Subgroups — C₁₀₇×#13210 #41039174

Order 27392.

510 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#360×C₁₀₇	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₁₀₇	—	2	1	no	—
6	C₂×#135×C₁₀₇	C₂²	1	2	yes	—
7	#532×C₁₀₇	—	2	1	no	—
8	C₂×M₃₂×C₁₀₇	C₂²	1	2	yes	—
9	#495×C₁₀₇	—	2	1	no	—
10	#495×C₁₀₇	C₄	1	2	yes	complemented
11	#494×C₁₀₇	C₄	1	1	yes	characteristic, complemented
12	#493×C₁₀₇	C₄	1	1	yes	characteristic, complemented
13	#492×C₁₀₇	—	2	1	no	—
14	C₂×#106×C₁₀₇	—	2	1	no	—
15	C₂×#106×C₁₀₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	#384×C₁₀₇	—	2	1	no	—
18	#360×C₁₀₇	—	2	8	no	—
19	#346×C₁₀₇	—	2	1	no	—
20	C₄×C₁₆×C₁₀₇	—	2	1	no	—
21	C₂×#41×C₁₀₇	—	2	2	no	—
22	#135×C₁₀₇	—	2	4	no	—
23	#135×C₁₀₇	—	2	4	no	—
24	#135×C₁₀₇	—	4	2	no	—
25	#135×C₁₀₇	—	4	2	no	—
26	C₂×Q₁₆×C₁₀₇	—	2	1	no	—
27	C₂×Q₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₂×SD₁₆×C₁₀₇	—	2	2	no	—
29	C₂×SD₁₆×C₁₀₇	—	2	2	no	—
30	C₂×D₈×C₁₀₇	—	2	1	no	—
31	C₂×D₈×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₀₇	—	2	1	no	—
35	#127×C₁₀₇	—	2	1	no	—
36	#124×C₁₀₇	—	2	2	no	—
37	#121×C₁₀₇	—	4	2	no	—
38	C₄×D₄×C₁₀₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₇	—	4	2	no	—
41	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
42	M₃₂×C₁₀₇	—	2	2	no	—
43	M₃₂×C₁₀₇	D₄	1	4	yes	complemented
44	C₂×C₁₆×C₁₀₇	—	2	2	no	—
45	#106×C₁₀₇	—	2	4	no	—
46	#106×C₁₀₇	—	4	2	no	—
47	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
51	#41×C₁₀₇	—	4	2	no	—
52	#41×C₁₀₇	—	4	2	no	—
53	#41×C₁₀₇	—	4	2	no	—
54	#41×C₁₀₇	—	4	2	no	—
55	C₂×Q₈×C₁₀₇	—	2	2	no	—
56	C₂×D₄×C₁₀₇	—	2	2	no	—
57	C₂×D₄×C₁₀₇	—	4	2	no	—
58	C₂×D₄×C₁₀₇	—	4	2	no	—
59	C₂²×C₄×C₁₀₇	—	4	2	no	—
60	C₂²×C₄×C₁₀₇	—	4	2	no	—
61	C₂²×C₄×C₁₀₇	D₈	1	2	yes	—
62	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₀₇	—	2	4	no	—
64	Q₁₆×C₁₀₇	—	4	2	no	—
65	SD₁₆×C₁₀₇	—	4	4	no	—
66	SD₁₆×C₁₀₇	—	4	4	no	—
67	D₈×C₁₀₇	—	2	4	no	—
68	D₈×C₁₀₇	—	4	2	no	—
69	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₀₇	D₈	1	2	yes	—
72	C₂×C₈×C₁₀₇	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₀₇	#41	1	2	yes	—
77	C₄⋊C₄×C₁₀₇	—	4	2	no	—
78	C₄⋊C₄×C₁₀₇	—	4	2	no	—
79	C₄⋊C₄×C₁₀₇	—	4	2	no	—
80	C₄⋊C₄×C₁₀₇	—	8	4	no	—
81	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
82	C₄²×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₁₀₇	SD₁₆	1	2	yes	—
85	C₁₆×C₁₀₇	—	2	4	no	—
86	C₂³×C₁₀₇	—	4	2	no	—
87	C₂³×C₁₀₇	C₂×D₈	1	1	yes	characteristic
88	Q₈×C₁₀₇	—	4	2	no	—
89	Q₈×C₁₀₇	—	4	2	no	—
90	D₄×C₁₀₇	—	4	2	no	—
91	D₄×C₁₀₇	—	4	2	no	—
92	D₄×C₁₀₇	—	8	2	no	—
93	D₄×C₁₀₇	—	8	2	no	—
94	D₄×C₁₀₇	—	8	4	no	—
