Subgroups — #28345493

Order 13696.

464 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₁₀₇	C₂	1	1	yes	maximal, characteristic
2	#20292705	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292701	C₂	1	2	yes	maximal, complemented
4	#20292696	C₂	1	2	yes	maximal, complemented
5	C₈×D₄₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292692	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292683	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292678	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292590	C₂	1	2	yes	maximal, complemented
10	C₂²⋊C₈×D₁₀₇	C₂	1	2	yes	maximal, complemented
11	#436×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₀₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₇	C₂²	1	2	yes	—
14	C₂×#13133802	C₂²	1	2	yes	complemented
15	C₂×C₈×D₁₀₇	C₂²	1	2	yes	complemented
16	C₂×C₈×D₁₀₇	C₂²	1	2	yes	complemented
17	C₄×D₄₂₈	C₂²	1	1	yes	characteristic
18	D₄×Dic₁₀₇	C₄	1	1	yes	characteristic
19	#16812071	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₀₇	C₂²	1	2	yes	—
21	(C₂²⋊D₁₀₇)⋊C₄	C₄	1	2	yes	—
22	C₄⋊C₄×D₁₀₇	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₁₀₇	C₄	1	2	yes	—
24	C₄²×D₁₀₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
26	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
27	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	complemented
29	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	complemented
33	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	complemented
34	#16812019	C₂²	1	2	yes	complemented
35	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	complemented
37	C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₈×C₁₀₇	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₀₇	C₂×C₄	1	2	yes	—
42	C₂×D₄₂₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₀₇	C₂×C₄	1	2	yes	—
44	D₄×D₁₀₇	—	2	4	no	—
45	C₂²×Dic₁₀₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₀₇	—	4	2	no	—
47	C₂×C₄×D₁₀₇	—	4	2	no	—
48	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₀₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₀₇	C₂³	1	2	yes	—
54	#13133802	—	4	2	no	—
55	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
58	C₈×D₁₀₇	—	4	2	no	—
59	C₈×D₁₀₇	—	4	2	no	—
60	C₄⋊Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
61	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
63	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
64	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
70	M₁₆×C₁₀₇	—	4	2	no	—
71	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
76	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
77	C₂³×C₁₀₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₇	—	2	4	no	—
79	C₂²×D₁₀₇	—	2	4	no	—
80	C₂²×D₁₀₇	—	4	2	no	—
81	C₂²×D₁₀₇	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₀₇	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₁₀₇	—	2	2	no	—
84	C₂×C₄×C₁₀₇	—	2	2	no	—
85	C₂×C₄×C₁₀₇	—	2	2	no	—
86	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₁₀₇	—	2	4	no	—
