Subgroups — #28345554

Order 13696.

322 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293181	C₂	1	1	yes	maximal, characteristic
2	#20292758	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292756	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292756	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292755	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292755	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₂₁₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292751	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292689	C₂	1	1	yes	maximal, characteristic
10	#20292687	C₂	1	1	yes	maximal, characteristic
11	#20292687	C₂	1	1	yes	maximal, characteristic
12	#20292686	C₂	1	1	yes	maximal, characteristic
13	#20292682	C₂	1	1	yes	maximal, characteristic
14	#20292679	C₂	1	1	yes	maximal, characteristic
15	#446×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
16	#16812108	C₄	1	1	yes	characteristic
17	#16812107	C₄	1	1	yes	characteristic
18	#16812099	C₄	1	1	yes	characteristic
19	#16812099	C₄	1	1	yes	characteristic
20	#16812089	C₄	1	1	yes	characteristic
21	#16812087	C₄	1	1	yes	characteristic
22	#16812087	C₄	1	1	yes	characteristic
23	#16812085	C₄	1	1	yes	characteristic
24	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁₄	C₂²	1	1	yes	characteristic
27	#16812066	C₂²	1	1	yes	characteristic
28	#16812066	C₂²	1	1	yes	characteristic
29	C₄²×D₁₀₇	C₂²	1	1	yes	characteristic
30	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
31	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
32	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
33	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
34	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
40	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
41	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
42	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
43	#16812020	C₂²	1	1	yes	characteristic
44	#16812020	C₂²	1	1	yes	characteristic
45	#16812019	C₂²	1	1	yes	characteristic
46	#16812019	C₂²	1	1	yes	characteristic
47	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
56	#97×C₁₀₇	C₂²	1	1	yes	characteristic
57	#97×C₁₀₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₄	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₂₁₄	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₂₁₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
72	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₁₀₇⋊C₄	C₂×C₄	1	1	yes	characteristic
82	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
89	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
99	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
100	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
101	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
102	C₂²×D₁₀₇	C₂²×C₄	1	1	yes	characteristic
103	Q₈×C₁₀₇	—	4	1	no	—
104	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
118	C₄×D₁₀₇	—	2	1	no	—
119	C₄×D₁₀₇	—	2	1	no	—
120	C₄×D₁₀₇	—	2	2	no	—
121	C₄×D₁₀₇	—	4	1	no	—
122	Dic₂₁₄	—	2	2	no	—
123	Dic₂₁₄	—	2	2	no	—
124	Dic₂₁₄	—	4	1	no	—
125	C₈×C₁₀₇	—	2	1	no	—
126	C₈×C₁₀₇	—	2	1	no	—
127	C₈×C₁₀₇	—	2	1	no	—
128	C₈×C₁₀₇	—	2	1	no	—
129	C₁₀₇⋊C₈	—	2	1	no	—
130	C₁₀₇⋊C₈	—	2	1	no	—
131	C₁₀₇⋊C₈	—	2	1	no	—
132	C₁₀₇⋊C₈	—	2	1	no	—
133	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×D₁₀₇	—	2	1	no	—
135	C₂×D₁₀₇	—	2	1	no	—
136	C₂×D₁₀₇	—	2	1	no	—
137	C₄×C₁₀₇	—	2	1	no	—
138	C₄×C₁₀₇	—	2	1	no	—
139	C₄×C₁₀₇	—	2	1	no	—
140	C₄×C₁₀₇	—	2	1	no	—
141	C₄×C₁₀₇	—	2	1	no	—
142	C₄×C₁₀₇	—	2	1	no	—
143	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₁₀₇	—	2	1	no	—
146	Dic₁₀₇	—	2	1	no	—
147	Dic₁₀₇	—	2	1	no	—
148	Dic₁₀₇	—	2	1	no	—
149	Dic₁₀₇	—	2	1	no	—
150	Dic₁₀₇	—	2	1	no	—
151	Dic₁₀₇	#131	1	2	yes	—
152	C₂×C₁₀₇	#519	1	1	yes	characteristic
153	C₂×C₁₀₇	C₂×#131	1	1	yes	characteristic
154	C₂×C₁₀₇	#568	1	1	yes	characteristic
155	D₁₀₇	—	4	1	no	—
156	#2985	—	107	1	no	maximal, Sylow
157	C₁₀₇	#2985	1	1	yes	characteristic, complemented, Sylow
158	#533	—	107	1	no	—
159	#446	—	107	1	no	—
160	#446	—	107	1	no	—
161	#446	—	107	1	no	—
162	#446	—	107	1	no	—
163	#446	—	107	1	no	—
164	#446	D₁₀₇	1	1	yes	characteristic, complemented