95	C₂×C₄×C₁₀₇	—	2	4	no	—
96	C₂×C₄×C₁₀₇	—	4	2	no	—
97	C₂×C₄×C₁₀₇	—	8	2	no	—
98	C₂×C₄×C₁₀₇	—	8	2	no	—
99	C₂×C₄×C₁₀₇	—	8	4	no	—
100	C₂×C₄×C₁₀₇	—	8	2	no	—
101	C₂×C₄×C₁₀₇	#102	1	2	yes	—
102	C₂×C₄×C₁₀₇	#102	1	2	yes	—
103	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₇	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₀₇	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₈×C₁₀₇	#102	1	2	yes	—
110	C₈×C₁₀₇	#102	1	2	yes	—
111	C₈×C₁₀₇	C₄×D₄	1	2	yes	—
112	C₈×C₁₀₇	#121	1	2	yes	—
113	C₂²×C₁₀₇	—	2	2	no	—
114	C₂²×C₁₀₇	—	8	4	no	—
115	C₂²×C₁₀₇	—	8	2	no	—
116	C₂²×C₁₀₇	C₂×#102	1	1	yes	characteristic
117	C₂²×C₁₀₇	#486	1	1	yes	characteristic
118	C₂²×C₁₀₇	#493	1	1	yes	characteristic
119	C₄×C₁₀₇	—	2	2	no	—
120	C₄×C₁₀₇	—	8	2	no	—
121	C₄×C₁₀₇	—	16	2	no	—
122	C₄×C₁₀₇	#360	1	4	yes	complemented
123	C₄×C₁₀₇	#390	1	1	yes	characteristic
124	C₄×C₁₀₇	C₂×#102	1	1	yes	characteristic
125	C₄×C₁₀₇	#486	1	1	yes	characteristic
126	C₄×C₁₀₇	#492	1	1	yes	characteristic
127	#13210	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₁₀₇	—	2	1	no	—
129	C₂×C₁₀₇	—	2	1	no	—
130	C₂×C₁₀₇	—	16	2	no	—
131	C₂×C₁₀₇	#1957	1	1	yes	characteristic
132	C₂×C₁₀₇	C₂×#360	1	2	yes	—
133	#3147	C₂×C₁₀₇	1	1	yes	characteristic
134	C₂×#360	C₂×C₁₀₇	1	4	yes	complemented
135	C₄×M₃₂	C₂×C₁₀₇	1	1	yes	characteristic, complemented
136	#1844	C₂×C₁₀₇	1	1	yes	characteristic, complemented
137	C₁₀₇	#13210	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#135	—	2	1	no	—
139	C₂×#135	C₂²×C₁₀₇	1	2	yes	—
140	#532	—	2	1	no	—
141	C₂×M₃₂	C₂²×C₁₀₇	1	2	yes	—
142	#495	—	2	1	no	—
143	#495	C₄×C₁₀₇	1	2	yes	complemented
144	#494	C₄×C₁₀₇	1	1	yes	characteristic, complemented
145	#493	C₄×C₁₀₇	1	1	yes	characteristic, complemented
146	#492	—	2	1	no	—
147	C₂×#106	—	2	1	no	—
148	C₂×#106	C₂²×C₁₀₇	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
150	#384	—	2	1	no	—
151	#360	—	2	8	no	—
152	#346	—	2	1	no	—
153	C₄×C₁₆	—	2	1	no	—
154	C₂×#41	—	2	2	no	—
155	#135	—	2	4	no	—
156	#135	—	2	4	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	C₂×C₄×C₁₀₇	1	2	yes	—
165	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	2	1	no	—
168	#127	—	2	1	no	—
169	#124	—	2	2	no	—
170	#121	—	4	2	no	—
171	C₄×D₄	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	4	2	no	—
174	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	D₄×C₁₀₇	1	4	yes	complemented
177	C₂×C₁₆	—	2	2	no	—
178	#106	—	2	4	no	—
179	#106	—	4	2	no	—
180	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
181	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
