Subgroups — #28374445

Order 13952.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295647	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295631	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295317	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295307	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295279	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295272	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295262	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295255	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295242	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295242	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295227	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295222	C₂	1	1	yes	maximal, characteristic, complemented
15	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182018	C₂²	1	1	yes	characteristic
17	C₂×#13182018	C₂²	1	1	yes	characteristic
18	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
19	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
20	#16812699	C₂²	1	1	yes	characteristic
21	#16812696	C₂²	1	1	yes	characteristic
22	#16812694	—	2	1	no	—
23	#16812691	C₂²	1	1	yes	characteristic
24	#16812687	C₂²	1	1	yes	characteristic
25	#16812685	—	2	1	no	—
26	#16812684	—	2	1	no	—
27	#16812684	—	2	1	no	—
28	#16812681	—	2	1	no	—
29	#16812678	C₂²	1	1	yes	characteristic
30	#16812677	—	2	1	no	—
31	#16812673	C₂²	1	1	yes	characteristic
32	#16812673	C₂²	1	1	yes	characteristic
33	#16812673	C₂²	1	1	yes	characteristic
34	#16812672	—	2	1	no	—
35	#16812672	—	2	1	no	—
36	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
37	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
38	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
39	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
40	#16812633	C₂²	1	1	yes	characteristic
41	#16812633	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812631	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812624	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812621	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	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#117×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	#13182018	—	4	1	no	—
73	#13182018	—	4	1	no	—
74	D₈₇₂	—	4	1	no	—
75	D₈₇₂	—	4	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	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	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	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₁₀₉	—	4	1	no	—
97	C₂×C₈×C₁₀₉	—	4	1	no	—
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
106	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	1	no	—
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₀₉	—	8	1	no	—
121	C₂²⋊D₁₀₉	—	8	1	no	—
122	C₂²⋊D₁₀₉	—	8	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	C₂×Dic₁₀₉	—	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	—	2	1	no	—
134	D₄₃₆	—	8	1	no	—
135	D₄₃₆	—	8	1	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	C₄×D₁₀₉	—	8	1	no	—
138	C₄×D₁₀₉	—	8	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	8	1	no	—
142	C₈×C₁₀₉	—	2	1	no	—
143	C₈×C₁₀₉	—	2	1	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	C₈×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	2	1	no	—
147	C₂²×C₁₀₉	—	2	1	no	—
148	C₂²×C₁₀₉	—	2	1	no	—
149	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
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	C₂×D₁₀₉	—	4	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	H(5,𝔽₂)	1	1	yes	characteristic
165	C₄×C₁₀₉	2₋¹⁺⁴	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	—	4	1	no	—
172	C₂×C₁₀₉	#537	1	1	yes	characteristic
173	C₂×C₁₀₉	#576	1	1	yes	characteristic
174	C₂×C₁₀₉	#577	1	1	yes	characteristic
175	D₁₀₉	—	8	1	no	—
176	D₁₀₉	—	8	1	no	—
177	D₁₀₉	—	8	1	no	—
178	#3257	—	109	1	no	maximal, Sylow
179	C₁₀₉	#3257	1	1	yes	characteristic, complemented, Sylow
180	#542	—	109	1	no	—
181	#540	—	109	1	no	—
182	#479	—	109	1	no	—
183	#478	—	109	1	no	—
184	#476	—	109	1	no	—
185	#474	—	109	1	no	—
186	#469	—	109	1	no	—
187	#468	—	109	1	no	—
188	#466	—	109	1	no	—
189	#465	—	109	1	no	—
190	#463	—	109	1	no	—
191	#463	—	109	1	no	—
192	#460	—	109	1	no	—
