Subgroups — C₁₀₇×#8172 #41034136

Order 27392.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#97×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₁₀₇	C₂	1	2	yes	maximal
5	C₁₀₇×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₁₀₇	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₀₇	—	2	1	no	—
9	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
10	C₂×#97×C₁₀₇	—	2	2	no	—
11	C₂×#97×C₁₀₇	—	2	1	no	—
12	C₂×C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₁₀₇	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₁₀₇	—	2	1	no	—
16	C₄×C₂²⋊C₄×C₁₀₇	—	2	2	no	—
17	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
18	#336×C₁₀₇	—	2	1	no	—
19	D₄⋊C₈×C₁₀₇	C₄	1	8	yes	complemented
20	C₂²×D₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₇	—	2	2	no	—
22	C₂³×C₄×C₁₀₇	—	2	1	no	—
23	C₂²×C₈×C₁₀₇	—	2	2	no	—
24	C₂²×C₈×C₁₀₇	—	2	2	no	—
25	C₄×D₄×C₁₀₇	—	2	4	no	—
26	C₄×D₄×C₁₀₇	—	2	4	no	—
27	C₄×D₄×C₁₀₇	—	2	2	no	—
28	C₄×D₄×C₁₀₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₀₇	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₁₀₇	—	2	1	no	—
33	C₂×C₄²×C₁₀₇	—	2	2	no	—
34	C₂×C₄²×C₁₀₇	—	2	1	no	—
35	C₂×C₄²×C₁₀₇	—	4	4	no	—
36	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
38	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₁₀₇	—	2	2	no	—
42	C₄⋊C₈×C₁₀₇	C₂×C₄	1	4	yes	complemented
43	#97×C₁₀₇	—	2	8	no	—
44	#97×C₁₀₇	—	4	2	no	—
45	C₄×C₈×C₁₀₇	—	2	2	no	—
46	C₄×C₈×C₁₀₇	C₂×C₄	1	4	yes	complemented
47	#95×C₁₀₇	—	2	2	no	—
48	#95×C₁₀₇	—	2	1	no	—
49	C₂⁴×C₁₀₇	—	2	1	no	—
50	C₂×D₄×C₁₀₇	—	2	1	no	—
51	C₂×D₄×C₁₀₇	—	2	2	no	—
52	C₂×D₄×C₁₀₇	C₄²	1	2	yes	—
53	C₂×D₄×C₁₀₇	M₁₆	1	4	yes	—
54	C₂²×C₄×C₁₀₇	—	2	8	no	—
55	C₂²×C₄×C₁₀₇	—	2	1	no	—
56	C₂²×C₄×C₁₀₇	—	2	2	no	—
57	C₂²×C₄×C₁₀₇	—	2	4	no	—
58	C₂²×C₄×C₁₀₇	—	4	4	no	—
59	C₂²×C₄×C₁₀₇	—	4	4	no	—
60	C₂²×C₄×C₁₀₇	—	4	2	no	—
61	C₂²×C₄×C₁₀₇	—	4	1	no	—
62	C₂²×C₄×C₁₀₇	—	4	1	no	—
63	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₀₇	—	2	8	no	—
69	C₂×C₈×C₁₀₇	—	2	4	no	—
70	C₂×C₈×C₁₀₇	—	2	4	no	—
71	C₂×C₈×C₁₀₇	—	4	2	no	—
72	C₂×C₈×C₁₀₇	—	4	2	no	—
73	C₄⋊C₄×C₁₀₇	—	2	1	no	—
74	C₄⋊C₄×C₁₀₇	—	2	2	no	—
75	C₄⋊C₄×C₁₀₇	C₄²	1	2	yes	—
76	C₄⋊C₄×C₁₀₇	M₁₆	1	4	yes	—
77	C₂²⋊C₄×C₁₀₇	—	2	8	no	—
78	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
79	C₄²×C₁₀₇	—	2	4	no	—
80	C₄²×C₁₀₇	—	2	2	no	—
81	C₄²×C₁₀₇	—	2	2	no	—
82	C₄²×C₁₀₇	—	2	1	no	—
83	C₄²×C₁₀₇	—	2	2	no	—
84	C₄²×C₁₀₇	—	2	1	no	—
85	C₄²×C₁₀₇	—	4	4	no	—
86	C₄²×C₁₀₇	—	4	2	no	—
87	C₄²×C₁₀₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₀₇	#41	1	2	yes	—
89	C₂³×C₁₀₇	—	2	2	no	—
90	C₂³×C₁₀₇	—	2	4	no	—
91	C₂³×C₁₀₇	—	4	1	no	—
92	C₂³×C₁₀₇	—	4	1	no	—
93	C₂³×C₁₀₇	—	4	2	no	—
94	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	D₄×C₁₀₇	—	2	2	no	—