183	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	D₈×C₁₀₇	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
196	Q₁₆	—	2	4	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	2	4	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
204	C₂×C₈	D₈×C₁₀₇	1	2	yes	—
205	C₂×C₈	SD₁₆×C₁₀₇	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
208	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
209	C₂×C₈	#41×C₁₀₇	1	2	yes	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	8	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
216	C₄²	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
217	C₄²	SD₁₆×C₁₀₇	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×D₈×C₁₀₇	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	#102×C₁₀₇	1	2	yes	—
235	C₂×C₄	#102×C₁₀₇	1	2	yes	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
237	C₂×C₄	#124×C₁₀₇	1	1	yes	characteristic
238	C₂×C₄	#127×C₁₀₇	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	1	yes	characteristic
241	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	1	yes	characteristic
242	C₈	#102×C₁₀₇	1	2	yes	—
243	C₈	#102×C₁₀₇	1	2	yes	—
244	C₈	C₄×D₄×C₁₀₇	1	2	yes	—
245	C₈	#121×C₁₀₇	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂×#102×C₁₀₇	1	1	yes	characteristic
250	C₂²	#486×C₁₀₇	1	1	yes	characteristic
251	C₂²	#493×C₁₀₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	16	2	no	—
255	C₄	#360×C₁₀₇	1	4	yes	complemented
256	C₄	#390×C₁₀₇	1	1	yes	characteristic
257	C₄	C₂×#102×C₁₀₇	1	1	yes	characteristic
258	C₄	#486×C₁₀₇	1	1	yes	characteristic
259	C₄	#492×C₁₀₇	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	C₁₀₇×#1957	1	1	yes	characteristic
264	C₂	C₂×#360×C₁₀₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₇
prime radical	C₁₀₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#360×C₁₀₇	C₂	semidirect
3. C₄×M₃₂×C₁₀₇	C₂	semidirect
4. C₁₀₇×#1844	C₂	semidirect
10. #495×C₁₀₇	C₄	semidirect
11. #494×C₁₀₇	C₄	semidirect
12. #493×C₁₀₇	C₄	semidirect
43. M₃₂×C₁₀₇	D₄	semidirect
122. C₄×C₁₀₇	#360	semidirect
127. #13210	C₁₀₇	direct (N × H)
134. C₂×#360	C₂×C₁₀₇	semidirect
135. C₄×M₃₂	C₂×C₁₀₇	semidirect
136. #1844	C₂×C₁₀₇	semidirect
137. C₁₀₇	#13210	direct (N × H)
143. #495	C₄×C₁₀₇	semidirect
144. #494	C₄×C₁₀₇	semidirect
145. #493	C₄×C₁₀₇	semidirect
176. M₃₂	D₄×C₁₀₇	semidirect
255. C₄	#360×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₇×#3147	2
C₂×#360×C₁₀₇	2
C₄×M₃₂×C₁₀₇	2
C₁₀₇×#1844	2
#13210	107

Subgroups — C107×#13210 #41039174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#13210 #41039174