Subgroups — C₁₀₇×#8660 #41034624

Order 27392.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₇×#1486	C₂	1	2	yes	maximal
5	C₁₀₇×#1301	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₀₇	—	2	1	no	—
7	C₂×C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₀₇	—	2	1	no	—
9	C₈×D₄×C₁₀₇	—	2	1	no	—
10	C₈×D₄×C₁₀₇	C₄	1	2	yes	—
11	C₈×D₄×C₁₀₇	C₄	1	2	yes	—
12	#431×C₁₀₇	C₂²	1	2	yes	—
13	#423×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₁₀₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	#341×C₁₀₇	—	2	1	no	—
18	#341×C₁₀₇	—	2	1	no	—
19	#335×C₁₀₇	—	2	1	no	—
20	#334×C₁₀₇	—	2	1	no	—
21	D₄⋊C₈×C₁₀₇	—	2	4	no	—
22	C₂²×D₄×C₁₀₇	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₇	—	2	1	no	—
24	C₂×M₁₆×C₁₀₇	—	2	2	no	—
25	C₂²×C₈×C₁₀₇	—	2	1	no	—
26	C₂²×C₈×C₁₀₇	—	2	2	no	—
27	C₂²×C₈×C₁₀₇	—	2	2	no	—
28	C₂²×C₈×C₁₀₇	—	4	4	no	—
29	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
31	C₄×D₄×C₁₀₇	—	2	1	no	—
32	C₄×D₄×C₁₀₇	—	2	1	no	—
33	C₄×D₄×C₁₀₇	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₁₀₇	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₀₇	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₇	—	2	1	no	—
37	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₇	—	2	1	no	—
39	C₄⋊C₈×C₁₀₇	—	2	2	no	—
40	C₄⋊C₈×C₁₀₇	—	2	2	no	—
41	C₄⋊C₈×C₁₀₇	—	2	2	no	—
42	C₄⋊C₈×C₁₀₇	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
44	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
45	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
46	C₄×C₈×C₁₀₇	—	2	2	no	—
47	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
51	C₂⁴×C₁₀₇	—	2	1	no	—
52	C₂×D₄×C₁₀₇	—	2	4	no	—
53	C₂×D₄×C₁₀₇	—	2	1	no	—
54	C₂×D₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₀₇	—	2	1	no	—
56	C₂²×C₄×C₁₀₇	—	2	2	no	—
57	C₂²×C₄×C₁₀₇	—	2	2	no	—
58	C₂²×C₄×C₁₀₇	—	4	2	no	—
59	C₂²×C₄×C₁₀₇	—	4	2	no	—
60	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₇	#41	1	1	yes	characteristic
63	M₁₆×C₁₀₇	—	4	2	no	—
64	M₁₆×C₁₀₇	—	4	2	no	—
65	C₂×C₈×C₁₀₇	—	2	2	no	—
66	C₂×C₈×C₁₀₇	—	2	2	no	—
67	C₂×C₈×C₁₀₇	—	2	2	no	—
68	C₂×C₈×C₁₀₇	—	4	4	no	—
69	C₂×C₈×C₁₀₇	—	4	4	no	—
70	C₂×C₈×C₁₀₇	—	4	2	no	—
71	C₂×C₈×C₁₀₇	—	4	2	no	—
72	C₂×C₈×C₁₀₇	—	4	2	no	—
73	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₇	—	2	1	no	—
78	C₄⋊C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
79	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
80	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
81	C₄²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₀₇	#41	1	1	yes	characteristic
85	C₂³×C₁₀₇	—	2	4	no	—
86	C₂³×C₁₀₇	—	2	2	no	—
87	C₂³×C₁₀₇	—	4	2	no	—
88	C₂³×C₁₀₇	—	4	2	no	—
89	C₂³×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
90	D₄×C₁₀₇	—	2	4	no	—
91	D₄×C₁₀₇	—	2	2	no	—
92	D₄×C₁₀₇	#108	1	4	yes	—
93	C₂×C₄×C₁₀₇	—	2	2	no	—
94	C₂×C₄×C₁₀₇	—	2	2	no	—
95	C₂×C₄×C₁₀₇	—	4	1	no	—
96	C₂×C₄×C₁₀₇	—	4	2	no	—
97	C₂×C₄×C₁₀₇	—	4	2	no	—
98	C₂×C₄×C₁₀₇	—	4	2	no	—
99	C₂×C₄×C₁₀₇	—	4	4	no	—
100	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₀₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₀₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₀₇	#135	1	2	yes	—
107	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₁₀₇	—	2	1	no	—
109	C₈×C₁₀₇	—	2	1	no	—
110	C₈×C₁₀₇	—	2	1	no	—
111	C₈×C₁₀₇	—	2	1	no	—
112	C₈×C₁₀₇	—	4	2	no	—
113	C₈×C₁₀₇	—	8	2	no	—
114	C₈×C₁₀₇	—	8	2	no	—
115	C₂²×C₁₀₇	—	2	1	no	—
116	C₂²×C₁₀₇	—	2	1	no	—
117	C₂²×C₁₀₇	—	2	4	no	—
118	C₂²×C₁₀₇	—	4	8	no	—