96	D₄×C₁₀₇	—	2	4	no	—
97	D₄×C₁₀₇	#97	1	4	yes	complemented
98	C₂×C₄×C₁₀₇	—	2	2	no	—
99	C₂×C₄×C₁₀₇	—	2	2	no	—
100	C₂×C₄×C₁₀₇	—	2	2	no	—
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	2	no	—
105	C₂×C₄×C₁₀₇	—	2	2	no	—
106	C₂×C₄×C₁₀₇	—	4	16	no	—
107	C₂×C₄×C₁₀₇	—	4	1	no	—
108	C₂×C₄×C₁₀₇	—	4	2	no	—
109	C₂×C₄×C₁₀₇	—	4	1	no	—
110	C₂×C₄×C₁₀₇	—	4	2	no	—
111	C₂×C₄×C₁₀₇	—	4	4	no	—
112	C₂×C₄×C₁₀₇	—	4	4	no	—
113	C₂×C₄×C₁₀₇	#104	1	4	yes	—
114	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₁₀₇	#137	1	1	yes	characteristic
123	C₈×C₁₀₇	—	2	8	no	—
124	C₈×C₁₀₇	—	4	4	no	—
125	C₂²×C₁₀₇	—	2	4	no	—
126	C₂²×C₁₀₇	—	4	4	no	—
127	C₂²×C₁₀₇	—	4	4	no	—
128	C₂²×C₁₀₇	—	4	2	no	—
129	C₂²×C₁₀₇	—	4	2	no	—
130	C₂²×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₁₀₇	#407	1	2	yes	—
132	C₂²×C₁₀₇	#417	1	1	yes	characteristic
133	C₂²×C₁₀₇	#418	1	1	yes	characteristic
134	C₂²×C₁₀₇	C₂×#104	1	2	yes	—
135	C₄×C₁₀₇	—	2	4	no	—
136	C₄×C₁₀₇	—	2	2	no	—
137	C₄×C₁₀₇	—	4	2	no	—
138	C₄×C₁₀₇	—	4	4	no	—
139	C₄×C₁₀₇	—	4	2	no	—
140	C₄×C₁₀₇	C₂×#97	1	1	yes	characteristic
141	C₄×C₁₀₇	C₄×M₁₆	1	1	yes	characteristic
142	C₄×C₁₀₇	#432	1	2	yes	—
143	C₄×C₁₀₇	#440	1	2	yes	—
144	C₄×C₁₀₇	#442	1	2	yes	—
145	#8172	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₁₀₇	—	4	4	no	—
147	C₂×C₁₀₇	#1489	1	2	yes	—
148	C₂×C₁₀₇	#1490	1	2	yes	—
149	C₂×C₁₀₇	#1749	1	1	yes	characteristic
150	C₂×C₁₀₇	C₄×#104	1	1	yes	characteristic
151	C₂×C₁₀₇	#1762	1	1	yes	characteristic
152	C₄²×D₄	C₂×C₁₀₇	1	1	yes	characteristic
153	#1767	C₂×C₁₀₇	1	1	yes	characteristic, complemented
154	C₄×#97	C₂×C₁₀₇	1	1	yes	characteristic, complemented
155	C₂×D₄⋊C₈	C₂×C₁₀₇	1	2	yes	—
156	#1279	C₂×C₁₀₇	1	2	yes	complemented
157	C₁₀₇	#8172	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×C₄×D₄	C₄×C₁₀₇	1	2	yes	—
159	C₂×C₄×D₄	C₂²×C₁₀₇	1	1	yes	characteristic
160	C₂²×C₄²	—	2	1	no	—
161	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
162	C₂×#97	—	2	2	no	—
163	C₂×#97	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₁₀₇	1	2	yes	—
165	C₄×C₄⋊C₄	C₄×C₁₀₇	1	2	yes	—
166	C₄×C₄⋊C₄	C₂²×C₁₀₇	1	1	yes	characteristic
167	C₄×C₂²⋊C₄	—	2	1	no	—
168	C₄×C₂²⋊C₄	—	2	2	no	—
169	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
170	#336	—	2	1	no	—
171	D₄⋊C₈	C₄×C₁₀₇	1	8	yes	complemented
172	C₂²×D₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₄×D₄	—	2	4	no	—
178	C₄×D₄	—	2	4	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₂×C₄×C₁₀₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₂²⋊C₄	—	2	1	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	4	4	no	—
188	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
189	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