165	C₈×Q₈	—	107	1	no	—
166	C₈×Q₈	—	107	1	no	—
167	#433	—	107	1	no	—
168	#433	—	107	1	no	—
169	#433	—	107	1	no	—
170	#433	—	107	1	no	—
171	#432	—	107	1	no	—
172	#431	—	107	1	no	—
173	C₄⋊Q₈	—	107	1	no	—
174	C₄⋊Q₈	—	107	1	no	—
175	#125	—	107	1	no	—
176	#125	—	107	1	no	—
177	C₂²⋊Q₈	—	107	1	no	—
178	C₂²⋊Q₈	—	107	1	no	—
179	C₂²⋊Q₈	—	107	1	no	—
180	C₂²⋊Q₈	—	107	1	no	—
181	C₄×Q₈	—	107	1	no	—
182	C₄×Q₈	—	107	1	no	—
183	C₄×Q₈	—	107	1	no	—
184	C₄×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
185	#117	—	107	1	no	—
186	#117	—	107	1	no	—
187	C₂×C₄²	—	107	1	no	—
188	C₄⋊C₈	—	107	1	no	—
189	C₄⋊C₈	—	107	1	no	—
190	C₄⋊C₈	—	107	1	no	—
191	C₄⋊C₈	—	107	1	no	—
192	C₄⋊C₈	—	107	1	no	—
193	C₄⋊C₈	—	107	1	no	—
194	C₄⋊C₈	—	107	1	no	—
195	C₄⋊C₈	—	107	1	no	—
196	C₄⋊C₈	—	107	1	no	—
197	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
198	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
200	C₂²⋊C₈	—	107	1	no	—
201	C₂²⋊C₈	—	107	1	no	—
202	C₂²⋊C₈	—	107	1	no	—
203	C₂²⋊C₈	—	107	1	no	—
204	#97	—	107	1	no	—
205	#97	—	107	1	no	—
206	#97	—	107	1	no	—
207	#97	—	107	1	no	—
208	#97	C₂×D₁₀₇	1	1	yes	characteristic
209	#97	C₂×D₁₀₇	1	1	yes	characteristic
210	C₄×C₈	—	107	1	no	—
211	C₄×C₈	—	107	1	no	—
212	C₄×C₈	—	107	1	no	—
213	C₄×C₈	—	107	1	no	—
214	C₄×C₈	—	107	1	no	—
215	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
216	C₂×Q₈	—	107	1	no	—
217	C₂×Q₈	—	107	1	no	—
218	C₂×Q₈	—	107	1	no	—
219	C₂×Q₈	C₄×D₁₀₇	1	1	yes	characteristic
220	C₂²×C₄	—	107	1	no	—
221	C₂²×C₄	—	107	1	no	—
222	C₂²×C₄	—	107	1	no	—
223	C₂×C₈	—	107	1	no	—
224	C₂×C₈	—	107	1	no	—
225	C₂×C₈	—	107	1	no	—
226	C₂×C₈	—	107	1	no	—
227	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
231	C₄⋊C₄	—	107	1	no	—
232	C₄⋊C₄	—	107	1	no	—
233	C₄⋊C₄	—	107	1	no	—
234	C₄⋊C₄	—	107	1	no	—
235	C₄⋊C₄	—	107	1	no	—
236	C₄⋊C₄	—	107	1	no	—
237	C₄⋊C₄	—	107	1	no	—
238	C₄⋊C₄	—	107	1	no	—
239	C₄⋊C₄	—	107	1	no	—
240	C₄⋊C₄	—	107	1	no	—
241	C₄⋊C₄	—	107	1	no	—
242	C₄⋊C₄	—	107	1	no	—
243	C₄⋊C₄	—	107	1	no	—
244	C₄⋊C₄	C₄×D₁₀₇	1	1	yes	characteristic
245	C₄⋊C₄	C₄×D₁₀₇	1	1	yes	characteristic
246	C₄⋊C₄	C₄×D₁₀₇	1	1	yes	characteristic
247	C₂²⋊C₄	—	107	1	no	—
248	C₂²⋊C₄	—	107	1	no	—
249	C₂²⋊C₄	—	107	1	no	—
250	C₂²⋊C₄	—	107	1	no	—
251	C₄²	—	107	1	no	—
252	C₄²	—	107	1	no	—
253	C₄²	—	107	1	no	—
254	C₄²	—	107	1	no	—
255	C₄²	—	107	1	no	—
256	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
257	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
258	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
259	C₂³	—	107	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	428	1	no	—
262	Q₈	—	214	2	no	—
263	Q₈	—	214	2	no	—
264	C₂×C₄	—	107	1	no	—
265	C₂×C₄	—	107	1	no	—
266	C₂×C₄	—	107	1	no	—
267	C₂×C₄	—	107	1	no	—
268	C₂×C₄	—	107	1	no	—
269	C₂×C₄	—	107	1	no	—
270	C₂×C₄	—	107	1	no	—
271	C₂×C₄	—	428	1	no	—
272	C₂×C₄	—	214	2	no	—
273	C₂×C₄	—	214	1	no	—
274	C₂×C₄	—	214	1	no	—
275	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	214	1	no	—
287	C₈	—	214	1	no	—
288	C₈	—	214	1	no	—
289	C₈	—	214	1	no	—
290	C₂²	—	214	1	no	—
291	C₂²	—	214	1	no	—
292	C₂²	—	214	1	no	—
293	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	107	2	no	—
301	C₄	—	214	1	no	—
302	C₄	—	214	1	no	—
303	C₄	—	214	1	no	—
304	C₄	—	214	1	no	—
305	C₄	—	214	1	no	—
306	C₄	—	214	1	no	—
307	C₄	#16812156	1	1	yes	characteristic
308	C₄	#16812157	1	1	yes	characteristic
309	C₂	—	428	1	no	—
310	C₂	#20293103	1	1	yes	characteristic
311	C₂	#131×D₁₀₇	1	1	yes	characteristic
312	C₂	#20293393	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#446×C₁₀₇	C₂
prime radical	C₂²×D₁₀₇	C₂²×C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#446	D₁₀₇
hypocenter	C₁₀₇	#2985
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #20292758	C₂	semidirect
3. #20292756	C₂	semidirect
4. #20292756	C₂	semidirect
5. #20292755	C₂	semidirect
6. #20292755	C₂	semidirect
7. C₈×Dic₂₁₄	C₂	semidirect
8. #20292751	C₂	semidirect
15. #446×C₁₀₇	C₂	semidirect
157. C₁₀₇	#2985	semidirect
164. #446	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293181	2
#20292758	2
#20292756	2
#20292755	2
C₈×Dic₂₁₄	2
#20292751	2
#20292689	2
#20292687	2
#20292686	2
#20292682	2
#20292679	2
#446×C₁₀₇	2
#2985	107