91	C₂×Dic₁₀₇	—	4	2	no	—
92	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
96	D₄₂₈	M₁₆	1	4	yes	complemented
97	C₄×D₁₀₇	—	2	2	no	—
98	C₄×D₁₀₇	—	2	2	no	—
99	C₄×D₁₀₇	—	2	2	no	—
100	C₄×D₁₀₇	—	2	2	no	—
101	C₄×D₁₀₇	—	2	1	no	—
102	C₄×D₁₀₇	—	2	1	no	—
103	C₄×D₁₀₇	—	4	2	no	—
104	C₄×D₁₀₇	—	4	2	no	—
105	C₈×C₁₀₇	—	2	1	no	—
106	C₈×C₁₀₇	—	2	2	no	—
107	C₈×C₁₀₇	—	2	1	no	—
108	C₁₀₇⋊C₈	—	2	1	no	—
109	C₁₀₇⋊C₈	—	2	2	no	—
110	C₁₀₇⋊C₈	—	2	1	no	—
111	C₂²×C₁₀₇	—	2	2	no	—
112	C₂²×C₁₀₇	—	2	2	no	—
113	C₂²×C₁₀₇	—	2	2	no	—
114	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₀₇	—	2	1	no	—
116	C₂×D₁₀₇	—	2	2	no	—
117	C₂×D₁₀₇	—	2	1	no	—
118	C₂×D₁₀₇	—	2	2	no	—
119	C₂×D₁₀₇	—	4	8	no	—
120	C₂×D₁₀₇	C₂×M₁₆	1	4	yes	—
121	C₂×D₁₀₇	#131	1	2	yes	—
122	C₄×C₁₀₇	—	2	2	no	—
123	C₄×C₁₀₇	—	2	1	no	—
124	C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
125	C₄×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₁₀₇	—	2	1	no	—
128	Dic₁₀₇	—	2	2	no	—
129	Dic₁₀₇	—	2	1	no	—
130	Dic₁₀₇	—	2	1	no	—
131	C₂×C₁₀₇	—	4	2	no	—
132	C₂×C₁₀₇	#518	1	1	yes	characteristic
133	C₂×C₁₀₇	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₁₀₇	C₂×#131	1	1	yes	characteristic
135	D₁₀₇	—	2	2	no	—
136	D₁₀₇	—	2	4	no	—
137	#2974	—	107	1	no	maximal, Sylow
138	C₁₀₇	#2974	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄×D₄	—	107	1	no	—
140	#436	—	107	1	no	—
141	#436	D₁₀₇	1	1	yes	characteristic, complemented
142	#435	—	107	2	no	—
143	#435	—	107	2	no	—
144	C₈×D₄	—	107	1	no	—
145	C₈×D₄	—	107	1	no	—
146	#432	—	107	1	no	—
147	#431	—	107	1	no	—
148	#407	—	107	2	no	—
149	C₂×C₂²⋊C₈	—	107	2	no	—
150	C₂²×D₄	—	107	1	no	—
151	C₂³×C₄	—	107	2	no	—
152	C₂×M₁₆	—	107	2	no	—
153	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	complemented
154	C₂²×C₈	—	107	2	no	—
155	C₂²×C₈	—	107	2	no	—
156	C₄×D₄	—	107	1	no	—
157	C₄×D₄	—	107	2	no	—
158	C₄×D₄	—	107	1	no	—
159	C₄×D₄	—	107	2	no	—
160	C₄×D₄	—	107	1	no	—
161	C₄×D₄	C₂×D₁₀₇	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	107	1	no	—
163	C₂×C₂²⋊C₄	—	107	2	no	—
164	C₂×C₄²	—	107	1	no	—
165	C₄⋊C₈	—	107	1	no	—
166	C₄⋊C₈	—	107	1	no	—
167	C₄⋊C₈	—	107	1	no	—
168	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
169	C₂²⋊C₈	—	107	2	no	—
170	C₂²⋊C₈	—	107	1	no	—
171	C₂²⋊C₈	—	107	2	no	—
172	C₂²⋊C₈	—	107	1	no	—
173	C₂²⋊C₈	—	107	2	no	—
174	C₂²⋊C₈	—	107	2	no	—
175	C₂²⋊C₈	C₂×D₁₀₇	1	2	yes	complemented
176	#97	—	107	2	no	—
177	C₄×C₈	—	107	1	no	—
178	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
179	C₂⁴	—	107	2	no	—
180	C₂×D₄	—	107	2	no	—
181	C₂×D₄	—	107	1	no	—
182	C₂×D₄	—	214	4	no	—
183	C₂×D₄	C₄×D₁₀₇	1	1	yes	characteristic
184	C₂²×C₄	—	107	2	no	—
185	C₂²×C₄	—	107	2	no	—
186	C₂²×C₄	—	107	2	no	—
187	C₂²×C₄	—	107	1	no	—
188	C₂²×C₄	—	107	1	no	—
189	C₂²×C₄	—	107	1	no	—
190	C₂²×C₄	—	107	2	no	—
191	C₂²×C₄	—	428	2	no	—
192	C₂²×C₄	—	428	2	no	—
193	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	428	2	no	—
196	C₂×C₈	—	107	1	no	—
197	C₂×C₈	—	107	1	no	—
198	C₂×C₈	—	107	2	no	—
199	C₂×C₈	—	428	2	no	—
200	C₂×C₈	—	428	2	no	—
