Subgroups — C₁₀₇×#10760 #41036724

Order 27392.

1170 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#413×C₁₀₇	C₂	1	2	yes	maximal
3	C₂×#341×C₁₀₇	C₂	1	2	yes	maximal, complemented
4	C₂×#338×C₁₀₇	C₂	1	2	yes	maximal, complemented
5	C₂²×#41×C₁₀₇	C₂²	1	1	yes	characteristic
6	C₂×#131×C₁₀₇	—	2	4	no	—
7	C₂×#131×C₁₀₇	—	2	4	no	—
8	C₂×#131×C₁₀₇	—	2	4	no	—
9	C₂²×M₁₆×C₁₀₇	C₄	1	2	yes	—
10	C₂²×M₁₆×C₁₀₇	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₀₇	C₄	1	2	yes	—
13	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	#413×C₁₀₇	—	2	4	no	—
15	#413×C₁₀₇	C₄	1	8	yes	—
16	C₂×#101×C₁₀₇	—	2	2	no	—
17	C₂×#100×C₁₀₇	—	2	2	no	—
18	#341×C₁₀₇	—	2	4	no	—
19	#338×C₁₀₇	—	2	4	no	—
20	C₂×#41×C₁₀₇	—	2	2	no	—
21	C₂×#41×C₁₀₇	—	2	2	no	—
22	C₂×#41×C₁₀₇	—	2	2	no	—
23	C₂×#41×C₁₀₇	—	2	2	no	—
24	C₂×#41×C₁₀₇	C₂×C₄	1	4	yes	—
25	C₂×#41×C₁₀₇	D₄	1	4	yes	—
26	C₂²×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₀₇	D₄	1	1	yes	characteristic
28	C₂²×D₄×C₁₀₇	Q₈	1	2	yes	—
29	C₂³×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
31	#131×C₁₀₇	—	2	32	no	—
32	C₂×M₁₆×C₁₀₇	—	2	4	no	—
33	C₂×M₁₆×C₁₀₇	—	2	2	no	—
34	C₂×M₁₆×C₁₀₇	—	2	2	no	—
35	C₂×M₁₆×C₁₀₇	—	2	4	no	—
36	C₂×M₁₆×C₁₀₇	—	2	4	no	—
37	C₂×M₁₆×C₁₀₇	—	2	4	no	—
38	C₂×M₁₆×C₁₀₇	—	2	4	no	—
39	C₂×M₁₆×C₁₀₇	—	2	4	no	—
40	C₂×M₁₆×C₁₀₇	—	2	4	no	—
41	C₂×M₁₆×C₁₀₇	—	2	4	no	—
42	C₂×M₁₆×C₁₀₇	C₂×C₄	1	8	yes	—
43	C₂×M₁₆×C₁₀₇	D₄	1	4	yes	—
44	C₂²×C₈×C₁₀₇	—	2	4	no	—
45	C₂²×C₈×C₁₀₇	—	2	2	no	—
46	C₂²×C₈×C₁₀₇	—	2	2	no	—
47	C₂²×C₈×C₁₀₇	—	2	4	no	—
48	C₂²×C₈×C₁₀₇	—	2	4	no	—
49	C₂²×C₈×C₁₀₇	—	2	4	no	—
50	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₇	D₄	1	4	yes	—
54	#101×C₁₀₇	—	2	8	no	—
55	#100×C₁₀₇	—	2	8	no	—
56	C₂⁴×C₁₀₇	C₄⋊C₄	1	2	yes	—
57	C₂⁴×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
58	#41×C₁₀₇	—	2	8	no	—
59	#41×C₁₀₇	—	2	8	no	—
60	#41×C₁₀₇	—	2	8	no	—
61	#41×C₁₀₇	—	2	8	no	—
62	C₂×Q₈×C₁₀₇	—	2	2	no	—
63	C₂×Q₈×C₁₀₇	—	2	2	no	—
64	C₂×Q₈×C₁₀₇	C₂²⋊C₄	1	4	yes	—
65	C₂×D₄×C₁₀₇	—	2	4	no	—
66	C₂×D₄×C₁₀₇	—	2	4	no	—
67	C₂×D₄×C₁₀₇	—	2	2	no	—
68	C₂×D₄×C₁₀₇	—	2	2	no	—
69	C₂×D₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
70	C₂×D₄×C₁₀₇	C₄⋊C₄	1	8	yes	—
71	C₂²×C₄×C₁₀₇	—	2	2	no	—
72	C₂²×C₄×C₁₀₇	—	2	2	no	—
73	C₂²×C₄×C₁₀₇	—	2	2	no	—
74	C₂²×C₄×C₁₀₇	—	2	2	no	—
75	C₂²×C₄×C₁₀₇	C₄²	1	4	yes	—
76	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₀₇	#41	1	2	yes	—
85	C₂²×C₄×C₁₀₇	#41	1	2	yes	—
86	M₁₆×C₁₀₇	—	2	16	no	—
87	M₁₆×C₁₀₇	—	2	8	no	—
88	M₁₆×C₁₀₇	—	2	8	no	—
89	M₁₆×C₁₀₇	—	2	8	no	—
90	M₁₆×C₁₀₇	—	4	4	no	—
91	M₁₆×C₁₀₇	—	4	4	no	—
92	C₂×C₈×C₁₀₇	—	2	16	no	—
93	C₂×C₈×C₁₀₇	—	2	1	no	—
94	C₂×C₈×C₁₀₇	—	2	1	no	—
95	C₂×C₈×C₁₀₇	—	2	1	no	—
96	C₂×C₈×C₁₀₇	—	2	2	no	—
97	C₂×C₈×C₁₀₇	—	2	2	no	—
98	C₂×C₈×C₁₀₇	—	2	1	no	—
99	C₂×C₈×C₁₀₇	—	2	8	no	—
100	C₂×C₈×C₁₀₇	—	2	8	no	—
101	C₂×C₈×C₁₀₇	—	4	4	no	—
102	C₂×C₈×C₁₀₇	—	4	4	no	—
103	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
104	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	4	yes	—
105	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
106	C₂³×C₁₀₇	—	2	2	no	—
107	C₂³×C₁₀₇	—	2	2	no	—
108	C₂³×C₁₀₇	—	4	2	no	—
109	C₂³×C₁₀₇	—	4	4	no	—
110	C₂³×C₁₀₇	#95	1	4	yes	—
