Subgroups — C₁₀₇×#7644 #41033608

Order 27392.

402 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1839	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1831	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#1272	C₂	1	2	yes	maximal
5	C₁₀₇×#1271	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₀₇	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₁₀₇	—	2	1	no	—
8	#391×C₁₀₇	C₄	1	1	yes	characteristic
9	#391×C₁₀₇	C₂²	1	1	yes	characteristic
10	#389×C₁₀₇	C₄	1	1	yes	characteristic
11	#389×C₁₀₇	C₂²	1	1	yes	characteristic
12	#387×C₁₀₇	—	2	1	no	—
13	#387×C₁₀₇	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₀₇	—	2	1	no	—
15	C₄×C₄⋊C₄×C₁₀₇	C₄	1	1	yes	characteristic
16	#376×C₁₀₇	—	2	1	no	—
17	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
18	#336×C₁₀₇	—	2	1	no	—
19	#336×C₁₀₇	—	2	1	no	—
20	#336×C₁₀₇	—	2	1	no	—
21	#336×C₁₀₇	C₂²	1	2	yes	—
22	#336×C₁₀₇	C₂²	1	2	yes	—
23	C₂²×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₇	—	2	2	no	—
25	C₂²×C₈×C₁₀₇	—	2	2	no	—
26	C₄×Q₈×C₁₀₇	—	2	2	no	—
27	C₄×Q₈×C₁₀₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₇	—	2	1	no	—
35	C₂×C₄²×C₁₀₇	—	2	1	no	—
36	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₁₀₇	—	4	2	no	—
44	#95×C₁₀₇	—	2	1	no	—
45	#95×C₁₀₇	—	2	1	no	—
46	#95×C₁₀₇	—	2	1	no	—
47	#95×C₁₀₇	—	2	1	no	—
48	#95×C₁₀₇	—	2	1	no	—
49	#95×C₁₀₇	—	2	1	no	—
50	#95×C₁₀₇	—	2	1	no	—
51	C₂×Q₈×C₁₀₇	—	2	1	no	—
52	C₂×Q₈×C₁₀₇	—	4	1	no	—
53	C₂×Q₈×C₁₀₇	—	4	1	no	—
54	C₂×Q₈×C₁₀₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₁₀₇	—	2	1	no	—
56	C₂²×C₄×C₁₀₇	—	2	1	no	—
57	C₂²×C₄×C₁₀₇	—	2	1	no	—
58	C₂²×C₄×C₁₀₇	—	2	1	no	—
59	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₀₇	—	4	2	no	—
67	C₂×C₈×C₁₀₇	—	4	2	no	—
68	C₂×C₈×C₁₀₇	—	4	2	no	—
69	C₂×C₈×C₁₀₇	—	4	2	no	—
70	C₂×C₈×C₁₀₇	—	4	2	no	—
71	C₂×C₈×C₁₀₇	—	4	2	no	—
72	C₄⋊C₄×C₁₀₇	—	2	4	no	—
73	C₄⋊C₄×C₁₀₇	—	2	1	no	—
74	C₄⋊C₄×C₁₀₇	—	4	1	no	—
75	C₄⋊C₄×C₁₀₇	—	4	1	no	—
76	C₄⋊C₄×C₁₀₇	—	4	2	no	—
77	C₄⋊C₄×C₁₀₇	—	4	2	no	—
78	C₄⋊C₄×C₁₀₇	M₁₆	1	2	yes	—
79	C₄²×C₁₀₇	—	2	4	no	—
80	C₄²×C₁₀₇	—	2	1	no	—
81	C₄²×C₁₀₇	—	2	1	no	—
82	C₄²×C₁₀₇	—	2	2	no	—
83	C₄²×C₁₀₇	—	4	1	no	—
84	C₄²×C₁₀₇	—	4	1	no	—
85	C₄²×C₁₀₇	—	4	1	no	—
86	C₄²×C₁₀₇	—	4	1	no	—
87	C₄²×C₁₀₇	—	4	2	no	—
88	C₄²×C₁₀₇	SD₁₆	1	2	yes	—
89	C₄²×C₁₀₇	Q₁₆	1	2	yes	—
90	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₁₀₇	—	4	2	no	—
92	Q₈×C₁₀₇	—	4	2	no	—
93	C₂×C₄×C₁₀₇	—	2	1	no	—
94	C₂×C₄×C₁₀₇	—	2	1	no	—
95	C₂×C₄×C₁₀₇	—	2	1	no	—
96	C₂×C₄×C₁₀₇	—	2	1	no	—
97	C₂×C₄×C₁₀₇	—	2	1	no	—
98	C₂×C₄×C₁₀₇	—	2	1	no	—
99	C₂×C₄×C₁₀₇	—	2	1	no	—
100	C₂×C₄×C₁₀₇	—	2	1	no	—
101	C₂×C₄×C₁₀₇	—	2	1	no	—
102	C₂×C₄×C₁₀₇	—	2	1	no	—
103	C₂×C₄×C₁₀₇	—	2	1	no	—
104	C₂×C₄×C₁₀₇	—	2	2	no	—
105	C₂×C₄×C₁₀₇	—	2	2	no	—
106	C₂×C₄×C₁₀₇	—	2	1	no	—
107	C₂×C₄×C₁₀₇	—	2	1	no	—
108	C₂×C₄×C₁₀₇	—	2	1	no	—
109	C₂×C₄×C₁₀₇	—	2	1	no	—
110	C₂×C₄×C₁₀₇	—	2	1	no	—
111	C₂×C₄×C₁₀₇	—	4	1	no	—
112	C₂×C₄×C₁₀₇	—	4	1	no	—
113	C₂×C₄×C₁₀₇	—	4	1	no	—
114	C₂×C₄×C₁₀₇	—	4	1	no	—
115	C₂×C₄×C₁₀₇	—	4	1	no	—
116	C₂×C₄×C₁₀₇	—	4	1	no	—
117	C₂×C₄×C₁₀₇	—	4	1	no	—
118	C₂×C₄×C₁₀₇	—	4	1	no	—
119	C₂×C₄×C₁₀₇	—	4	1	no	—
120	C₂×C₄×C₁₀₇	—	4	1	no	—
121	C₂×C₄×C₁₀₇	#103	1	2	yes	—
122	C₂×C₄×C₁₀₇	#103	1	2	yes	—
123	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₀₇	#135	1	2	yes	—
128	C₈×C₁₀₇	—	8	2	no	—
129	C₈×C₁₀₇	—	8	2	no	—
130	C₂²×C₁₀₇	#407	1	1	yes	characteristic