119	C₂²×C₁₀₇	—	4	2	no	—
120	C₂²×C₁₀₇	—	4	2	no	—
121	C₂²×C₁₀₇	#380	1	1	yes	characteristic
122	C₂²×C₁₀₇	#478	1	2	yes	—
123	C₄×C₁₀₇	—	2	1	no	—
124	C₄×C₁₀₇	—	2	1	no	—
125	C₄×C₁₀₇	—	2	1	no	—
126	C₄×C₁₀₇	—	4	2	no	—
127	C₄×C₁₀₇	#416	1	1	yes	characteristic
128	C₄×C₁₀₇	#418	1	1	yes	characteristic
129	C₄×C₁₀₇	C₂×#108	1	2	yes	—
130	C₄×C₁₀₇	#463	1	2	yes	—
131	#8660	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₀₇	—	2	2	no	—
133	C₂×C₁₀₇	—	4	4	no	—
134	C₂×C₁₀₇	#1804	1	1	yes	characteristic
135	C₂×C₁₀₇	#1810	1	1	yes	characteristic
136	C₂×C₁₀₇	#1814	1	1	yes	characteristic
137	C₂×C₈×D₄	C₂×C₁₀₇	1	1	yes	characteristic
138	#1837	C₂×C₁₀₇	1	1	yes	characteristic, complemented
139	#1564	C₂×C₁₀₇	1	1	yes	characteristic, complemented
140	#1486	C₂×C₁₀₇	1	2	yes	—
141	#1301	C₂×C₁₀₇	1	2	yes	complemented
142	C₁₀₇	#8660	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³×C₈	—	2	1	no	—
144	C₂×C₄×D₄	C₂²×C₁₀₇	1	1	yes	characteristic
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	C₄×C₁₀₇	1	2	yes	—
148	C₈×D₄	C₄×C₁₀₇	1	2	yes	—
149	#431	C₂²×C₁₀₇	1	2	yes	—
150	#423	C₂²×C₁₀₇	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	1	yes	characteristic
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
154	#341	—	2	1	no	—
155	#341	—	2	1	no	—
156	#335	—	2	1	no	—
157	#334	—	2	1	no	—
158	D₄⋊C₈	—	2	4	no	—
159	C₂²×D₄	Q₈×C₁₀₇	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	4	4	no	—
166	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂×C₄×C₁₀₇	1	2	yes	—
171	C₄×D₄	D₄×C₁₀₇	1	2	yes	—
172	C₂×C₄⋊C₄	Q₈×C₁₀₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂×C₄×C₁₀₇	1	2	yes	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
186	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
187	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
199	C₂²×C₄	#41×C₁₀₇	1	1	yes	characteristic
200	M₁₆	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
221	C₄²	#41×C₁₀₇	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	#108×C₁₀₇	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
239	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
240	C₂×C₄	#121×C₁₀₇	1	1	yes	characteristic
241	C₂×C₄	#121×C₁₀₇	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
243	C₂×C₄	#135×C₁₀₇	1	2	yes	—
244	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	8	2	no	—
251	C₈	—	8	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#380×C₁₀₇	1	1	yes	characteristic
259	C₂²	#478×C₁₀₇	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	#416×C₁₀₇	1	1	yes	characteristic
265	C₄	#418×C₁₀₇	1	1	yes	characteristic
266	C₄	C₂×#108×C₁₀₇	1	2	yes	—
267	C₄	#463×C₁₀₇	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	C₁₀₇×#1804	1	1	yes	characteristic
271	C₂	C₁₀₇×#1810	1	1	yes	characteristic
272	C₂	C₁₀₇×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₇	#121
commutator	C₄²	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	#380
prime radical	C₂²×D₄×C₁₀₇	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₇×#1837	C₂	semidirect
3. C₁₀₇×#1564	C₂	semidirect
5. C₁₀₇×#1301	C₂	semidirect
131. #8660	C₁₀₇	direct (N × H)
138. #1837	C₂×C₁₀₇	semidirect
139. #1564	C₂×C₁₀₇	semidirect
141. #1301	C₂×C₁₀₇	semidirect
142. C₁₀₇	#8660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₀₇	2
C₁₀₇×#1837	2
C₁₀₇×#1564	2
C₁₀₇×#1486	2
C₁₀₇×#1301	2
#8660	107

Subgroups — C107×#8660 #41034624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8660 #41034624