Subgroups — #28376100

Order 13952.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295761	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295749	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295420	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295415	C₂	1	1	yes	maximal, characteristic
5	#20295410	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295242	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295242	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295232	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295227	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295139	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182018	C₂²	1	1	yes	characteristic
20	C₂×#13182018	C₂²	1	1	yes	characteristic
21	C₂×#13182018	C₂²	1	1	yes	characteristic
22	C₂×#13182018	C₂²	1	1	yes	characteristic
23	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
24	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
25	#16812699	C₂²	1	1	yes	characteristic
26	#16812696	C₂²	1	1	yes	characteristic
27	#16812694	—	2	1	no	—
28	#16812694	—	2	1	no	—
29	#16812688	—	2	1	no	—
30	#16812687	C₂²	1	1	yes	characteristic
31	#16812687	C₂²	1	1	yes	characteristic
32	#16812687	C₂²	1	1	yes	characteristic
33	#16812687	C₂²	1	1	yes	characteristic
34	#16812684	—	2	1	no	—
35	#16812684	—	2	1	no	—
36	#16812675	—	2	1	no	—
37	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
38	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
40	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
41	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
42	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
43	#16812661	—	2	1	no	—
44	#16812657	—	2	1	no	—
45	#16812631	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812624	C₂²	1	1	yes	characteristic
49	#16812621	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#97×C₁₀₉	C₂²	1	1	yes	characteristic
57	#97×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	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×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	Dic₄₃₆	—	2	2	no	—
71	Dic₄₃₆	—	4	1	no	—
72	#13182018	—	2	2	no	—
73	#13182018	—	4	1	no	—
74	#13182018	—	4	1	no	—
75	#13182018	—	4	1	no	—
76	#13182018	—	4	1	no	—
77	D₈₇₂	—	2	2	no	—
78	D₈₇₂	—	4	1	no	—
79	C₄⋊Dic₁₀₉	—	2	1	no	—
80	C₄⋊Dic₁₀₉	—	2	1	no	—
81	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	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	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	#13182006	—	2	1	no	—
95	#13182006	—	2	1	no	—
96	#13182006	—	2	1	no	—
97	C₄×Dic₁₀₉	—	2	1	no	—
98	C₄×Dic₁₀₉	—	2	1	no	—
99	C₄×Dic₁₀₉	—	2	1	no	—
100	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
108	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	C₂²×D₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	4	1	no	—
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
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	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	D₄₃₆	—	2	1	no	—
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	C₄×D₁₀₉	—	4	2	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	C₄×D₁₀₉	—	8	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	2	1	no	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	—	2	1	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	C₈×C₁₀₉	—	2	1	no	—
147	C₈×C₁₀₉	#41	1	2	yes	—
148	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	4	1	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	C₄×C₁₀₉	—	2	1	no	—
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	2	1	no	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	2	1	no	—
164	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	C₂×C₁₀₉	#550	1	1	yes	characteristic
172	C₂×C₁₀₉	#575	1	1	yes	characteristic
173	C₂×C₁₀₉	#577	1	1	yes	characteristic
174	D₁₀₉	—	8	1	no	—
175	D₁₀₉	—	8	1	no	—
176	D₁₀₉	—	8	1	no	—
177	#3401	—	109	1	no	maximal, Sylow
178	C₁₀₉	#3401	1	1	yes	characteristic, complemented, Sylow
179	#559	—	109	1	no	—
180	#556	—	109	1	no	—
181	#497	—	109	1	no	—
182	#496	—	109	1	no	—
183	#495	—	109	1	no	—
184	#464	—	109	1	no	—
185	#464	—	109	1	no	—
186	#463	—	109	1	no	—
187	#463	—	109	1	no	—
188	#461	—	109	1	no	—
189	#460	—	109	1	no	—
190	#446	D₁₀₉	1	1	yes	characteristic, complemented
191	#442	—	109	1	no	—
