Subgroups — C₁₀₇×#8022 #41033986

Order 27392.

786 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#1294	C₂	1	4	yes	maximal
5	C₂×#117×C₁₀₇	—	2	1	no	—
6	C₂²×C₄²×C₁₀₇	—	2	1	no	—
7	C₂²×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
8	#432×C₁₀₇	—	2	2	no	—
9	#432×C₁₀₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈×C₁₀₇	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₁₀₇	C₂²	1	2	yes	—
12	C₂×#97×C₁₀₇	—	2	1	no	—
13	#384×C₁₀₇	C₄	1	2	yes	complemented
14	#382×C₁₀₇	—	2	1	no	—
15	#380×C₁₀₇	—	2	1	no	—
16	#380×C₁₀₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₀₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₁₀₇	C₄	1	2	yes	complemented
19	#376×C₁₀₇	—	2	2	no	—
20	#376×C₁₀₇	C₄	1	4	yes	complemented
21	C₂×#95×C₁₀₇	—	2	1	no	—
22	C₂×#95×C₁₀₇	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₀₇	—	2	2	no	—
24	C₂³×C₄×C₁₀₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₇	—	2	2	no	—
28	#117×C₁₀₇	—	4	4	no	—
29	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₁₀₇	—	2	2	no	—
33	C₂×C₄²×C₁₀₇	—	2	8	no	—
34	C₂×C₄²×C₁₀₇	—	2	4	no	—
35	C₂×C₄²×C₁₀₇	—	2	2	no	—
36	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
41	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₀₇	Q₈	1	2	yes	—
44	C₄⋊C₈×C₁₀₇	—	2	4	no	—
45	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
46	C₂²⋊C₈×C₁₀₇	C₂×C₄	1	4	yes	—
47	#97×C₁₀₇	—	2	4	no	—
48	#95×C₁₀₇	—	2	4	no	—
49	#95×C₁₀₇	—	2	2	no	—
50	#95×C₁₀₇	—	2	2	no	—
51	#95×C₁₀₇	—	2	2	no	—
52	#95×C₁₀₇	C₂×C₄	1	4	yes	—
53	C₂⁴×C₁₀₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₇	—	2	2	no	—
55	C₂²×C₄×C₁₀₇	—	2	2	no	—
56	C₂²×C₄×C₁₀₇	—	2	4	no	—
57	C₂²×C₄×C₁₀₇	—	2	8	no	—
58	C₂²×C₄×C₁₀₇	—	2	4	no	—
59	C₂²×C₄×C₁₀₇	—	2	4	no	—
60	C₂²×C₄×C₁₀₇	—	4	2	no	—
61	C₂²×C₄×C₁₀₇	—	4	4	no	—
62	C₂²×C₄×C₁₀₇	—	4	2	no	—
63	C₂²×C₄×C₁₀₇	—	4	2	no	—
64	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₁₀₇	—	2	4	no	—
79	C₂×C₈×C₁₀₇	—	4	4	no	—
80	C₄⋊C₄×C₁₀₇	—	4	4	no	—
81	C₄⋊C₄×C₁₀₇	—	4	4	no	—
82	C₄⋊C₄×C₁₀₇	—	4	4	no	—
83	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
84	C₄²×C₁₀₇	—	2	16	no	—
85	C₄²×C₁₀₇	—	4	4	no	—
86	C₄²×C₁₀₇	C₄²	1	2	yes	—
87	C₄²×C₁₀₇	C₄²	1	2	yes	—
88	C₄²×C₁₀₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₁₀₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₁₀₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₁₀₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₁₀₇	C₄⋊C₄	1	2	yes	—
93	C₄²×C₁₀₇	C₄⋊C₄	1	2	yes	—
94	C₂³×C₁₀₇	—	2	2	no	—
95	C₂³×C₁₀₇	—	2	2	no	—
96	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
100	C₂³×C₁₀₇	#137	1	2	yes	—
101	C₂×C₄×C₁₀₇	—	2	2	no	—
102	C₂×C₄×C₁₀₇	—	2	2	no	—
103	C₂×C₄×C₁₀₇	—	2	4	no	—
