	1	no	—
178	D₁₀₉	—	8	1	no	—
179	D₁₀₉	—	8	1	no	—
180	#3119	—	109	1	no	maximal, Sylow
181	C₁₀₉	#3119	1	1	yes	characteristic, complemented, Sylow
182	#540	—	109	1	no	—
183	#520	—	109	1	no	—
184	#481	—	109	1	no	—
185	#480	—	109	1	no	—
186	#479	—	109	1	no	—
187	#478	D₁₀₉	1	1	yes	characteristic, complemented
188	#460	—	109	1	no	—
189	#459	—	109	1	no	—
190	#449	—	109	1	no	—
191	#449	—	109	1	no	—
192	#442	—	109	1	no	—
193	#440	—	109	1	no	—
194	C₄×SD₁₆	—	109	1	no	—
195	C₄×D₈	—	109	1	no	—
196	#433	—	109	1	no	—
197	C₂×#41	—	109	1	no	—
198	C₂×SD₁₆	—	109	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂×D₈	—	109	1	no	—
201	C₂×D₈	—	109	1	no	—
202	#127	—	109	1	no	—
203	#125	C₂×D₁₀₉	1	1	yes	characteristic
204	#123	—	218	1	no	—
205	#123	—	218	1	no	—
206	#121	—	109	1	no	—
207	#121	—	109	1	no	—
208	#121	—	218	1	no	—
209	#121	—	218	1	no	—
210	C₄×Q₈	—	109	1	no	—
211	C₄×Q₈	—	109	1	no	—
212	C₄×D₄	—	109	1	no	—
213	C₄×D₄	—	109	1	no	—
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	—	218	1	no	—
216	C₄×D₄	—	218	1	no	—
217	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
218	#117	—	109	1	no	—
219	#117	—	218	1	no	—
220	C₂×C₄⋊C₄	—	109	1	no	—
221	C₂×C₄⋊C₄	—	218	1	no	—
222	#107	—	109	1	no	—
223	#107	C₂×D₁₀₉	1	1	yes	characteristic
224	#106	—	109	1	no	—
225	#106	C₂×D₁₀₉	1	1	yes	characteristic
226	C₄⋊C₈	—	109	1	no	—
227	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	#103	—	109	1	no	—
229	#103	—	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	C₂²⋊C₈	—	109	1	no	—
237	C₂²⋊C₈	—	109	1	no	—
238	#97	—	109	1	no	—
239	C₄×C₈	—	109	1	no	—
240	#41	—	218	2	no	—
241	#41	—	218	2	no	—
242	C₂×Q₈	—	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	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	109	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	SD₁₆	—	218	2	no	—
259	SD₁₆	—	218	2	no	—
260	D₈	—	218	2	no	—
261	D₈	—	218	2	no	—
262	C₂×C₈	—	109	1	no	—
263	C₂×C₈	—	109	1	no	—
264	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	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	—	109	1	no	—
279	C₂²⋊C₄	—	109	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₄²	—	109	1	no	—
286	C₄²	—	218	1	no	—
287	C₄²	—	218	1	no	—
288	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	109	1	no	—
291	C₂³	—	218	1	no	—
292	C₂³	—	218	1	no	—
293	Q₈	—	109	2	no	—
294	Q₈	—	218	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	872	1	no	—
297	D₄	—	872	1	no	—
298	D₄	—	872	1	no	—
299	D₄	—	872	1	no	—
300	D₄	—	218	4	no	—
301	D₄	—	218	1	no	—
302	D₄	—	218	1	no	—
303	D₄	—	218	1	no	—
304	D₄	—	218	1	no	—
305	D₄	#13182030	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	872	1	no	—
311	C₂×C₄	—	872	1	no	—
312	C₂×C₄	—	872	1	no	—
313	C₂×C₄	—	872	1	no	—
314	C₂×C₄	—	109	1	no	—
315	C₂×C₄	—	109	1	no	—
316	C₂×C₄	—	109	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	—	218	4	no	—
319	C₂×C₄	—	218	1	no	—
320	C₂×C₄	—	218	1	no	—
321	C₂×C₄	—	218	1	no	—
322	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	218	1	no	—
328	C₈	—	218	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	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	1	no	—
339	C₂²	—	218	1	no	—
340	C₂²	—	218	1	no	—
341	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	436	1	no	—
348	C₄	—	218	1	no	—
349	C₄	—	218	2	no	—
350	C₄	—	218	1	no	—
351	C₄	—	218	1	no	—
352	C₄	#41×D₁₀₉	1	1	yes	characteristic
353	C₄	C₂×#13182030	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	872	1	no	—
356	C₂	—	872	1	no	—
357	C₂	—	436	1	no	—
358	C₂	#20295539	1	1	yes	characteristic
359	C₂	#20295830	1	1	yes	characteristic
360	C₂	#20295833	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	#3119
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. #20295632	C₂	semidirect
2. #20295505	C₂	semidirect
3. #20295346	C₂	semidirect
4. #20295341	C₂	semidirect
5. #20295330	C₂	semidirect
6. #20295227	C₂	semidirect
7. #20295221	C₂	semidirect
8. #20295172	C₂	semidirect
9. #20295172	C₂	semidirect
10. #20295135	C₂	semidirect
11. #20295125	C₂	semidirect
12. #20295113	C₂	semidirect
13. #20295106	C₂	semidirect
14. #20295086	C₂	semidirect
15. #478×C₁₀₉	C₂	semidirect
181. C₁₀₉	#3119	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
#20295632	2
#20295505	2
#20295346	2
#20295341	2
#20295330	2
#20295227	2
#20295221	2
#20295172	2
#20295135	2
#20295125	2
#20295113	2
#20295106	2
#20295086	2
#478×C₁₀₉	2
#3119	109