Subgroups — #28376107

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295749	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295421	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295418	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295408	C₂	1	1	yes	maximal, characteristic
5	#20295248	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295241	C₂	1	1	yes	maximal, characteristic
8	#20295238	C₂	1	1	yes	maximal, characteristic
9	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295233	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295126	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295124	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
15	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182019	C₂²	1	1	yes	characteristic
19	C₂×#13182019	C₂²	1	1	yes	characteristic
20	C₂×#13182017	C₂²	1	1	yes	characteristic
21	C₂×#13182017	C₂²	1	1	yes	characteristic
22	C₂×#13182016	C₂²	1	1	yes	characteristic
23	C₂×#13182016	C₂²	1	1	yes	characteristic
24	C₂×#13182014	C₂²	1	1	yes	characteristic
25	#16812699	C₂²	1	1	yes	characteristic
26	#16812694	—	2	1	no	—
27	#16812694	—	2	1	no	—
28	#16812687	C₂²	1	1	yes	characteristic
29	#16812687	C₂²	1	1	yes	characteristic
30	#16812675	—	2	1	no	—
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	#16812657	—	2	1	no	—
35	#16812632	C₂²	1	1	yes	characteristic
36	C₄⋊#11724784	C₂²	1	1	yes	characteristic
37	C₄⋊#11724784	C₂²	1	1	yes	characteristic
38	C₄⋊#11724784	C₂²	1	1	yes	characteristic
39	#16812623	C₂²	1	1	yes	characteristic
40	#16812623	C₂²	1	1	yes	characteristic
41	#16812622	C₂²	1	1	yes	characteristic
42	#16812622	C₂²	1	1	yes	characteristic
43	#16812620	C₂²	1	1	yes	characteristic
44	#16812619	C₂²	1	1	yes	characteristic
45	#16812610	C₂²	1	1	yes	characteristic
46	#16812610	C₂²	1	1	yes	characteristic
47	C₄×#11724784	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
49	#126×C₁₀₉	—	2	1	no	—
50	#126×C₁₀₉	—	2	1	no	—
51	#124×C₁₀₉	C₂²	1	1	yes	characteristic
52	#124×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
58	#117×C₁₀₉	—	2	1	no	—
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₀₉	—	2	1	no	—
63	#13182019	—	2	2	no	—
64	#13182019	—	4	1	no	—
65	#13182019	—	4	1	no	—
66	#13182019	—	4	1	no	—
67	#13182017	—	2	2	no	—
68	#13182017	—	4	1	no	—
69	#13182016	—	2	2	no	—
70	#13182016	—	4	1	no	—
71	#13182014	—	4	1	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182006	—	2	1	no	—
82	#13182006	—	2	1	no	—
83	#13182006	—	2	1	no	—
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	C₄×Dic₁₀₉	—	2	1	no	—
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
89	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	—	2	1	no	—
93	C₄⋊C₄×C₁₀₉	—	2	1	no	—
94	C₄⋊C₄×C₁₀₉	—	2	1	no	—
95	C₄⋊C₄×C₁₀₉	—	2	1	no	—
96	C₄⋊C₄×C₁₀₉	—	2	1	no	—
97	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
102	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
103	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	—	2	1	no	—
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	C₂³×C₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	4	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	—	2	1	no	—
136	C₄×D₁₀₉	—	4	2	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	#11724784	—	2	1	no	—
142	#11724784	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	#11724784	#41	1	2	yes	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
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	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	2	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	H(5,𝔽₂)	1	1	yes	characteristic
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	—	8	1	no	—
167	C₂×C₁₀₉	#550	1	1	yes	characteristic
168	C₂×C₁₀₉	#575	1	1	yes	characteristic
169	C₂×C₁₀₉	#578	1	1	yes	characteristic
170	D₁₀₉	—	8	1	no	—
171	#3402	—	109	1	no	maximal, Sylow
172	C₁₀₉	#3402	1	1	yes	characteristic, complemented, Sylow
173	#556	—	109	1	no	—
174	#556	D₁₀₉	1	1	yes	characteristic, complemented
175	#497	—	109	1	no	—
176	#497	—	109	1	no	—
177	#495	—	109	1	no	—
178	#464	—	109	1	no	—
179	#464	—	109	1	no	—
180	#463	—	109	1	no	—
181	#463	—	109	1	no	—
182	#462	—	109	1	no	—
