Subgroups — C₁₀₇×#10672 #41036636

Order 27392.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#1957	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1957	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1957	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₂×#107×C₁₀₇	—	2	2	no	—
10	C₂×#106×C₁₀₇	—	2	2	no	—
11	C₂×#102×C₁₀₇	—	2	2	no	—
12	C₂×#102×C₁₀₇	—	2	2	no	—
13	C₂×#102×C₁₀₇	—	2	2	no	—
14	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	#384×C₁₀₇	C₂²	1	1	yes	characteristic
18	#384×C₁₀₇	C₂²	1	1	yes	characteristic
19	#384×C₁₀₇	C₂²	1	1	yes	characteristic
20	#340×C₁₀₇	—	2	2	no	—
21	C₂²×D₄×C₁₀₇	—	2	1	no	—
22	C₂²×D₄×C₁₀₇	—	2	1	no	—
23	C₂²×D₄×C₁₀₇	—	2	1	no	—
24	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
25	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
26	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
27	#127×C₁₀₇	—	2	1	no	—
28	#127×C₁₀₇	—	2	1	no	—
29	#127×C₁₀₇	—	2	1	no	—
30	#127×C₁₀₇	Q₈	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
40	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
41	#107×C₁₀₇	—	4	4	no	—
42	#106×C₁₀₇	—	4	4	no	—
43	#102×C₁₀₇	—	4	4	no	—
44	#102×C₁₀₇	—	4	4	no	—
45	#102×C₁₀₇	—	4	4	no	—
46	C₄×C₈×C₁₀₇	—	2	1	no	—
47	C₄×C₈×C₁₀₇	—	2	1	no	—
48	C₄×C₈×C₁₀₇	—	2	1	no	—
49	C₄×C₈×C₁₀₇	—	2	1	no	—
50	C₄×C₈×C₁₀₇	—	2	1	no	—
51	C₄×C₈×C₁₀₇	—	2	1	no	—
52	C₂⁴×C₁₀₇	—	4	1	no	—
53	C₂×D₄×C₁₀₇	—	2	2	no	—
54	C₂×D₄×C₁₀₇	—	2	2	no	—
55	C₂×D₄×C₁₀₇	—	2	2	no	—
56	C₂×D₄×C₁₀₇	—	4	1	no	—
57	C₂×D₄×C₁₀₇	—	4	1	no	—
58	C₂×D₄×C₁₀₇	—	4	1	no	—
59	C₂×D₄×C₁₀₇	—	4	2	no	—
60	C₂×D₄×C₁₀₇	—	4	2	no	—
61	C₂×D₄×C₁₀₇	—	4	2	no	—
62	C₂×D₄×C₁₀₇	—	4	2	no	—
63	C₂×D₄×C₁₀₇	—	4	2	no	—
64	C₂×D₄×C₁₀₇	—	4	2	no	—
65	C₂²×C₄×C₁₀₇	—	4	1	no	—
66	C₂²×C₄×C₁₀₇	—	4	1	no	—
67	C₂²×C₄×C₁₀₇	—	4	1	no	—
68	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₀₇	—	2	2	no	—
72	C₂×C₈×C₁₀₇	—	2	2	no	—
73	C₂×C₈×C₁₀₇	—	2	2	no	—
74	C₂×C₈×C₁₀₇	—	2	2	no	—
75	C₂×C₈×C₁₀₇	—	2	2	no	—
76	C₂×C₈×C₁₀₇	—	2	2	no	—
77	C₂×C₈×C₁₀₇	—	2	2	no	—
78	C₂×C₈×C₁₀₇	D₈	1	4	yes	—
79	C₂×C₈×C₁₀₇	SD₁₆	1	4	yes	—
80	C₄⋊C₄×C₁₀₇	—	4	2	no	—
81	C₄⋊C₄×C₁₀₇	—	4	2	no	—
82	C₄⋊C₄×C₁₀₇	—	4	2	no	—
83	C₄⋊C₄×C₁₀₇	—	4	2	no	—
84	C₄⋊C₄×C₁₀₇	—	4	2	no	—
85	C₄⋊C₄×C₁₀₇	—	4	2	no	—
86	C₄⋊C₄×C₁₀₇	—	4	2	no	—
87	C₄⋊C₄×C₁₀₇	—	4	2	no	—
88	C₄⋊C₄×C₁₀₇	—	4	2	no	—
89	C₄²×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₁₀₇	#41	1	1	yes	characteristic
91	C₄²×C₁₀₇	#41	1	1	yes	characteristic
92	C₄²×C₁₀₇	#41	1	1	yes	characteristic
93	C₂³×C₁₀₇	—	4	2	no	—
94	C₂³×C₁₀₇	—	8	1	no	—
95	C₂³×C₁₀₇	—	8	1	no	—
96	C₂³×C₁₀₇	—	8	1	no	—
97	C₂³×C₁₀₇	—	8	1	no	—
98	C₂³×C₁₀₇	—	8	1	no	—
99	C₂³×C₁₀₇	—	8	1	no	—
100	C₂³×C₁₀₇	#127	1	1	yes	characteristic
101	D₄×C₁₀₇	—	4	4	no	—
102	D₄×C₁₀₇	—	4	4	no	—
103	D₄×C₁₀₇	—	4	4	no	—
104	D₄×C₁₀₇	—	8	2	no	—
105	D₄×C₁₀₇	—	8	2	no	—
106	D₄×C₁₀₇	—	8	2	no	—
107	C₂×C₄×C₁₀₇	—	8	1	no	—
108	C₂×C₄×C₁₀₇	—	8	1	no	—