104	C₂×C₄×C₁₀₇	—	2	1	no	—
105	C₂×C₄×C₁₀₇	—	2	1	no	—
106	C₂×C₄×C₁₀₇	—	2	1	no	—
107	C₂×C₄×C₁₀₇	—	2	1	no	—
108	C₂×C₄×C₁₀₇	—	2	2	no	—
109	C₂×C₄×C₁₀₇	—	2	4	no	—
110	C₂×C₄×C₁₀₇	—	2	4	no	—
111	C₂×C₄×C₁₀₇	—	2	4	no	—
112	C₂×C₄×C₁₀₇	—	2	8	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	4	no	—
118	C₂×C₄×C₁₀₇	—	4	8	no	—
119	C₂×C₄×C₁₀₇	—	4	8	no	—
120	C₂×C₄×C₁₀₇	—	4	4	no	—
121	C₂×C₄×C₁₀₇	—	4	4	no	—
122	C₂×C₄×C₁₀₇	—	4	4	no	—
123	C₂×C₄×C₁₀₇	—	4	4	no	—
124	C₂×C₄×C₁₀₇	#95	1	2	yes	—
125	C₂×C₄×C₁₀₇	#95	1	2	yes	—
126	C₂×C₄×C₁₀₇	#95	1	2	yes	—
127	C₂×C₄×C₁₀₇	#95	1	2	yes	—
128	C₂×C₄×C₁₀₇	#104	1	4	yes	—
129	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₁₀₇	—	4	4	no	—
134	C₂²×C₁₀₇	—	2	1	no	—
135	C₂²×C₁₀₇	—	2	1	no	—
136	C₂²×C₁₀₇	—	2	1	no	—
137	C₂²×C₁₀₇	—	2	1	no	—
138	C₂²×C₁₀₇	—	2	4	no	—
139	C₂²×C₁₀₇	—	2	4	no	—
140	C₂²×C₁₀₇	C₂×#95	1	1	yes	characteristic
141	C₂²×C₁₀₇	#410	1	2	yes	—
142	C₂²×C₁₀₇	#417	1	2	yes	—
143	C₂²×C₁₀₇	#418	1	1	yes	characteristic
144	C₂²×C₁₀₇	#419	1	1	yes	characteristic
145	C₂²×C₁₀₇	C₂×#104	1	2	yes	—
146	C₂²×C₁₀₇	#428	1	2	yes	—
147	C₄×C₁₀₇	—	2	2	no	—
148	C₄×C₁₀₇	—	2	2	no	—
149	C₄×C₁₀₇	—	2	2	no	—
150	C₄×C₁₀₇	—	2	1	no	—
151	C₄×C₁₀₇	—	2	1	no	—
152	C₄×C₁₀₇	—	2	1	no	—
153	C₄×C₁₀₇	—	2	1	no	—
154	C₄×C₁₀₇	—	4	8	no	—
155	C₄×C₁₀₇	—	8	4	no	—
156	C₄×C₁₀₇	#339	1	4	yes	—
157	#8022	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₁₀₇	—	2	2	no	—
159	C₂×C₁₀₇	—	2	2	no	—
160	C₂×C₁₀₇	#1505	1	2	yes	—
161	C₂×C₁₀₇	#1506	1	2	yes	—
162	C₂×C₁₀₇	C₂×#339	1	1	yes	characteristic
163	C₂×C₁₀₇	#1736	1	1	yes	characteristic
164	C₂×C₁₀₇	#1740	1	1	yes	characteristic
165	C₂×#432	C₂×C₁₀₇	1	1	yes	characteristic
166	#2285	C₂×C₁₀₇	1	1	yes	characteristic
167	C₂×#376	C₂×C₁₀₇	1	1	yes	characteristic
168	#1294	C₂×C₁₀₇	1	4	yes	—
169	C₁₀₇	#8022	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×#117	—	2	1	no	—
171	C₂²×C₄²	—	2	1	no	—
172	C₂²×C₄²	C₂²×C₁₀₇	1	1	yes	characteristic
173	#432	—	2	2	no	—
174	#432	C₄×C₁₀₇	1	4	yes	complemented
175	C₂×C₄⋊C₈	—	2	1	no	—
176	C₂×C₂²⋊C₈	C₂²×C₁₀₇	1	2	yes	—
177	C₂×#97	—	2	1	no	—
178	#384	C₄×C₁₀₇	1	2	yes	complemented
179	#382	—	2	1	no	—
180	#380	—	2	1	no	—
181	#380	C₂²×C₁₀₇	1	2	yes	—
182	C₄×C₄⋊C₄	—	2	1	no	—
183	C₄×C₄⋊C₄	C₄×C₁₀₇	1	2	yes	complemented
184	#376	—	2	2	no	—
185	#376	C₄×C₁₀₇	1	4	yes	complemented
186	C₂×#95	—	2	1	no	—
187	C₂×#95	C₂²×C₁₀₇	1	2	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂³×C₄	D₄×C₁₀₇	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₁₀₇	1	1	yes	characteristic
191	C₂³×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
192	C₂²×C₈	—	2	2	no	—
193	#117	—	4	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	—	2	8	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
