Subgroups — #433×D₁₀₇ #28345083

Order 13696.

412 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₁₀₇	C₂	1	1	yes	maximal, characteristic
2	#20292689	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292688	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292687	C₂	1	2	yes	maximal, complemented
5	#20292686	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292685	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₈×D₁₀₇	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₈×D₁₀₇	C₂	1	2	yes	maximal
9	#97×D₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#433×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₀₇	C₂²	1	2	yes	—
14	C₂×C₈×D₁₀₇	C₂²	1	2	yes	—
15	#16812068	C₂²	1	1	yes	characteristic
16	#16812067	C₄	1	2	yes	—
17	#16812066	C₂²	1	1	yes	characteristic
18	#16812066	C₂²	1	1	yes	characteristic
19	#16812065	C₄	1	2	yes	—
20	C₄⋊C₄×D₁₀₇	C₄	1	2	yes	—
21	C₂²⋊C₄×D₁₀₇	C₄	1	2	yes	—
22	C₄²×D₁₀₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₀₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	complemented
25	Dic₁₀₇⋊C₈	C₂²	1	2	yes	complemented
26	Dic₁₀₇⋊C₈	C₂²	1	2	yes	complemented
27	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
30	#16812020	C₂²	1	1	yes	characteristic, complemented
31	#16812019	C₂²	1	2	yes	complemented
32	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₈×Dic₁₀₇	C₂²	1	2	yes	complemented
34	#117×C₁₀₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	complemented
36	C₂²⋊C₈×C₁₀₇	C₂²	1	2	yes	—
37	#97×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₁₀₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₀₇	—	2	4	no	—
42	C₂×C₄×D₁₀₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₀₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
50	C₈×D₁₀₇	—	2	4	no	—
51	C₈×D₁₀₇	—	2	4	no	—
52	C₄⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
53	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
56	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₀₇	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₀₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
64	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
67	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
68	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
69	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₀₇	—	2	2	no	—
71	C₂²×D₁₀₇	—	2	4	no	—
72	C₂²×D₁₀₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₁₀₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₀₇	—	2	2	no	—
75	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₁₀₇	—	2	2	no	—
80	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₀₇	—	2	2	no	—
85	C₄×D₁₀₇	—	2	4	no	—
86	C₄×D₁₀₇	—	2	4	no	—
87	C₄×D₁₀₇	—	2	2	no	—
88	C₄×D₁₀₇	—	2	2	no	—
89	C₄×D₁₀₇	#41	1	4	yes	—
90	C₈×C₁₀₇	—	2	2	no	—
91	C₈×C₁₀₇	—	2	2	no	—
92	C₁₀₇⋊C₈	—	2	2	no	—
93	C₁₀₇⋊C₈	—	2	2	no	—
94	C₂²×C₁₀₇	—	2	1	no	—
95	C₂²×C₁₀₇	—	2	2	no	—
96	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₀₇	—	2	1	no	—
98	C₂×D₁₀₇	—	2	2	no	—
99	C₂×D₁₀₇	—	4	4	no	—
100	C₂×D₁₀₇	#117	1	2	yes	—
101	C₂×D₁₀₇	#131	1	4	yes	—
102	C₄×C₁₀₇	—	2	1	no	—
103	C₄×C₁₀₇	—	2	2	no	—
104	C₄×C₁₀₇	—	2	2	no	—
105	C₄×C₁₀₇	C₂×#41	1	2	yes	—
106	Dic₁₀₇	—	2	1	no	—
107	Dic₁₀₇	—	2	2	no	—
108	Dic₁₀₇	—	2	2	no	—
109	Dic₁₀₇	#117	1	2	yes	—
110	C₂×C₁₀₇	—	4	1	no	—
111	C₂×C₁₀₇	C₂×#117	1	1	yes	characteristic
112	C₂×C₁₀₇	C₂×#131	1	2	yes	—
113	D₁₀₇	—	4	1	no	—
114	D₁₀₇	#433	1	4	yes	direct product factor
115	C₂×#433	—	107	1	no	maximal, Sylow
116	C₁₀₇	C₂×#433	1	1	yes	characteristic, complemented, Sylow
117	C₂×#117	—	107	1	no	—
118	#433	—	107	1	no	—
119	#433	—	107	2	no	—
120	#433	—	107	2	no	—
121	#433	—	107	1	no	—
122	#433	—	107	1	no	—
123	#433	D₁₀₇	1	1	yes	characteristic, direct product factor
124	C₂×C₄⋊C₈	—	107	2	no	—
125	C₂×C₂²⋊C₈	—	107	2	no	—
126	C₂×#97	—	107	1	no	—
127	C₂×C₄×C₈	—	107	1	no	—