109	C₂×C₄×C₁₀₇	—	8	1	no	—
110	C₂×C₄×C₁₀₇	—	8	1	no	—
111	C₂×C₄×C₁₀₇	—	8	1	no	—
112	C₂×C₄×C₁₀₇	—	8	1	no	—
113	C₂×C₄×C₁₀₇	—	8	1	no	—
114	C₂×C₄×C₁₀₇	—	8	1	no	—
115	C₂×C₄×C₁₀₇	—	8	1	no	—
116	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₀₇	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₀₇	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₀₇	#123	1	1	yes	characteristic
122	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₁₀₇	#135	1	2	yes	—
125	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₁₀₇	#137	1	2	yes	—
128	C₈×C₁₀₇	—	2	4	no	—
129	C₈×C₁₀₇	—	2	4	no	—
130	C₈×C₁₀₇	—	4	2	no	—
131	C₂²×C₁₀₇	—	8	2	no	—
132	C₂²×C₁₀₇	—	8	2	no	—
133	C₂²×C₁₀₇	—	8	2	no	—
134	C₂²×C₁₀₇	—	16	1	no	—
135	C₂²×C₁₀₇	—	16	1	no	—
136	C₂²×C₁₀₇	—	16	1	no	—
137	C₂²×C₁₀₇	—	16	1	no	—
138	C₂²×C₁₀₇	#399	1	1	yes	characteristic
139	C₂²×C₁₀₇	#492	1	1	yes	characteristic
140	C₂²×C₁₀₇	#493	1	1	yes	characteristic
141	C₂²×C₁₀₇	#495	1	1	yes	characteristic
142	C₂²×C₁₀₇	#496	1	1	yes	characteristic
143	C₂²×C₁₀₇	#496	1	1	yes	characteristic
144	C₂²×C₁₀₇	#497	1	1	yes	characteristic
145	C₄×C₁₀₇	—	16	1	no	—
146	C₄×C₁₀₇	—	16	1	no	—
147	C₄×C₁₀₇	—	16	1	no	—
148	C₄×C₁₀₇	#474	1	2	yes	—
149	C₄×C₁₀₇	#476	1	2	yes	—
150	C₄×C₁₀₇	#478	1	2	yes	—
151	C₄×C₁₀₇	#480	1	2	yes	—
152	C₄×C₁₀₇	#481	1	2	yes	—
153	C₄×C₁₀₇	#482	1	2	yes	—
154	#10672	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₁₀₇	—	16	2	no	—
156	C₂×C₁₀₇	#1706	1	1	yes	characteristic
157	C₂×C₁₀₇	#1714	1	1	yes	characteristic
158	C₂×C₁₀₇	#1720	1	1	yes	characteristic
159	C₂×C₁₀₇	#1721	1	1	yes	characteristic
160	C₂×C₁₀₇	#2092	1	1	yes	characteristic
161	C₂×C₁₀₇	#2093	1	1	yes	characteristic
162	C₂×C₁₀₇	#2094	1	1	yes	characteristic
163	#1957	C₂×C₁₀₇	1	1	yes	characteristic
164	#1957	C₂×C₁₀₇	1	1	yes	characteristic
165	#1957	C₂×C₁₀₇	1	1	yes	characteristic
166	#1845	C₂×C₁₀₇	1	1	yes	characteristic, complemented
167	#1844	C₂×C₁₀₇	1	1	yes	characteristic, complemented
168	#1834	C₂×C₁₀₇	1	1	yes	characteristic, complemented
169	#1273	C₂×C₁₀₇	1	1	yes	characteristic, complemented
170	C₁₀₇	#10672	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#127	C₂²×C₁₀₇	1	1	yes	characteristic
172	C₂×#107	—	2	2	no	—
173	C₂×#106	—	2	2	no	—
174	C₂×#102	—	2	2	no	—
175	C₂×#102	—	2	2	no	—
176	C₂×#102	—	2	2	no	—
177	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
180	#384	C₂²×C₁₀₇	1	1	yes	characteristic
181	#384	C₂²×C₁₀₇	1	1	yes	characteristic
182	#384	C₂²×C₁₀₇	1	1	yes	characteristic
183	#340	—	2	2	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
188	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
189	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
190	#127	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	—	2	1	no	—
193	#127	Q₈×C₁₀₇	1	2	yes	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
204	#107	—	4	4	no	—
205	#106	—	4	4	no	—