203	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
206	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
208	C₂×C₄²	Q₈×C₁₀₇	1	2	yes	—
209	C₄⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	C₂×C₄×C₁₀₇	1	4	yes	—
212	#97	—	2	4	no	—
213	#95	—	2	4	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	C₂×C₄×C₁₀₇	1	4	yes	—
218	C₂⁴	C₂×D₄×C₁₀₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
230	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
235	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
236	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₄²	—	2	16	no	—
250	C₄²	—	4	4	no	—
251	C₄²	C₄²×C₁₀₇	1	2	yes	—
252	C₄²	C₄²×C₁₀₇	1	2	yes	—
253	C₄²	C₂²⋊C₄×C₁₀₇	1	2	yes	—
254	C₄²	C₂²⋊C₄×C₁₀₇	1	2	yes	—
255	C₄²	C₄⋊C₄×C₁₀₇	1	2	yes	—
256	C₄²	C₄⋊C₄×C₁₀₇	1	2	yes	—
257	C₄²	C₄⋊C₄×C₁₀₇	1	2	yes	—
258	C₄²	C₄⋊C₄×C₁₀₇	1	2	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
263	C₂³	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
264	C₂³	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
265	C₂³	#137×C₁₀₇	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	#95×C₁₀₇	1	2	yes	—
290	C₂×C₄	#95×C₁₀₇	1	2	yes	—
291	C₂×C₄	#95×C₁₀₇	1	2	yes	—
292	C₂×C₄	#95×C₁₀₇	1	2	yes	—
293	C₂×C₄	#104×C₁₀₇	1	4	yes	—
294	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
298	C₈	—	4	4	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	C₂×#95×C₁₀₇	1	1	yes	characteristic
306	C₂²	#410×C₁₀₇	1	2	yes	—
307	C₂²	#417×C₁₀₇	1	2	yes	—
308	C₂²	#418×C₁₀₇	1	1	yes	characteristic
309	C₂²	#419×C₁₀₇	1	1	yes	characteristic
310	C₂²	C₂×#104×C₁₀₇	1	2	yes	—
311	C₂²	#428×C₁₀₇	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	8	no	—
320	C₄	—	8	4	no	—
321	C₄	#339×C₁₀₇	1	4	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	C₁₀₇×#1505	1	2	yes	—
325	C₂	C₁₀₇×#1506	1	2	yes	—
326	C₂	C₂×#339×C₁₀₇	1	1	yes	characteristic
327	C₂	C₁₀₇×#1736	1	1	yes	characteristic
328	C₂	C₁₀₇×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₇
prime radical	C₂⁴×C₁₀₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂²×C₄²×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. #432×C₁₀₇	C₄	semidirect
13. #384×C₁₀₇	C₄	semidirect
18. C₄×C₄⋊C₄×C₁₀₇	C₄	semidirect
20. #376×C₁₀₇	C₄	semidirect
157. #8022	C₁₀₇	direct (N × H)
169. C₁₀₇	#8022	direct (N × H)
174. #432	C₄×C₁₀₇	semidirect
178. #384	C₄×C₁₀₇	semidirect
183. C₄×C₄⋊C₄	C₄×C₁₀₇	semidirect
185. #376	C₄×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₀₇	2
C₁₀₇×#2285	2
C₂×#376×C₁₀₇	2
C₁₀₇×#1294	2
#8022	107

Subgroups — C107×#8022 #41033986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8022 #41033986