Subgroups — C₁₀₇×#10767 #41036731

Order 27392.

1148 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#413×C₁₀₇	C₂	1	2	yes	maximal
3	C₁₀₇×#1738	C₂	1	4	yes	maximal, complemented
4	C₂²×#41×C₁₀₇	C₂²	1	1	yes	characteristic
5	#568×C₁₀₇	—	2	8	no	—
6	C₂×#131×C₁₀₇	—	2	4	no	—
7	C₂²×M₁₆×C₁₀₇	C₄	1	4	yes	—
8	C₂²×M₁₆×C₁₀₇	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₁₀₇	C₂²	1	2	yes	—
10	#413×C₁₀₇	—	2	4	no	—
11	#413×C₁₀₇	C₄	1	8	yes	—
12	C₂×#101×C₁₀₇	—	2	2	no	—
13	C₂×#100×C₁₀₇	—	2	2	no	—
14	#341×C₁₀₇	—	2	8	no	—
15	C₂×#41×C₁₀₇	—	2	4	no	—
16	C₂×#41×C₁₀₇	—	2	4	no	—
17	C₂×#41×C₁₀₇	C₂×C₄	1	4	yes	—
18	C₂×#41×C₁₀₇	D₄	1	4	yes	—
19	C₂²×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
20	C₂²×D₄×C₁₀₇	D₄	1	1	yes	characteristic
21	C₂²×D₄×C₁₀₇	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
24	#131×C₁₀₇	—	2	32	no	—
25	C₂×M₁₆×C₁₀₇	—	2	4	no	—
26	C₂×M₁₆×C₁₀₇	—	2	2	no	—
27	C₂×M₁₆×C₁₀₇	—	2	4	no	—
28	C₂×M₁₆×C₁₀₇	—	2	16	no	—
29	C₂×M₁₆×C₁₀₇	—	2	4	no	—
30	C₂×M₁₆×C₁₀₇	—	2	4	no	—
31	C₂×M₁₆×C₁₀₇	—	2	4	no	—
32	C₂×M₁₆×C₁₀₇	—	2	4	no	—
33	C₂×M₁₆×C₁₀₇	—	2	4	no	—
34	C₂×M₁₆×C₁₀₇	C₂×C₄	1	8	yes	—
35	C₂×M₁₆×C₁₀₇	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₁₀₇	D₄	1	8	yes	—
37	C₂²×C₈×C₁₀₇	—	2	4	no	—
38	C₂²×C₈×C₁₀₇	—	2	2	no	—
39	C₂²×C₈×C₁₀₇	—	2	4	no	—
40	#101×C₁₀₇	—	2	8	no	—
41	#100×C₁₀₇	—	2	8	no	—
42	C₂⁴×C₁₀₇	C₄⋊C₄	1	2	yes	—
43	C₂⁴×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
44	#41×C₁₀₇	—	2	8	no	—
45	#41×C₁₀₇	—	2	8	no	—
46	#41×C₁₀₇	—	2	8	no	—
47	#41×C₁₀₇	—	2	8	no	—
48	C₂×Q₈×C₁₀₇	—	2	4	no	—
49	C₂×Q₈×C₁₀₇	C₂²⋊C₄	1	4	yes	—
50	C₂×D₄×C₁₀₇	—	2	8	no	—
51	C₂×D₄×C₁₀₇	—	2	4	no	—
52	C₂×D₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
53	C₂×D₄×C₁₀₇	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₁₀₇	—	2	4	no	—
55	C₂²×C₄×C₁₀₇	—	2	4	no	—
56	C₂²×C₄×C₁₀₇	C₄²	1	4	yes	—
57	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₁₀₇	#41	1	4	yes	—
65	M₁₆×C₁₀₇	—	2	16	no	—
66	M₁₆×C₁₀₇	—	2	4	no	—
67	M₁₆×C₁₀₇	—	2	8	no	—
68	M₁₆×C₁₀₇	—	2	8	no	—
69	M₁₆×C₁₀₇	—	2	8	no	—
70	M₁₆×C₁₀₇	—	4	4	no	—
71	M₁₆×C₁₀₇	—	4	4	no	—
72	M₁₆×C₁₀₇	C₂²⋊C₄	1	8	yes	—
73	C₂×C₈×C₁₀₇	—	2	16	no	—
74	C₂×C₈×C₁₀₇	—	2	4	no	—
75	C₂×C₈×C₁₀₇	—	2	8	no	—
76	C₂×C₈×C₁₀₇	—	2	8	no	—
77	C₂×C₈×C₁₀₇	—	4	4	no	—
78	C₂×C₈×C₁₀₇	—	4	4	no	—
79	C₂³×C₁₀₇	—	2	4	no	—
80	C₂³×C₁₀₇	—	8	1	no	—
81	C₂³×C₁₀₇	—	8	2	no	—
82	C₂³×C₁₀₇	#95	1	4	yes	—
83	C₂³×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₁₀₇	C₂²⋊Q₈	1	4	yes	—