201	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
204	C₄⋊C₄	—	107	2	no	—
205	C₄⋊C₄	—	107	1	no	—
206	C₄⋊C₄	C₄×D₁₀₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	107	2	no	—
208	C₂²⋊C₄	—	107	2	no	—
209	C₂²⋊C₄	—	107	2	no	—
210	C₂²⋊C₄	C₄×D₁₀₇	1	2	yes	—
211	C₄²	—	107	1	no	—
212	C₄²	—	107	1	no	—
213	C₄²	—	107	1	no	—
214	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
215	C₂³	—	107	2	no	—
216	C₂³	—	107	1	no	—
217	C₂³	—	428	2	no	—
218	C₂³	—	214	4	no	—
219	C₂³	—	214	4	no	—
220	C₂³	C₂×C₄×D₁₀₇	1	2	yes	—
221	D₄	—	2	2	no	—
222	D₄	—	107	4	no	—
223	D₄	—	214	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	107	1	no	—
227	C₂×C₄	—	107	1	no	—
228	C₂×C₄	—	107	1	no	—
229	C₂×C₄	—	107	2	no	—
230	C₂×C₄	—	428	2	no	—
231	C₂×C₄	—	428	2	no	—
232	C₂×C₄	—	428	2	no	—
233	C₂×C₄	—	214	2	no	—
234	C₂×C₄	—	214	2	no	—
235	C₂×C₄	—	214	2	no	—
236	C₂×C₄	—	214	1	no	—
237	C₂×C₄	—	214	2	no	—
238	C₂×C₄	—	214	1	no	—
239	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	214	1	no	—
247	C₈	—	214	2	no	—
248	C₈	—	214	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	107	4	no	—
253	C₂²	—	107	2	no	—
254	C₂²	—	428	8	no	—
255	C₂²	—	214	1	no	—
256	C₂²	—	214	2	no	—
257	C₂²	—	214	1	no	—
258	C₂²	—	214	2	no	—
259	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	214	1	no	—
263	C₄	—	214	1	no	—
264	C₄	—	214	1	no	—
265	C₄	—	214	2	no	—
266	C₄	M₁₆×D₁₀₇	1	2	yes	—
267	C₄	#16812154	1	1	yes	characteristic
268	C₄	#16812155	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	214	2	no	—
271	C₂	—	214	4	no	—
272	C₂	#20293099	1	1	yes	characteristic
273	C₂	C₂×M₁₆×D₁₀₇	1	1	yes	characteristic
274	C₂	#131×D₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#436×C₁₀₇	C₂
prime radical	C₂×D₄×D₁₀₇	C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#436	D₁₀₇
hypocenter	C₁₀₇	#2974
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #20292705	C₂	semidirect
3. #20292701	C₂	semidirect
4. #20292696	C₂	semidirect
5. C₈×D₄₂₈	C₂	semidirect
6. #20292692	C₂	semidirect
7. #20292683	C₂	semidirect
8. #20292678	C₂	semidirect
9. #20292590	C₂	semidirect
10. C₂²⋊C₈×D₁₀₇	C₂	semidirect
11. #436×C₁₀₇	C₂	semidirect
14. C₂×#13133802	C₂²	semidirect
15. C₂×C₈×D₁₀₇	C₂²	semidirect
16. C₂×C₈×D₁₀₇	C₂²	semidirect
25. C₄⋊(C₁₀₇⋊C₈)	C₂²	semidirect
26. Dic₁₀₇⋊C₈	C₂²	semidirect
27. Dic₁₀₇⋊C₈	C₂²	semidirect
28. C₂²⋊(C₁₀₇⋊C₈)	C₂²	semidirect
32. (C₂×D₁₀₇)⋊C₈	C₂²	semidirect
33. (C₂×D₁₀₇)⋊C₈	C₂²	semidirect
34. #16812019	C₂²	semidirect
35. C₄×C₁₀₇⋊C₈	C₂²	semidirect
36. C₂×M₁₆×C₁₀₇	C₂²	semidirect
38. C₄⋊C₈×C₁₀₇	C₂²	semidirect
39. C₂²⋊C₈×C₁₀₇	C₂²	semidirect
40. C₄×C₈×C₁₀₇	C₂²	semidirect
96. D₄₂₈	M₁₆	semidirect
138. C₁₀₇	#2974	semidirect
141. #436	D₁₀₇	semidirect
153. C₂×M₁₆	C₂×D₁₀₇	semidirect
168. C₄⋊C₈	C₂×D₁₀₇	semidirect
175. C₂²⋊C₈	C₂×D₁₀₇	semidirect
178. C₄×C₈	C₂×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₁₀₇	2
#20292705	2
#20292701	2
#20292696	2
C₈×D₄₂₈	2
#20292692	2
#20292683	2
#20292678	2
#20292590	2
C₂²⋊C₈×D₁₀₇	2
#436×C₁₀₇	2
#2974	107