131	C₂²×C₁₀₇	#410	1	1	yes	characteristic
132	C₂²×C₁₀₇	#413	1	1	yes	characteristic
133	C₂²×C₁₀₇	C₂×#103	1	1	yes	characteristic
134	C₂²×C₁₀₇	#416	1	1	yes	characteristic
135	C₂²×C₁₀₇	#421	1	1	yes	characteristic
136	C₂²×C₁₀₇	#421	1	1	yes	characteristic
137	C₄×C₁₀₇	—	2	2	no	—
138	C₄×C₁₀₇	—	2	1	no	—
139	C₄×C₁₀₇	—	2	1	no	—
140	C₄×C₁₀₇	—	2	2	no	—
141	C₄×C₁₀₇	—	2	1	no	—
142	C₄×C₁₀₇	—	2	1	no	—
143	C₄×C₁₀₇	—	2	2	no	—
144	C₄×C₁₀₇	—	4	2	no	—
145	C₄×C₁₀₇	—	4	1	no	—
146	C₄×C₁₀₇	—	4	2	no	—
147	C₄×C₁₀₇	—	4	1	no	—
148	C₄×C₁₀₇	—	8	1	no	—
149	C₄×C₁₀₇	—	8	1	no	—
150	#7644	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₀₇	#1464	1	1	yes	characteristic
152	C₂×C₁₀₇	#1487	1	1	yes	characteristic
153	C₂×C₁₀₇	#1488	1	1	yes	characteristic
154	C₂×C₁₀₇	#1518	1	1	yes	characteristic
155	C₂×C₁₀₇	#1521	1	1	yes	characteristic
156	C₂×C₁₀₇	#1535	1	1	yes	characteristic
157	C₂×C₁₀₇	#1542	1	1	yes	characteristic
158	#2295	C₂×C₁₀₇	1	1	yes	characteristic
159	#1839	C₂×C₁₀₇	1	1	yes	characteristic
160	#1831	C₂×C₁₀₇	1	1	yes	characteristic
161	#1272	C₂×C₁₀₇	1	2	yes	—
162	#1271	C₂×C₁₀₇	1	2	yes	—
163	C₁₀₇	#7644	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×C₄×Q₈	C₄×C₁₀₇	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	2	1	no	—
166	#391	C₄×C₁₀₇	1	1	yes	characteristic
167	#391	C₂²×C₁₀₇	1	1	yes	characteristic
168	#389	C₄×C₁₀₇	1	1	yes	characteristic
169	#389	C₂²×C₁₀₇	1	1	yes	characteristic
170	#387	—	2	1	no	—
171	#387	—	2	1	no	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	C₄×C₁₀₇	1	1	yes	characteristic
174	#376	—	2	1	no	—
175	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
176	#336	—	2	1	no	—
177	#336	—	2	1	no	—
178	#336	—	2	1	no	—
179	#336	C₂²×C₁₀₇	1	2	yes	—
180	#336	C₂²×C₁₀₇	1	2	yes	—
181	C₂²×Q₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
201	C₄⋊C₈	—	4	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	M₁₆×C₁₀₇	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	M₁₆×C₁₀₇	1	2	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	SD₁₆×C₁₀₇	1	2	yes	—
247	C₄²	Q₁₆×C₁₀₇	1	2	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	#103×C₁₀₇	1	2	yes	—
280	C₂×C₄	#103×C₁₀₇	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
282	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
283	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₁₆×C₁₀₇	1	1	yes	characteristic
285	C₂×C₄	#135×C₁₀₇	1	2	yes	—
286	C₈	—	8	2	no	—
287	C₈	—	8	2	no	—
288	C₂²	#407×C₁₀₇	1	1	yes	characteristic
289	C₂²	#410×C₁₀₇	1	1	yes	characteristic
290	C₂²	#413×C₁₀₇	1	1	yes	characteristic
291	C₂²	C₂×#103×C₁₀₇	1	1	yes	characteristic
292	C₂²	#416×C₁₀₇	1	1	yes	characteristic
293	C₂²	#421×C₁₀₇	1	1	yes	characteristic
294	C₂²	#421×C₁₀₇	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	C₁₀₇×#1464	1	1	yes	characteristic
309	C₂	C₁₀₇×#1487	1	1	yes	characteristic
310	C₂	C₁₀₇×#1488	1	1	yes	characteristic
311	C₂	C₁₀₇×#1518	1	1	yes	characteristic
312	C₂	C₁₀₇×#1521	1	1	yes	characteristic
313	C₂	C₁₀₇×#1535	1	1	yes	characteristic
314	C₂	C₁₀₇×#1542	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
150. #7644	C₁₀₇	direct (N × H)
163. C₁₀₇	#7644	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₇×#2295	2
C₁₀₇×#1839	2
C₁₀₇×#1831	2
C₁₀₇×#1272	2
C₁₀₇×#1271	2
#7644	107

Subgroups — C107×#7644 #41033608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7644 #41033608