86	Q₈×C₁₀₇	—	2	8	no	—
87	D₄×C₁₀₇	—	2	16	no	—
88	D₄×C₁₀₇	—	2	8	no	—
89	C₂×C₄×C₁₀₇	—	2	4	no	—
90	C₂×C₄×C₁₀₇	—	2	8	no	—
91	C₂×C₄×C₁₀₇	—	2	4	no	—
92	C₂×C₄×C₁₀₇	—	2	4	no	—
93	C₂×C₄×C₁₀₇	#95	1	4	yes	—
94	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₀₇	#117	1	2	yes	—
99	C₂×C₄×C₁₀₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₁₀₇	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₁₀₇	#120	1	4	yes	—
102	C₂×C₄×C₁₀₇	#121	1	4	yes	—
103	C₂×C₄×C₁₀₇	#123	1	4	yes	—
104	C₈×C₁₀₇	—	2	8	no	—
105	C₈×C₁₀₇	—	4	8	no	—
106	C₂²×C₁₀₇	—	2	4	no	—
107	C₂²×C₁₀₇	—	4	1	no	—
108	C₂²×C₁₀₇	—	4	2	no	—
109	C₂²×C₁₀₇	—	4	4	no	—
110	C₂²×C₁₀₇	—	4	4	no	—
111	C₂²×C₁₀₇	—	8	4	no	—
112	C₂²×C₁₀₇	C₂×#95	1	1	yes	characteristic
113	C₂²×C₁₀₇	#380	1	2	yes	—
114	C₂²×C₁₀₇	#385	1	4	yes	—
115	C₄×C₁₀₇	—	2	4	no	—
116	C₄×C₁₀₇	C₄×C₂²⋊C₄	1	2	yes	—
117	C₄×C₁₀₇	#379	1	1	yes	characteristic
118	C₄×C₁₀₇	#381	1	1	yes	characteristic
119	C₄×C₁₀₇	#386	1	4	yes	—
120	#10767	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₀₇	—	2	1	no	—
122	C₂×C₁₀₇	—	2	2	no	—
123	C₂×C₁₀₇	—	2	4	no	—
124	C₂×C₁₀₇	—	4	4	no	—
125	C₂×C₁₀₇	C₂²⋊#95	1	1	yes	characteristic
126	C₂×#568	C₂×C₁₀₇	1	1	yes	characteristic
127	C₂×#413	C₂×C₁₀₇	1	2	yes	—
128	#1738	C₂×C₁₀₇	1	4	yes	complemented
129	C₁₀₇	#10767	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×#41	C₂²×C₁₀₇	1	1	yes	characteristic
131	#568	—	2	8	no	—
132	C₂×#131	—	2	4	no	—
133	C₂²×M₁₆	C₄×C₁₀₇	1	4	yes	—
134	C₂²×M₁₆	C₂²×C₁₀₇	1	4	yes	—
135	C₂²×M₁₆	C₂²×C₁₀₇	1	2	yes	—
136	#413	—	2	4	no	—
137	#413	C₄×C₁₀₇	1	8	yes	—
138	C₂×#101	—	2	2	no	—
139	C₂×#100	—	2	2	no	—
140	#341	—	2	8	no	—
141	C₂×#41	—	2	4	no	—
142	C₂×#41	—	2	4	no	—
143	C₂×#41	C₂×C₄×C₁₀₇	1	4	yes	—
144	C₂×#41	D₄×C₁₀₇	1	4	yes	—
145	C₂²×Q₈	D₄×C₁₀₇	1	1	yes	characteristic
146	C₂²×D₄	D₄×C₁₀₇	1	1	yes	characteristic
147	C₂²×D₄	Q₈×C₁₀₇	1	2	yes	—
148	C₂³×C₄	C₂×C₄×C₁₀₇	1	2	yes	—
149	C₂³×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
150	#131	—	2	32	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	16	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	C₂×C₄×C₁₀₇	1	8	yes	—
161	C₂×M₁₆	C₂×C₄×C₁₀₇	1	4	yes	—
162	C₂×M₁₆	D₄×C₁₀₇	1	8	yes	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	4	no	—
166	#101	—	2	8	no	—
167	#100	—	2	8	no	—
168	C₂⁴	C₄⋊C₄×C₁₀₇	1	2	yes	—
169	C₂⁴	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
170	#41	—	2	8	no	—
171	#41	—	2	8	no	—
172	#41	—	2	8	no	—