191	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	C₂×C₄×C₁₀₇	1	4	yes	complemented
195	#97	—	2	8	no	—
196	#97	—	4	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	C₂×C₄×C₁₀₇	1	4	yes	complemented
199	#95	—	2	2	no	—
200	#95	—	2	1	no	—
201	C₂⁴	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₄²×C₁₀₇	1	2	yes	—
205	C₂×D₄	M₁₆×C₁₀₇	1	4	yes	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	C₄²×C₁₀₇	1	2	yes	—
228	C₄⋊C₄	M₁₆×C₁₀₇	1	4	yes	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	—	4	2	no	—
239	C₄²	C₂²×C₄×C₁₀₇	1	2	yes	—
240	C₄²	#41×C₁₀₇	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
247	D₄	—	2	2	no	—
248	D₄	—	2	4	no	—
249	D₄	#97×C₁₀₇	1	4	yes	complemented
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	#104×C₁₀₇	1	4	yes	—
266	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
267	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
268	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
270	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
271	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
272	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
273	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	1	yes	characteristic
274	C₂×C₄	#137×C₁₀₇	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
283	C₂²	#407×C₁₀₇	1	2	yes	—
284	C₂²	#417×C₁₀₇	1	1	yes	characteristic
285	C₂²	#418×C₁₀₇	1	1	yes	characteristic
286	C₂²	C₂×#104×C₁₀₇	1	2	yes	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	C₂×#97×C₁₀₇	1	1	yes	characteristic
293	C₄	C₄×M₁₆×C₁₀₇	1	1	yes	characteristic
294	C₄	#432×C₁₀₇	1	2	yes	—
295	C₄	#440×C₁₀₇	1	2	yes	—
296	C₄	#442×C₁₀₇	1	2	yes	—
297	C₂	—	4	4	no	—
298	C₂	C₁₀₇×#1489	1	2	yes	—
299	C₂	C₁₀₇×#1490	1	2	yes	—
300	C₂	C₁₀₇×#1749	1	1	yes	characteristic
301	C₂	C₄×#104×C₁₀₇	1	1	yes	characteristic
302	C₂	C₁₀₇×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle	C₂³×C₁₀₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₀₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₇×#1767	C₂	semidirect
3. C₄×#97×C₁₀₇	C₂	semidirect
5. C₁₀₇×#1279	C₂	semidirect
19. D₄⋊C₈×C₁₀₇	C₄	semidirect
42. C₄⋊C₈×C₁₀₇	C₂×C₄	semidirect
46. C₄×C₈×C₁₀₇	C₂×C₄	semidirect
97. D₄×C₁₀₇	#97	semidirect
145. #8172	C₁₀₇	direct (N × H)
153. #1767	C₂×C₁₀₇	semidirect
154. C₄×#97	C₂×C₁₀₇	semidirect
156. #1279	C₂×C₁₀₇	semidirect
157. C₁₀₇	#8172	direct (N × H)
171. D₄⋊C₈	C₄×C₁₀₇	semidirect
194. C₄⋊C₈	C₂×C₄×C₁₀₇	semidirect
198. C₄×C₈	C₂×C₄×C₁₀₇	semidirect
249. D₄	#97×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₁₀₇	2
C₁₀₇×#1767	2
C₄×#97×C₁₀₇	2
C₂×D₄⋊C₈×C₁₀₇	2
C₁₀₇×#1279	2
#8172	107

Subgroups — C107×#8172 #41034136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8172 #41034136