183	#462	—	109	1	no	—
184	#446	—	109	1	no	—
185	#440	—	109	1	no	—
186	#440	—	109	1	no	—
187	C₄×Q₁₆	—	109	1	no	—
188	C₂×Q₁₆	—	109	1	no	—
189	C₂×Q₁₆	—	109	1	no	—
190	C₂×Q₁₆	—	109	1	no	—
191	C₂×Q₁₆	—	109	1	no	—
192	C₂×SD₁₆	—	109	1	no	—
193	C₂×SD₁₆	—	109	1	no	—
194	C₂×SD₁₆	—	109	1	no	—
195	C₂×SD₁₆	—	109	1	no	—
196	C₂×D₈	—	109	1	no	—
197	C₄⋊Q₈	—	109	1	no	—
198	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
199	#126	—	2	1	no	—
200	#126	—	2	1	no	—
201	#126	—	218	1	no	—
202	#126	—	218	1	no	—
203	#124	—	109	1	no	—
204	#124	—	109	1	no	—
205	#124	C₂×D₁₀₉	1	1	yes	characteristic
206	#124	C₂×D₁₀₉	1	1	yes	characteristic
207	C₂²⋊Q₈	—	2	1	no	—
208	C₂²⋊Q₈	—	218	1	no	—
209	C₄×Q₈	—	109	1	no	—
210	C₄×Q₈	—	109	1	no	—
211	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
212	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
213	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
214	C₄×D₄	—	109	1	no	—
215	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
216	#117	—	2	1	no	—
217	#117	—	218	1	no	—
218	#107	—	109	1	no	—
219	C₄⋊C₈	—	109	1	no	—
220	C₄⋊C₈	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	#103	—	109	1	no	—
223	#103	—	109	1	no	—
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#102	—	109	1	no	—
227	#102	—	109	1	no	—
228	#97	—	109	1	no	—
229	#97	—	109	1	no	—
230	C₄×C₈	—	109	1	no	—
231	C₂×Q₈	—	109	1	no	—
232	C₂×Q₈	—	109	1	no	—
233	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
234	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
236	C₂×D₄	—	109	1	no	—
237	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	218	1	no	—
240	Q₁₆	—	436	1	no	—
241	Q₁₆	—	436	1	no	—
242	Q₁₆	—	436	1	no	—
243	Q₁₆	—	218	2	no	—
244	SD₁₆	—	436	1	no	—
245	SD₁₆	—	436	1	no	—
246	SD₁₆	—	218	2	no	—
247	SD₁₆	—	218	2	no	—
248	D₈	—	436	1	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	—	2	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	218	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	—	2	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
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	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	218	1	no	—
279	C₄²	—	218	1	no	—
280	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
281	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
282	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	218	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	218	1	no	—
291	Q₈	—	218	1	no	—
292	Q₈	—	218	1	no	—
293	Q₈	—	218	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	218	1	no	—
297	D₄	—	218	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	436	2	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	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
313	C₈	—	109	2	no	—
314	C₈	—	218	1	no	—
315	C₈	—	218	1	no	—
316	C₈	—	218	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	436	1	no	—
321	C₂²	—	436	1	no	—
322	C₂²	—	436	1	no	—
323	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	436	1	no	—
334	C₄	—	436	1	no	—
335	C₄	—	436	1	no	—
336	C₄	#41×D₁₀₉	1	1	yes	characteristic
337	C₄	#16812745	1	1	yes	characteristic
338	C₂	—	8	1	no	—
339	C₂	—	872	1	no	—
340	C₂	#20295695	1	1	yes	characteristic
341	C₂	#20295824	1	1	yes	characteristic
342	C₂	#20295841	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#556×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182014	C₂²
hypercenter	#556	D₁₀₉
hypocenter	C₁₀₉	#3402
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

Complements

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

N	H ≅ G/N	type
1. #20295749	C₂	semidirect
2. #20295421	C₂	semidirect
3. #20295418	C₂	semidirect
5. #20295248	C₂	semidirect
6. #20295245	C₂	semidirect
9. #20295236	C₂	semidirect
10. #20295233	C₂	semidirect
11. #20295157	C₂	semidirect
12. #20295126	C₂	semidirect
13. #20295124	C₂	semidirect
14. C₄×#13182019	C₂	semidirect
15. #556×C₁₀₉	C₂	semidirect
172. C₁₀₉	#3402	semidirect
174. #556	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295749	2
#20295421	2
#20295418	2
#20295408	2
#20295248	2
#20295245	2
#20295241	2
#20295238	2
#20295236	2
#20295233	2
#20295157	2
#20295126	2
#20295124	2
C₄×#13182019	2
#556×C₁₀₉	2
#3402	109