Subgroups — C₁₀₇×#8585 #41034549

Order 27392.

684 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#127×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1842	C₂	1	2	yes	maximal, complemented
3	C₂×#331×C₁₀₇	C₂	1	2	yes	maximal
4	C₁₀₇×#1296	C₂	1	2	yes	maximal, complemented
5	C₂×#127×C₁₀₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×D₄×C₁₀₇	—	2	2	no	—
7	C₂×C₄×D₄×C₁₀₇	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₁₀₇	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₀₇	—	2	2	no	—
10	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	4	yes	—
11	#390×C₁₀₇	—	2	2	no	—
12	#390×C₁₀₇	C₄	1	2	yes	—
13	#384×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
15	#331×C₁₀₇	—	2	4	no	—
16	C₂²×D₄×C₁₀₇	—	2	2	no	—
17	C₂²×D₄×C₁₀₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₀₇	—	2	2	no	—
19	C₂²×C₈×C₁₀₇	—	2	4	no	—
20	#127×C₁₀₇	—	2	2	no	—
21	#127×C₁₀₇	D₄	1	2	yes	—
22	#127×C₁₀₇	Q₈	1	2	yes	—
23	C₄×D₄×C₁₀₇	—	4	4	no	—
24	C₄×D₄×C₁₀₇	—	4	4	no	—
25	C₄×D₄×C₁₀₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊C₄×C₁₀₇	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₁₀₇	—	2	4	no	—
30	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
33	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
34	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₁₀₇	—	2	4	no	—
36	C₄⋊C₈×C₁₀₇	—	2	4	no	—
37	C₄⋊C₈×C₁₀₇	—	4	4	no	—
38	C₄⋊C₈×C₁₀₇	—	4	4	no	—
39	C₂⁴×C₁₀₇	—	2	2	no	—
40	C₂×D₄×C₁₀₇	—	2	2	no	—
41	C₂×D₄×C₁₀₇	—	2	4	no	—
42	C₂×D₄×C₁₀₇	—	2	4	no	—
43	C₂×D₄×C₁₀₇	—	4	2	no	—
44	C₂×D₄×C₁₀₇	—	4	2	no	—
45	C₂×D₄×C₁₀₇	—	4	2	no	—
46	C₂×D₄×C₁₀₇	—	4	2	no	—
47	C₂×D₄×C₁₀₇	—	4	2	no	—
48	C₂×D₄×C₁₀₇	—	4	2	no	—
49	C₂×D₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₁₀₇	—	2	2	no	—
51	C₂²×C₄×C₁₀₇	—	4	4	no	—
52	C₂²×C₄×C₁₀₇	—	4	8	no	—
53	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₀₇	—	4	4	no	—
59	C₂×C₈×C₁₀₇	—	4	4	no	—
60	C₂×C₈×C₁₀₇	—	4	4	no	—
61	C₄⋊C₄×C₁₀₇	—	4	2	no	—
62	C₄⋊C₄×C₁₀₇	—	4	2	no	—
63	C₄⋊C₄×C₁₀₇	—	4	2	no	—
64	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
65	C₂²⋊C₄×C₁₀₇	—	4	8	no	—
66	C₄²×C₁₀₇	—	2	2	no	—
67	C₄²×C₁₀₇	—	2	2	no	—
68	C₄²×C₁₀₇	—	2	2	no	—
69	C₄²×C₁₀₇	—	4	1	no	—
70	C₄²×C₁₀₇	D₈	1	4	yes	—
71	C₄²×C₁₀₇	SD₁₆	1	4	yes	—
72	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₁₀₇	#41	1	2	yes	—
75	C₂³×C₁₀₇	—	2	4	no	—
76	C₂³×C₁₀₇	—	4	2	no	—
77	C₂³×C₁₀₇	—	4	2	no	—
78	C₂³×C₁₀₇	—	4	4	no	—
79	C₂³×C₁₀₇	—	4	4	no	—
80	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	D₄×C₁₀₇	—	4	4	no	—
82	D₄×C₁₀₇	—	4	4	no	—
83	D₄×C₁₀₇	—	4	4	no	—
84	D₄×C₁₀₇	—	4	4	no	—
85	D₄×C₁₀₇	—	4	4	no	—
86	D₄×C₁₀₇	—	4	4	no	—
87	C₂×C₄×C₁₀₇	—	2	1	no	—
88	C₂×C₄×C₁₀₇	—	2	2	no	—
89	C₂×C₄×C₁₀₇	—	2	1	no	—
90	C₂×C₄×C₁₀₇	—	2	2	no	—
91	C₂×C₄×C₁₀₇	—	2	1	no	—
92	C₂×C₄×C₁₀₇	—	2	2	no	—
93	C₂×C₄×C₁₀₇	—	4	2	no	—
94	C₂×C₄×C₁₀₇	—	4	4	no	—
95	C₂×C₄×C₁₀₇	—	8	8	no	—
96	C₂×C₄×C₁₀₇	#100	1	2	yes	—
97	C₂×C₄×C₁₀₇	#102	1	4	yes	—
98	C₂×C₄×C₁₀₇	#102	1	4	yes	—
99	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₀₇	#121	1	2	yes	—
