Subgroups — #28375632

Order 13952.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295722	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295425	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295350	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295347	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295339	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295337	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295323	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295277	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295277	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182014	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295104	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295069	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295066	C₂	1	1	yes	maximal, characteristic, complemented
15	#553×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182014	C₂²	1	1	yes	characteristic
17	C₂×#13182013	C₂²	1	1	yes	characteristic
18	C₂×#13182013	C₂²	1	1	yes	characteristic
19	#16812694	—	2	1	no	—
20	#16812691	C₂²	1	1	yes	characteristic
21	#16812681	—	2	1	no	—
22	#16812673	C₂²	1	1	yes	characteristic
23	#16812673	C₂²	1	1	yes	characteristic
24	#16812671	—	2	1	no	—
25	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
26	#16812659	—	2	1	no	—
27	#16812658	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
29	#16812632	C₂²	1	1	yes	characteristic
30	#16812632	C₂²	1	1	yes	characteristic
31	#16812632	C₂²	1	1	yes	characteristic
32	#16812632	C₂²	1	1	yes	characteristic
33	#16812632	C₂²	1	1	yes	characteristic
34	#16812630	C₂²	1	1	yes	characteristic
35	#16812630	C₂²	1	1	yes	characteristic
36	#16812630	C₂²	1	1	yes	characteristic
37	#16812630	C₂²	1	1	yes	characteristic
38	C₄⋊#11724784	C₂²	1	1	yes	characteristic
39	#16812620	C₂²	1	1	yes	characteristic
40	#16812620	C₂²	1	1	yes	characteristic
41	#16812620	C₂²	1	1	yes	characteristic
42	#16812619	C₂²	1	1	yes	characteristic
43	#16812619	C₂²	1	1	yes	characteristic
44	#16812619	C₂²	1	1	yes	characteristic
45	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
46	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
47	C₄×#11724784	C₂²	1	1	yes	characteristic
48	#126×C₁₀₉	—	2	1	no	—
49	#125×C₁₀₉	C₂²	1	1	yes	characteristic
50	#123×C₁₀₉	—	2	1	no	—
51	#121×C₁₀₉	—	2	1	no	—
52	#121×C₁₀₉	C₂²	1	1	yes	characteristic
53	#121×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	—	2	1	no	—
55	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
57	#117×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	#13182014	—	2	2	no	—
65	#13182013	—	2	2	no	—
66	#13182013	—	2	2	no	—
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	C₂²⋊Dic₁₀₉	—	2	1	no	—
78	#13182006	—	2	1	no	—
79	#13182006	—	2	1	no	—
80	#13182006	—	2	1	no	—
81	C₄×Dic₁₀₉	—	2	1	no	—
82	C₂×D₄×C₁₀₉	—	2	1	no	—
83	C₂×D₄×C₁₀₉	—	2	1	no	—
84	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
85	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₀₉	—	2	1	no	—
87	C₂²×C₄×C₁₀₉	—	2	1	no	—
88	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₁₀₉	—	2	1	no	—
91	C₄⋊C₄×C₁₀₉	—	2	1	no	—
92	C₄⋊C₄×C₁₀₉	—	2	1	no	—
93	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
98	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
99	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
100	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
103	C₄²×C₁₀₉	—	2	1	no	—
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂³×C₁₀₉	—	2	1	no	—
106	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₁₀₉	—	2	1	no	—
109	D₄×C₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	—	2	1	no	—
111	D₄×C₁₀₉	—	4	1	no	—
112	D₄×C₁₀₉	—	4	2	no	—
113	D₄×C₁₀₉	—	8	1	no	—
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	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	8	1	no	—
122	C₂×C₄×C₁₀₉	—	8	1	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₀₉	—	4	2	no	—
129	C₂²⋊D₁₀₉	—	8	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	8	1	no	—
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	—	2	1	no	—
136	C₄×D₁₀₉	—	8	1	no	—
137	#11724784	—	2	1	no	—
138	#11724784	—	2	1	no	—
139	#11724784	—	2	1	no	—
140	#11724784	#41	1	2	yes	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	1	no	—
144	C₂²×C₁₀₉	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	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	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	2	1	no	—