111	C₂³×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
114	C₂³×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
115	Q₈×C₁₀₇	—	2	8	no	—
116	D₄×C₁₀₇	—	2	16	no	—
117	D₄×C₁₀₇	—	2	8	no	—
118	C₂×C₄×C₁₀₇	—	2	2	no	—
119	C₂×C₄×C₁₀₇	—	2	2	no	—
120	C₂×C₄×C₁₀₇	—	2	4	no	—
121	C₂×C₄×C₁₀₇	—	2	4	no	—
122	C₂×C₄×C₁₀₇	—	2	4	no	—
123	C₂×C₄×C₁₀₇	—	2	4	no	—
124	C₂×C₄×C₁₀₇	#95	1	4	yes	—
125	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₀₇	C₄×D₄	1	4	yes	—
133	C₂×C₄×C₁₀₇	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₁₀₇	#120	1	4	yes	—
135	C₂×C₄×C₁₀₇	#121	1	2	yes	—
136	C₂×C₄×C₁₀₇	#121	1	2	yes	—
137	C₂×C₄×C₁₀₇	#123	1	4	yes	—
138	C₈×C₁₀₇	—	2	2	no	—
139	C₈×C₁₀₇	—	2	4	no	—
140	C₈×C₁₀₇	—	2	2	no	—
141	C₈×C₁₀₇	—	4	8	no	—
142	C₂²×C₁₀₇	—	2	2	no	—
143	C₂²×C₁₀₇	—	2	4	no	—
144	C₂²×C₁₀₇	—	2	4	no	—
145	C₂²×C₁₀₇	—	4	2	no	—
146	C₂²×C₁₀₇	—	4	2	no	—
147	C₂²×C₁₀₇	—	4	4	no	—
148	C₂²×C₁₀₇	—	4	4	no	—
149	C₂²×C₁₀₇	—	4	4	no	—
150	C₂²×C₁₀₇	C₂×#95	1	1	yes	characteristic
151	C₂²×C₁₀₇	#380	1	1	yes	characteristic
152	C₂²×C₁₀₇	#380	1	1	yes	characteristic
153	C₂²×C₁₀₇	#385	1	4	yes	—
154	C₄×C₁₀₇	—	2	4	no	—
155	C₄×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₁₀₇	#379	1	1	yes	characteristic
158	C₄×C₁₀₇	#381	1	1	yes	characteristic
159	C₄×C₁₀₇	#386	1	4	yes	—
160	#10760	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₁₀₇	—	2	1	no	—
162	C₂×C₁₀₇	—	2	1	no	—
163	C₂×C₁₀₇	—	2	4	no	—
164	C₂×C₁₀₇	—	4	4	no	—
165	C₂×C₁₀₇	C₂²⋊#95	1	1	yes	characteristic
166	C₂×C₁₀₇	#1813	1	2	yes	—
167	C₂²×#131	C₂×C₁₀₇	1	1	yes	characteristic
168	C₂×#413	C₂×C₁₀₇	1	2	yes	—
169	C₂×#341	C₂×C₁₀₇	1	2	yes	complemented
170	C₂×#338	C₂×C₁₀₇	1	2	yes	complemented
171	C₁₀₇	#10760	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×#41	C₂²×C₁₀₇	1	1	yes	characteristic
173	C₂×#131	—	2	4	no	—
174	C₂×#131	—	2	4	no	—
175	C₂×#131	—	2	4	no	—
176	C₂²×M₁₆	C₄×C₁₀₇	1	2	yes	—
177	C₂²×M₁₆	C₂²×C₁₀₇	1	4	yes	—
178	C₂²×M₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
179	C₂³×C₈	C₄×C₁₀₇	1	2	yes	—
180	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
181	#413	—	2	4	no	—
182	#413	C₄×C₁₀₇	1	8	yes	—
183	C₂×#101	—	2	2	no	—
184	C₂×#100	—	2	2	no	—
185	#341	—	2	4	no	—
186	#338	—	2	4	no	—
187	C₂×#41	—	2	2	no	—
188	C₂×#41	—	2	2	no	—
189	C₂×#41	—	2	2	no	—
190	C₂×#41	—	2	2	no	—
191	C₂×#41	C₂×C₄×C₁₀₇	1	4	yes	—
192	C₂×#41	D₄×C₁₀₇	1	4	yes	—
193	C₂²×Q₈	D₄×C₁₀₇	1	1	yes	characteristic
194	C₂²×D₄	D₄×C₁₀₇	1	1	yes	characteristic
195	C₂²×D₄	Q₈×C₁₀₇	1	2	yes	—
196	C₂³×C₄	C₂×C₄×C₁₀₇	1	2	yes	—
197	C₂³×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
198	#131	—	2	32	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	4	no	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	4	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	C₂×C₄×C₁₀₇	1	8	yes	—
210	C₂×M₁₆	D₄×C₁₀₇	1	4	yes	—
211	C₂²×C₈	—	2	4	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	2	4	no	—
217	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
218	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
219	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₁₀₇	1	4	yes	—
221	#101	—	2	8	no	—
222	#100	—	2	8	no	—