128	C₂³×C₄	—	107	1	no	—
129	C₂²×C₈	—	107	2	no	—
130	C₂²×C₈	—	107	2	no	—
131	#117	—	107	1	no	—
132	#117	—	107	2	no	—
133	#117	—	107	2	no	—
134	#117	—	107	1	no	—
135	#117	—	107	1	no	—
136	#117	C₂×D₁₀₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	107	2	no	—
138	C₂×C₂²⋊C₄	—	107	2	no	—
139	C₂×C₄²	—	107	1	no	—
140	C₂×C₄²	—	107	1	no	—
141	C₄⋊C₈	—	107	2	no	—
142	C₄⋊C₈	—	107	2	no	—
143	C₄⋊C₈	—	107	2	no	—
144	C₄⋊C₈	C₂×D₁₀₇	1	2	yes	complemented
145	C₂²⋊C₈	—	107	2	no	—
146	C₂²⋊C₈	—	107	2	no	—
147	C₂²⋊C₈	—	107	2	no	—
148	C₂²⋊C₈	C₂×D₁₀₇	1	2	yes	—
149	#97	—	107	2	no	—
150	#97	—	107	1	no	—
151	#97	C₂×D₁₀₇	1	1	yes	characteristic, complemented
152	C₄×C₈	—	107	2	no	—
153	C₄×C₈	—	107	1	no	—
154	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic, complemented
155	C₂⁴	—	107	1	no	—
156	C₂²×C₄	—	107	1	no	—
157	C₂²×C₄	—	107	1	no	—
158	C₂²×C₄	—	107	2	no	—
159	C₂²×C₄	—	107	2	no	—
160	C₂²×C₄	—	107	1	no	—
161	C₂²×C₄	—	107	1	no	—
162	C₂²×C₄	—	107	1	no	—
163	C₂²×C₄	—	214	4	no	—
164	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
165	C₂×C₈	—	107	2	no	—
166	C₂×C₈	—	107	2	no	—
167	C₂×C₈	—	214	4	no	—
168	C₂×C₈	—	214	4	no	—
169	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
170	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
171	C₄⋊C₄	—	107	2	no	—
172	C₄⋊C₄	—	107	2	no	—
173	C₄⋊C₄	—	107	2	no	—
174	C₄⋊C₄	C₄×D₁₀₇	1	2	yes	—
175	C₂²⋊C₄	—	107	2	no	—
176	C₂²⋊C₄	—	107	2	no	—
177	C₂²⋊C₄	—	107	2	no	—
178	C₂²⋊C₄	C₄×D₁₀₇	1	2	yes	—
179	C₄²	—	107	2	no	—
180	C₄²	—	107	2	no	—
181	C₄²	—	107	1	no	—
182	C₄²	—	107	1	no	—
183	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
184	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
185	C₂³	—	107	1	no	—
186	C₂³	—	107	1	no	—
187	C₂³	—	214	2	no	—
188	C₂³	—	214	4	no	—
189	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	107	1	no	—
192	C₂×C₄	—	107	4	no	—
193	C₂×C₄	—	107	2	no	—
194	C₂×C₄	—	107	2	no	—
195	C₂×C₄	—	107	1	no	—
196	C₂×C₄	—	214	2	no	—
197	C₂×C₄	—	214	2	no	—
198	C₂×C₄	—	214	4	no	—
199	C₂×C₄	—	214	4	no	—
200	C₂×C₄	—	214	2	no	—
201	C₂×C₄	—	214	2	no	—
202	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	214	2	no	—
209	C₈	—	214	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	107	4	no	—
213	C₂²	—	107	2	no	—
214	C₂²	—	428	4	no	—
215	C₂²	—	214	2	no	—
216	C₂²	—	214	1	no	—
217	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	107	2	no	—
222	C₄	—	214	2	no	—
223	C₄	—	214	2	no	—
224	C₄	—	214	1	no	—
225	C₄	#41×D₁₀₇	1	2	yes	—
226	C₂	—	4	1	no	—
227	C₂	—	107	4	no	—
228	C₂	—	428	1	no	—
229	C₂	#117×D₁₀₇	1	1	yes	characteristic
230	C₂	#131×D₁₀₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	#433×C₁₀₇	C₂
prime radical	C₂³×D₁₀₇	C₂×C₄
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	#433	D₁₀₇
hypocenter	C₁₀₇	C₂×#433
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #20292689	C₂	semidirect
3. #20292688	C₂	semidirect
4. #20292687	C₂	semidirect
5. #20292686	C₂	semidirect
6. #20292685	C₂	semidirect
7. C₄⋊C₈×D₁₀₇	C₂	semidirect
9. #97×D₁₀₇	C₂	semidirect
10. C₄×C₈×D₁₀₇	C₂	semidirect
11. #433×C₁₀₇	C₂	semidirect
24. C₄⋊(C₁₀₇⋊C₈)	C₂²	semidirect
25. Dic₁₀₇⋊C₈	C₂²	semidirect
26. Dic₁₀₇⋊C₈	C₂²	semidirect
30. #16812020	C₂²	semidirect
31. #16812019	C₂²	semidirect
32. C₄×C₁₀₇⋊C₈	C₂²	semidirect
33. C₈×Dic₁₀₇	C₂²	semidirect
35. C₄⋊C₈×C₁₀₇	C₂²	semidirect
37. #97×C₁₀₇	C₂²	semidirect
38. C₄×C₈×C₁₀₇	C₂²	semidirect
114. D₁₀₇	#433	direct (N × H)
116. C₁₀₇	C₂×#433	semidirect
123. #433	D₁₀₇	direct (N × H)
144. C₄⋊C₈	C₂×D₁₀₇	semidirect
151. #97	C₂×D₁₀₇	semidirect
154. 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
#117×D₁₀₇	2
#20292689	2
#20292688	2
#20292687	2
#20292686	2
#20292685	2
C₄⋊C₈×D₁₀₇	2
C₂²⋊C₈×D₁₀₇	2
#97×D₁₀₇	2
C₄×C₈×D₁₀₇	2
#433×C₁₀₇	2
C₂×#433	107

Subgroups — #433×D107 #28345083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #433×D₁₀₇ #28345083