Subgroups — C₁₀₇×#8157 #41034121

Order 27392.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1458	C₂	1	2	yes	maximal
3	C₁₀₇×#1452	C₂	1	2	yes	maximal, complemented
4	C₁₀₇×#1281	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₇	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₇	C₄	1	1	yes	characteristic
7	C₂×#117×C₁₀₇	C₂²	1	1	yes	characteristic
8	#431×C₁₀₇	C₂²	1	4	yes	—
9	#405×C₁₀₇	—	2	4	no	—
10	C₂×#97×C₁₀₇	C₄	1	1	yes	characteristic
11	C₂×#97×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	#324×C₁₀₇	C₄	1	4	yes	—
15	#323×C₁₀₇	C₄	1	4	yes	—
16	#322×C₁₀₇	—	2	4	no	—
17	C₂³×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₇	—	2	2	no	—
19	C₂×M₁₆×C₁₀₇	—	2	2	no	—
20	C₂×M₁₆×C₁₀₇	—	2	2	no	—
21	C₂²×C₈×C₁₀₇	—	2	2	no	—
22	C₂²×C₈×C₁₀₇	—	2	2	no	—
23	C₂²×C₈×C₁₀₇	—	2	2	no	—
24	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	#117×C₁₀₇	—	2	2	no	—
29	#117×C₁₀₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₇	—	2	4	no	—
35	C₂²⋊C₈×C₁₀₇	C₂×C₄	1	8	yes	—
36	#97×C₁₀₇	—	2	2	no	—
37	#97×C₁₀₇	—	2	2	no	—
38	C₄×C₈×C₁₀₇	—	2	2	no	—
39	C₄×C₈×C₁₀₇	—	2	4	no	—
40	C₄×C₈×C₁₀₇	D₄	1	4	yes	—
41	C₂⁴×C₁₀₇	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₇	—	2	1	no	—
43	C₂²×C₄×C₁₀₇	—	2	1	no	—
44	C₂²×C₄×C₁₀₇	—	2	1	no	—
45	C₂²×C₄×C₁₀₇	—	2	1	no	—
46	C₂²×C₄×C₁₀₇	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₀₇	M₁₆	1	2	yes	—
49	C₂²×C₄×C₁₀₇	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₁₀₇	—	2	8	no	—
53	C₂×C₈×C₁₀₇	—	2	8	no	—
54	C₂×C₈×C₁₀₇	—	2	1	no	—
55	C₂×C₈×C₁₀₇	—	2	1	no	—
56	C₂×C₈×C₁₀₇	—	2	1	no	—
57	C₂×C₈×C₁₀₇	—	2	1	no	—
58	C₂×C₈×C₁₀₇	—	2	1	no	—
59	C₂×C₈×C₁₀₇	—	2	1	no	—
60	C₂×C₈×C₁₀₇	—	2	1	no	—
61	C₂×C₈×C₁₀₇	—	2	1	no	—
62	C₂×C₈×C₁₀₇	—	2	8	no	—
63	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₀₇	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₀₇	M₁₆	1	2	yes	—
67	C₄⋊C₄×C₁₀₇	—	2	4	no	—
68	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
69	C₄²×C₁₀₇	—	2	1	no	—
70	C₄²×C₁₀₇	—	2	1	no	—
71	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
72	C₄²×C₁₀₇	#41	1	2	yes	—
73	C₂³×C₁₀₇	—	2	1	no	—
74	C₂³×C₁₀₇	—	2	1	no	—
75	C₂³×C₁₀₇	—	4	2	no	—
76	C₂³×C₁₀₇	#97	1	2	yes	—
77	C₂³×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₀₇	—	2	1	no	—
79	C₂×C₄×C₁₀₇	—	2	1	no	—
80	C₂×C₄×C₁₀₇	—	2	2	no	—
81	C₂×C₄×C₁₀₇	—	2	2	no	—
82	C₂×C₄×C₁₀₇	—	2	2	no	—
83	C₂×C₄×C₁₀₇	—	2	2	no	—
84	C₂×C₄×C₁₀₇	—	2	2	no	—
85	C₂×C₄×C₁₀₇	—	4	2	no	—
86	C₂×C₄×C₁₀₇	—	4	2	no	—
87	C₂×C₄×C₁₀₇	#97	1	2	yes	—
88	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₁₀₇	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₁₀₇	—	2	2	no	—
96	C₈×C₁₀₇	—	2	2	no	—
97	C₈×C₁₀₇	—	2	2	no	—
98	C₈×C₁₀₇	—	2	2	no	—
99	C₈×C₁₀₇	—	4	8	no	—
100	C₂²×C₁₀₇	—	2	2	no	—
101	C₂²×C₁₀₇	—	2	2	no	—
102	C₂²×C₁₀₇	—	4	2	no	—
103	C₂²×C₁₀₇	—	4	2	no	—
104	C₂²×C₁₀₇	—	4	2	no	—
105	C₂²×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₁₀₇	C₂×#97	1	1	yes	characteristic
107	C₂²×C₁₀₇	C₄×M₁₆	1	1	yes	characteristic
108	C₄×C₁₀₇	—	2	2	no	—
109	C₄×C₁₀₇	—	2	4	no	—
110	C₄×C₁₀₇	—	4	2	no	—
111	C₄×C₁₀₇	#407	1	1	yes	characteristic
112	C₄×C₁₀₇	#407	1	1	yes	characteristic
113	C₄×C₁₀₇	#432	1	1	yes	characteristic
114	C₄×C₁₀₇	#432	1	1	yes	characteristic