192	#441	—	109	1	no	—
193	C₄×SD₁₆	—	109	1	no	—
194	C₂×Q₁₆	—	109	1	no	—
195	C₂×Q₁₆	—	109	1	no	—
196	C₂×SD₁₆	—	109	1	no	—
197	C₂×SD₁₆	—	109	1	no	—
198	C₂×SD₁₆	—	109	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂×SD₁₆	—	109	1	no	—
201	C₂×D₈	—	109	1	no	—
202	C₂×D₈	—	109	1	no	—
203	C₄⋊Q₈	—	109	1	no	—
204	#127	—	109	1	no	—
205	#126	—	218	1	no	—
206	#126	—	218	1	no	—
207	#124	—	109	1	no	—
208	#124	—	109	1	no	—
209	#124	—	109	1	no	—
210	#124	—	109	1	no	—
211	#124	—	218	1	no	—
212	#123	—	218	1	no	—
213	#123	—	218	1	no	—
214	C₂²⋊Q₈	—	218	1	no	—
215	C₄×Q₈	—	109	1	no	—
216	C₄×Q₈	—	109	1	no	—
217	C₄×Q₈	—	109	1	no	—
218	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	109	1	no	—
221	C₄×D₄	—	109	1	no	—
222	C₄×D₄	—	218	1	no	—
223	#117	—	218	1	no	—
224	#106	—	109	1	no	—
225	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
226	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	#103	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	—	109	1	no	—
231	#102	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#97	C₂×D₁₀₉	1	1	yes	characteristic
235	#97	C₂×D₁₀₉	1	1	yes	characteristic
236	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
237	C₂×Q₈	—	109	1	no	—
238	C₂×Q₈	—	109	1	no	—
239	C₂×Q₈	—	109	1	no	—
240	C₂×Q₈	D₄₃₆	1	1	yes	characteristic
241	C₂×D₄	—	109	1	no	—
242	C₂×D₄	—	109	1	no	—
243	C₂×D₄	—	109	1	no	—
244	C₂×D₄	—	218	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	C₂²×C₄	—	218	1	no	—
249	Q₁₆	—	436	1	no	—
250	Q₁₆	—	218	2	no	—
251	SD₁₆	—	436	1	no	—
252	SD₁₆	—	436	1	no	—
253	SD₁₆	—	436	1	no	—
254	SD₁₆	—	436	1	no	—
255	SD₁₆	—	218	2	no	—
256	D₈	—	436	1	no	—
257	D₈	—	218	2	no	—
258	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	—	109	1	no	—
263	C₄⋊C₄	—	109	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	D₄₃₆	1	1	yes	characteristic
270	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
271	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
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	C₂²⋊C₄	—	218	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	—	218	1	no	—
284	C₄²	—	218	1	no	—
285	C₄²	—	218	1	no	—
286	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
287	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
288	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
289	C₂³	—	218	1	no	—
290	C₂³	—	218	1	no	—
291	C₂³	—	218	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	218	1	no	—
294	Q₈	—	218	1	no	—
295	Q₈	—	218	1	no	—
296	Q₈	—	218	1	no	—
297	Q₈	—	218	1	no	—
298	Q₈	—	218	1	no	—
299	D₄	—	872	1	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	D₄	—	218	1	no	—
307	C₂×C₄	—	872	1	no	—
308	C₂×C₄	—	872	1	no	—
309	C₂×C₄	—	436	2	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	—	218	1	no	—
315	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
316	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	#13182030	1	2	yes	—
326	C₂²	—	436	1	no	—
327	C₂²	—	436	1	no	—
328	C₂²	—	436	1	no	—
329	C₂²	—	436	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	C₂²×D₄₃₆	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	436	1	no	—
343	C₄	—	436	1	no	—
344	C₄	—	436	1	no	—
345	C₄	—	436	1	no	—
346	C₄	—	436	1	no	—
347	C₄	C₂×#13182030	1	1	yes	characteristic
348	C₄	#16812745	1	1	yes	characteristic
349	C₂	—	872	1	no	—
350	C₂	—	872	1	no	—
351	C₂	—	872	1	no	—
352	C₂	#20295698	1	1	yes	characteristic
353	C₂	#20295826	1	1	yes	characteristic
354	C₂	#20295838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#446×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295415	C₂
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	#3401
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #20295761	C₂	semidirect
2. #20295749	C₂	semidirect
3. #20295420	C₂	semidirect
5. #20295410	C₂	semidirect
6. #20295249	C₂	semidirect
7. #20295249	C₂	semidirect
8. #20295242	C₂	semidirect
9. #20295242	C₂	semidirect
10. #20295232	C₂	semidirect
11. #20295227	C₂	semidirect
12. #20295139	C₂	semidirect
13. #20295134	C₂	semidirect
14. C₄×#13182018	C₂	semidirect
15. #446×C₁₀₉	C₂	semidirect
178. C₁₀₉	#3401	semidirect
190. #446	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295761	2
#20295749	2
#20295420	2
#20295415	2
#20295410	2
#20295249	2
#20295242	2
#20295232	2
#20295227	2
#20295139	2
#20295134	2
C₄×#13182018	2
#446×C₁₀₉	2
#3401	109