223	C₂⁴	C₄⋊C₄×C₁₀₇	1	2	yes	—
224	C₂⁴	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
225	#41	—	2	8	no	—
226	#41	—	2	8	no	—
227	#41	—	2	8	no	—
228	#41	—	2	8	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	C₂²⋊C₄×C₁₀₇	1	4	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂²⋊C₄×C₁₀₇	1	4	yes	—
237	C₂×D₄	C₄⋊C₄×C₁₀₇	1	8	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₄²×C₁₀₇	1	4	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	4	yes	—
244	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₁₀₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
251	C₂²×C₄	#41×C₁₀₇	1	2	yes	—
252	C₂²×C₄	#41×C₁₀₇	1	2	yes	—
253	M₁₆	—	2	16	no	—
254	M₁₆	—	2	8	no	—
255	M₁₆	—	2	8	no	—
256	M₁₆	—	2	8	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	4	4	no	—
259	C₂×C₈	—	2	16	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	8	no	—
267	C₂×C₈	—	2	8	no	—
268	C₂×C₈	—	4	4	no	—
269	C₂×C₈	—	4	4	no	—
270	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
271	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	4	yes	—
272	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	4	no	—
277	C₂³	#95×C₁₀₇	1	4	yes	—
278	C₂³	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
279	C₂³	C₂×C₄⋊C₄×C₁₀₇	1	2	yes	—
280	C₂³	C₂²⋊Q₈×C₁₀₇	1	2	yes	—
281	C₂³	C₂²⋊Q₈×C₁₀₇	1	2	yes	—
282	Q₈	—	2	8	no	—
283	D₄	—	2	16	no	—
284	D₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	#95×C₁₀₇	1	4	yes	—
292	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	2	yes	—
297	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
298	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₁₀₇	1	4	yes	—
300	C₂×C₄	C₄×D₄×C₁₀₇	1	4	yes	—
301	C₂×C₄	#120×C₁₀₇	1	4	yes	—
302	C₂×C₄	#121×C₁₀₇	1	2	yes	—
303	C₂×C₄	#121×C₁₀₇	1	2	yes	—
304	C₂×C₄	#123×C₁₀₇	1	4	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	8	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	C₂×#95×C₁₀₇	1	1	yes	characteristic
318	C₂²	#380×C₁₀₇	1	1	yes	characteristic
319	C₂²	#380×C₁₀₇	1	1	yes	characteristic
320	C₂²	#385×C₁₀₇	1	4	yes	—
321	C₄	—	2	4	no	—
322	C₄	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
323	C₄	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
324	C₄	#379×C₁₀₇	1	1	yes	characteristic
325	C₄	#381×C₁₀₇	1	1	yes	characteristic
326	C₄	#386×C₁₀₇	1	4	yes	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	4	no	—
330	C₂	—	4	4	no	—
331	C₂	C₁₀₇×C₂²⋊#95	1	1	yes	characteristic
332	C₂	C₁₀₇×#1813	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	#117
commutator	C₂×C₄	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	#380
prime radical	C₂²×#41×C₁₀₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂²×M₁₆×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#341×C₁₀₇	C₂	semidirect
4. C₂×#338×C₁₀₇	C₂	semidirect
160. #10760	C₁₀₇	direct (N × H)
169. C₂×#341	C₂×C₁₀₇	semidirect
170. C₂×#338	C₂×C₁₀₇	semidirect
171. C₁₀₇	#10760	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₀₇	2
C₂×#413×C₁₀₇	2
C₂×#341×C₁₀₇	2
C₂×#338×C₁₀₇	2
#10760	107

Subgroups — C107×#10760 #41036724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#10760 #41036724