206	#102	—	4	4	no	—
207	#102	—	4	4	no	—
208	#102	—	4	4	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₂⁴	—	4	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	D₈×C₁₀₇	1	4	yes	—
242	C₂×C₈	SD₁₆×C₁₀₇	1	4	yes	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄²	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
253	C₄²	#41×C₁₀₇	1	1	yes	characteristic
254	C₄²	#41×C₁₀₇	1	1	yes	characteristic
255	C₄²	#41×C₁₀₇	1	1	yes	characteristic
256	C₂³	—	4	2	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	8	1	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	—	8	1	no	—
262	C₂³	—	8	1	no	—
263	C₂³	#127×C₁₀₇	1	1	yes	characteristic
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	8	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
280	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
282	C₂×C₄	#123×C₁₀₇	1	1	yes	characteristic
283	C₂×C₄	#123×C₁₀₇	1	1	yes	characteristic
284	C₂×C₄	#123×C₁₀₇	1	1	yes	characteristic
285	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
286	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	2	yes	—
287	C₂×C₄	#135×C₁₀₇	1	2	yes	—
288	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
289	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
290	C₂×C₄	#137×C₁₀₇	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	16	1	no	—
298	C₂²	—	16	1	no	—
299	C₂²	—	16	1	no	—
300	C₂²	—	16	1	no	—
301	C₂²	#399×C₁₀₇	1	1	yes	characteristic
302	C₂²	#492×C₁₀₇	1	1	yes	characteristic
303	C₂²	#493×C₁₀₇	1	1	yes	characteristic
304	C₂²	#495×C₁₀₇	1	1	yes	characteristic
305	C₂²	#496×C₁₀₇	1	1	yes	characteristic
306	C₂²	#496×C₁₀₇	1	1	yes	characteristic
307	C₂²	#497×C₁₀₇	1	1	yes	characteristic
308	C₄	—	16	1	no	—
309	C₄	—	16	1	no	—
310	C₄	—	16	1	no	—
311	C₄	#474×C₁₀₇	1	2	yes	—
312	C₄	#476×C₁₀₇	1	2	yes	—
313	C₄	#478×C₁₀₇	1	2	yes	—
314	C₄	#480×C₁₀₇	1	2	yes	—
315	C₄	#481×C₁₀₇	1	2	yes	—
316	C₄	#482×C₁₀₇	1	2	yes	—
317	C₂	—	16	2	no	—
318	C₂	C₁₀₇×#1706	1	1	yes	characteristic
319	C₂	C₁₀₇×#1714	1	1	yes	characteristic
320	C₂	C₁₀₇×#1720	1	1	yes	characteristic
321	C₂	C₁₀₇×#1721	1	1	yes	characteristic
322	C₂	C₁₀₇×#2092	1	1	yes	characteristic
323	C₂	C₁₀₇×#2093	1	1	yes	characteristic
324	C₂	C₁₀₇×#2094	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
prime radical	C₂×#127×C₁₀₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₇×#1845	C₂	semidirect
5. C₁₀₇×#1844	C₂	semidirect
6. C₁₀₇×#1834	C₂	semidirect
7. C₁₀₇×#1273	C₂	semidirect
154. #10672	C₁₀₇	direct (N × H)
166. #1845	C₂×C₁₀₇	semidirect
167. #1844	C₂×C₁₀₇	semidirect
168. #1834	C₂×C₁₀₇	semidirect
169. #1273	C₂×C₁₀₇	semidirect
170. C₁₀₇	#10672	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×#1957	2
C₁₀₇×#1845	2
C₁₀₇×#1844	2
C₁₀₇×#1834	2
C₁₀₇×#1273	2
#10672	107

Subgroups — C107×#10672 #41036636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#10672 #41036636