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	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	—	4	1	no	—
167	C₂×C₁₀₉	—	4	1	no	—
168	C₂×C₁₀₉	—	8	1	no	—
169	C₂×C₁₀₉	#547	1	1	yes	characteristic
170	C₂×C₁₀₉	#575	1	1	yes	characteristic
171	C₂×C₁₀₉	#576	1	1	yes	characteristic
172	D₁₀₉	—	8	1	no	—
173	#3355	—	109	1	no	maximal, Sylow
174	C₁₀₉	#3355	1	1	yes	characteristic, complemented, Sylow
175	#553	—	109	1	no	—
176	#553	D₁₀₉	1	1	yes	characteristic, complemented
177	#499	—	109	1	no	—
178	#482	—	109	1	no	—
179	#482	—	109	1	no	—
180	#480	—	109	1	no	—
181	#480	—	109	1	no	—
182	#478	—	109	1	no	—
183	#478	—	109	1	no	—
184	#469	—	109	1	no	—
185	#469	—	109	1	no	—
186	C₄×D₈	—	109	1	no	—
187	#436	—	109	1	no	—
188	#428	—	109	1	no	—
189	#428	—	109	1	no	—
190	C₂×D₈	—	109	1	no	—
191	C₂×M₁₆	—	109	1	no	—
192	C₂×M₁₆	—	109	1	no	—
193	#126	—	2	1	no	—
194	#126	—	218	1	no	—
195	#125	—	109	1	no	—
196	#125	C₂×D₁₀₉	1	1	yes	characteristic
197	#123	—	2	1	no	—
198	#123	—	218	1	no	—
199	#121	—	2	1	no	—
200	#121	—	109	1	no	—
201	#121	—	109	1	no	—
202	#121	—	218	1	no	—
203	#121	C₂×D₁₀₉	1	1	yes	characteristic
204	#121	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	109	1	no	—
207	C₄×D₄	—	218	1	no	—
208	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
209	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
210	#117	—	109	1	no	—
211	#117	C₂×D₁₀₉	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	109	1	no	—
213	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
214	#107	—	109	1	no	—
215	#107	—	109	1	no	—
216	#107	—	109	1	no	—
217	#107	—	109	1	no	—
218	#107	—	109	1	no	—
219	#106	—	109	1	no	—
220	#106	—	109	1	no	—
221	#106	—	109	1	no	—
222	#106	—	109	1	no	—
223	C₄⋊C₈	—	109	1	no	—
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	#102	—	109	1	no	—
227	#102	—	109	1	no	—
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	C₂²⋊C₈	—	109	1	no	—
231	C₂²⋊C₈	—	109	1	no	—
232	C₄×C₈	—	109	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	109	1	no	—
236	C₂×D₄	—	218	1	no	—
237	C₂×D₄	—	218	1	no	—
238	C₂×D₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
239	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	C₂²×C₄	—	218	1	no	—
244	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
246	D₈	—	218	2	no	—
247	M₁₆	—	218	2	no	—
248	M₁₆	—	218	2	no	—
249	C₂×C₈	—	109	1	no	—
250	C₂×C₈	—	109	1	no	—
251	C₂×C₈	—	109	1	no	—
252	C₂×C₈	—	109	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	109	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
275	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
276	C₄²	—	2	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	218	1	no	—
281	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	8	1	no	—
288	D₄	—	872	1	no	—
289	D₄	—	436	2	no	—
290	D₄	—	218	1	no	—
291	D₄	—	218	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	872	1	no	—
302	C₂×C₄	—	872	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
311	C₈	—	109	2	no	—
312	C₈	—	218	1	no	—
313	C₈	—	218	1	no	—
314	C₈	—	218	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	436	1	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	436	1	no	—
336	C₄	—	436	1	no	—
337	C₄	—	436	1	no	—
338	C₄	#41×D₁₀₉	1	1	yes	characteristic
339	C₄	#16812746	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	4	1	no	—
342	C₂	—	8	1	no	—
343	C₂	—	872	1	no	—
344	C₂	#20295675	1	1	yes	characteristic
345	C₂	#20295824	1	1	yes	characteristic
346	C₂	#20295829	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#553×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#553	D₁₀₉
hypocenter	C₁₀₉	#3355
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #553

Complements

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

N	H ≅ G/N	type
1. #20295722	C₂	semidirect
2. #20295425	C₂	semidirect
3. #20295350	C₂	semidirect
4. #20295347	C₂	semidirect
5. #20295339	C₂	semidirect
6. #20295337	C₂	semidirect
7. #20295326	C₂	semidirect
8. #20295323	C₂	semidirect
9. #20295277	C₂	semidirect
10. #20295277	C₂	semidirect
11. C₄×#13182014	C₂	semidirect
12. #20295104	C₂	semidirect
13. #20295069	C₂	semidirect
14. #20295066	C₂	semidirect
15. #553×C₁₀₉	C₂	semidirect
174. C₁₀₉	#3355	semidirect
176. #553	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295722	2
#20295425	2
#20295350	2
#20295347	2
#20295339	2
#20295337	2
#20295326	2
#20295323	2
#20295277	2
C₄×#13182014	2
#20295104	2
#20295069	2
#20295066	2
#553×C₁₀₉	2
#3355	109