193	#459	—	109	1	no	—
194	#424	D₁₀₉	1	1	yes	characteristic, complemented
195	C₂×SD₁₆	—	109	1	no	—
196	C₂×SD₁₆	—	109	1	no	—
197	C₂×D₈	—	109	1	no	—
198	C₂×D₈	—	109	1	no	—
199	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
200	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
201	C₄⋊Q₈	—	109	1	no	—
202	#127	—	109	1	no	—
203	#126	—	218	1	no	—
204	#125	—	109	1	no	—
205	#124	—	109	1	no	—
206	#124	—	218	1	no	—
207	#123	—	218	1	no	—
208	#123	—	218	1	no	—
209	#123	—	218	1	no	—
210	C₂²⋊Q₈	—	109	1	no	—
211	C₂²⋊Q₈	—	218	1	no	—
212	#121	—	109	1	no	—
213	#121	—	109	1	no	—
214	#121	—	109	1	no	—
215	#121	—	218	1	no	—
216	#121	—	218	1	no	—
217	C₄×Q₈	—	109	1	no	—
218	C₄×D₄	—	109	1	no	—
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	109	1	no	—
221	#117	C₂×D₁₀₉	1	1	yes	characteristic
222	#107	—	109	1	no	—
223	#107	—	109	1	no	—
224	#106	—	109	1	no	—
225	#106	—	109	1	no	—
226	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
229	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
230	#103	—	109	1	no	—
231	#103	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#102	—	109	1	no	—
235	#102	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	C₂×Q₈	—	109	1	no	—
239	C₂×Q₈	—	218	1	no	—
240	C₂×D₄	—	109	1	no	—
241	C₂×D₄	—	109	1	no	—
242	C₂×D₄	—	109	1	no	—
243	C₂×D₄	—	218	1	no	—
244	C₂×D₄	—	218	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	—	218	1	no	—
248	C₂²×C₄	—	218	1	no	—
249	C₂²×C₄	—	218	1	no	—
250	C₂²×C₄	—	218	1	no	—
251	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	SD₁₆	—	436	1	no	—
253	SD₁₆	—	436	1	no	—
254	D₈	—	436	1	no	—
255	D₈	—	436	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	4	1	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	—	109	1	no	—
265	C₄⋊C₄	—	109	1	no	—
266	C₄⋊C₄	—	218	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
272	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
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	D₄₃₆	1	1	yes	characteristic
283	C₂²⋊C₄	D₄₃₆	1	1	yes	characteristic
284	C₄²	—	218	1	no	—
285	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
286	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
287	C₂³	—	218	1	no	—
288	C₂³	—	218	1	no	—
289	C₂³	—	218	1	no	—
290	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
291	Q₈	—	872	1	no	—
292	Q₈	—	218	1	no	—
293	Q₈	—	218	1	no	—
294	D₄	—	872	1	no	—
295	D₄	—	872	1	no	—
296	D₄	—	872	1	no	—
297	D₄	—	872	1	no	—
298	D₄	—	872	1	no	—
299	D₄	—	218	1	no	—
300	D₄	—	218	1	no	—
301	D₄	—	218	1	no	—
302	D₄	—	218	1	no	—
303	D₄	—	218	1	no	—
304	D₄	—	218	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	872	1	no	—
308	C₂×C₄	—	872	1	no	—
309	C₂×C₄	—	872	1	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	C₂³×D₁₀₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	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	—	436	1	no	—
336	C₂²	—	436	1	no	—
337	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	436	1	no	—
344	C₄	—	436	1	no	—
345	C₄	—	436	1	no	—
346	C₄	—	436	1	no	—
347	C₄	—	436	1	no	—
348	C₄	#16812744	1	1	yes	characteristic
349	C₄	#16812747	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	872	1	no	—
352	C₂	—	872	1	no	—
353	C₂	—	872	1	no	—
354	C₂	#20295607	1	1	yes	characteristic
355	C₂	#20295831	1	1	yes	characteristic
356	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	#424×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#3257
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

Complements

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

N	H ≅ G/N	type
1. #20295647	C₂	semidirect
2. #20295631	C₂	semidirect
3. #20295334	C₂	semidirect
4. #20295327	C₂	semidirect
5. #20295317	C₂	semidirect
6. #20295307	C₂	semidirect
7. #20295279	C₂	semidirect
8. #20295272	C₂	semidirect
9. #20295262	C₂	semidirect
10. #20295255	C₂	semidirect
11. #20295242	C₂	semidirect
12. #20295242	C₂	semidirect
13. #20295227	C₂	semidirect
14. #20295222	C₂	semidirect
15. #424×C₁₀₉	C₂	semidirect
179. C₁₀₉	#3257	semidirect
194. #424	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295647	2
#20295631	2
#20295334	2
#20295327	2
#20295317	2
#20295307	2
#20295279	2
#20295272	2
#20295262	2
#20295255	2
#20295242	2
#20295227	2
#20295222	2
#424×C₁₀₉	2
#3257	109