173	#41	—	2	8	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	C₂²⋊C₄×C₁₀₇	1	4	yes	—
176	C₂×D₄	—	2	8	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	C₂²⋊C₄×C₁₀₇	1	4	yes	—
179	C₂×D₄	C₄⋊C₄×C₁₀₇	1	8	yes	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₄²×C₁₀₇	1	4	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	4	yes	—
184	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₁₀₇	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₁₀₇	1	4	yes	—
190	C₂²×C₄	#41×C₁₀₇	1	4	yes	—
191	M₁₆	—	2	16	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	8	no	—
194	M₁₆	—	2	8	no	—
195	M₁₆	—	2	8	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	4	no	—
198	M₁₆	C₂²⋊C₄×C₁₀₇	1	8	yes	—
199	C₂×C₈	—	2	16	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	8	1	no	—
207	C₂³	—	8	2	no	—
208	C₂³	#95×C₁₀₇	1	4	yes	—
209	C₂³	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
210	C₂³	C₂×C₄⋊C₄×C₁₀₇	1	2	yes	—
211	C₂³	C₂²⋊Q₈×C₁₀₇	1	4	yes	—
212	Q₈	—	2	8	no	—
213	D₄	—	2	16	no	—
214	D₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	#95×C₁₀₇	1	4	yes	—
220	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	2	yes	—
224	C₂×C₄	#117×C₁₀₇	1	2	yes	—
225	C₂×C₄	C₄×D₄×C₁₀₇	1	4	yes	—
226	C₂×C₄	C₄×D₄×C₁₀₇	1	4	yes	—
227	C₂×C₄	#120×C₁₀₇	1	4	yes	—
228	C₂×C₄	#121×C₁₀₇	1	4	yes	—
229	C₂×C₄	#123×C₁₀₇	1	4	yes	—
230	C₈	—	2	8	no	—
231	C₈	—	4	8	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	8	4	no	—
238	C₂²	C₂×#95×C₁₀₇	1	1	yes	characteristic
239	C₂²	#380×C₁₀₇	1	2	yes	—
240	C₂²	#385×C₁₀₇	1	4	yes	—
241	C₄	—	2	4	no	—
242	C₄	C₄×C₂²⋊C₄×C₁₀₇	1	2	yes	—
243	C₄	#379×C₁₀₇	1	1	yes	characteristic
244	C₄	#381×C₁₀₇	1	1	yes	characteristic
245	C₄	#386×C₁₀₇	1	4	yes	—
246	C₂	—	2	1	no	—
247	C₂	—	2	2	no	—
248	C₂	—	2	4	no	—
249	C₂	—	4	4	no	—
250	C₂	C₁₀₇×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₇	#379
commutator	C₂×C₄	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₇
socle	C₂×C₁₀₇	C₂²⋊#95
prime radical	C₂²×#41×C₁₀₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₇ ◁ C₂³×C₄×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₇×#1738	C₂	semidirect
120. #10767	C₁₀₇	direct (N × H)
128. #1738	C₂×C₁₀₇	semidirect
129. C₁₀₇	#10767	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×#568×C₁₀₇	2
C₂×#413×C₁₀₇	2
C₁₀₇×#1738	2
#10767	107

Subgroups — C107×#10767 #41036731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#10767 #41036731