115	#8157	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₁₀₇	—	2	1	no	—
117	C₂×C₁₀₇	—	2	1	no	—
118	C₂×C₁₀₇	—	4	2	no	—
119	C₂×C₁₀₇	#1749	1	1	yes	characteristic
120	C₂×C₁₀₇	#1757	1	2	yes	—
121	C₂×#405	C₂×C₁₀₇	1	1	yes	characteristic
122	#1458	C₂×C₁₀₇	1	2	yes	—
123	#1452	C₂×C₁₀₇	1	2	yes	complemented
124	#1281	C₂×C₁₀₇	1	2	yes	complemented
125	C₁₀₇	#8157	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×M₁₆	C₄×C₁₀₇	1	1	yes	characteristic
127	C₂³×C₈	C₄×C₁₀₇	1	1	yes	characteristic
128	C₂×#117	C₂²×C₁₀₇	1	1	yes	characteristic
129	#431	C₂²×C₁₀₇	1	4	yes	—
130	#405	—	2	4	no	—
131	C₂×#97	C₄×C₁₀₇	1	1	yes	characteristic
132	C₂×#97	C₂²×C₁₀₇	1	1	yes	characteristic
133	C₂×C₄×C₈	C₄×C₁₀₇	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
135	#324	C₄×C₁₀₇	1	4	yes	—
136	#323	C₄×C₁₀₇	1	4	yes	—
137	#322	—	2	4	no	—
138	C₂³×C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
148	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
149	#117	—	2	2	no	—
150	#117	—	2	2	no	—
151	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
153	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	C₂²⋊C₈	C₂×C₄×C₁₀₇	1	8	yes	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	4	no	—
161	C₄×C₈	D₄×C₁₀₇	1	4	yes	—
162	C₂⁴	C₄²×C₁₀₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₄²×C₁₀₇	1	1	yes	characteristic
168	C₂²×C₄	C₄²×C₁₀₇	1	2	yes	—
169	C₂²×C₄	M₁₆×C₁₀₇	1	2	yes	—
170	C₂²×C₄	M₁₆×C₁₀₇	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₀₇	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
185	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
186	C₂×C₈	M₁₆×C₁₀₇	1	2	yes	—
187	C₂×C₈	M₁₆×C₁₀₇	1	2	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
193	C₄²	#41×C₁₀₇	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#97×C₁₀₇	1	2	yes	—
198	C₂³	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	#97×C₁₀₇	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
210	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
211	C₂×C₄	C₄×D₄×C₁₀₇	1	4	yes	—
212	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
213	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
214	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
215	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	4	8	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
227	C₂²	C₂×#97×C₁₀₇	1	1	yes	characteristic
228	C₂²	C₄×M₁₆×C₁₀₇	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	#407×C₁₀₇	1	1	yes	characteristic
233	C₄	#407×C₁₀₇	1	1	yes	characteristic
234	C₄	#432×C₁₀₇	1	1	yes	characteristic
235	C₄	#432×C₁₀₇	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	4	2	no	—
239	C₂	C₁₀₇×#1749	1	1	yes	characteristic
240	C₂	C₁₀₇×#1757	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₇	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₇×#1452	C₂	semidirect
4. C₁₀₇×#1281	C₂	semidirect
115. #8157	C₁₀₇	direct (N × H)
123. #1452	C₂×C₁₀₇	semidirect
124. #1281	C₂×C₁₀₇	semidirect
125. C₁₀₇	#8157	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#405×C₁₀₇	2
C₁₀₇×#1458	2
C₁₀₇×#1452	2
C₁₀₇×#1281	2
#8157	107

Subgroups — C107×#8157 #41034121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8157 #41034121