102	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
105	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₁₀₇	#137	1	2	yes	—
107	C₈×C₁₀₇	—	8	4	no	—
108	C₂²×C₁₀₇	—	4	4	no	—
109	C₂²×C₁₀₇	—	4	8	no	—
110	C₂²×C₁₀₇	—	8	2	no	—
111	C₂²×C₁₀₇	—	8	4	no	—
112	C₂²×C₁₀₇	—	8	2	no	—
113	C₂²×C₁₀₇	#380	1	1	yes	characteristic
114	C₂²×C₁₀₇	C₂×#100	1	1	yes	characteristic
115	C₂²×C₁₀₇	#413	1	1	yes	characteristic
116	C₂²×C₁₀₇	C₂×#102	1	2	yes	—
117	C₂²×C₁₀₇	#417	1	2	yes	—
118	C₄×C₁₀₇	—	2	1	no	—
119	C₄×C₁₀₇	—	2	1	no	—
120	C₄×C₁₀₇	—	2	4	no	—
121	C₄×C₁₀₇	—	4	1	no	—
122	C₄×C₁₀₇	—	8	2	no	—
123	C₄×C₁₀₇	#331	1	4	yes	complemented
124	C₄×C₁₀₇	#459	1	2	yes	—
125	C₄×C₁₀₇	#461	1	2	yes	—
126	C₄×C₁₀₇	#474	1	2	yes	—
127	C₄×C₁₀₇	#476	1	2	yes	—
128	#8585	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₇	—	8	4	no	—
130	C₂×C₁₀₇	C₂×#331	1	1	yes	characteristic
131	C₂×C₁₀₇	#1547	1	2	yes	—
132	C₂×C₁₀₇	#1560	1	1	yes	characteristic
133	C₂×C₁₀₇	#1800	1	1	yes	characteristic
134	C₂×C₁₀₇	#1802	1	2	yes	—
135	C₄×#127	C₂×C₁₀₇	1	1	yes	characteristic
136	#1842	C₂×C₁₀₇	1	2	yes	complemented
137	C₂×#331	C₂×C₁₀₇	1	2	yes	—
138	#1296	C₂×C₁₀₇	1	2	yes	complemented
139	C₁₀₇	#8585	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#127	C₂²×C₁₀₇	1	1	yes	characteristic
141	C₂×C₄×D₄	—	2	2	no	—
142	C₂×C₄×D₄	C₄×C₁₀₇	1	2	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	—	2	2	no	—
145	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	4	yes	—
146	#390	—	2	2	no	—
147	#390	C₄×C₁₀₇	1	2	yes	—
148	#384	C₂²×C₁₀₇	1	1	yes	characteristic
149	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
150	#331	—	2	4	no	—
151	C₂²×D₄	—	2	2	no	—
152	C₂²×D₄	C₂×C₄×C₁₀₇	1	2	yes	—
153	C₂³×C₄	—	2	2	no	—
154	C₂²×C₈	—	2	4	no	—
155	#127	—	2	2	no	—
156	#127	D₄×C₁₀₇	1	2	yes	—
157	#127	Q₈×C₁₀₇	1	2	yes	—
158	C₄×D₄	—	4	4	no	—
159	C₄×D₄	—	4	4	no	—
160	C₄×D₄	—	4	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	—	2	2	no	—
165	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
168	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
169	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	4	4	no	—
173	C₄⋊C₈	—	4	4	no	—
174	C₂⁴	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	C₄⋊C₄×C₁₀₇	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	4	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	D₈×C₁₀₇	1	4	yes	—
206	C₄²	SD₁₆×C₁₀₇	1	4	yes	—
207	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
208	C₄²	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
209	C₄²	#41×C₁₀₇	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	#100×C₁₀₇	1	2	yes	—
232	C₂×C₄	#102×C₁₀₇	1	4	yes	—
233	C₂×C₄	#102×C₁₀₇	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
235	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
236	C₂×C₄	#121×C₁₀₇	1	2	yes	—
237	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
239	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	2	yes	—
240	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
241	C₂×C₄	#137×C₁₀₇	1	2	yes	—
242	C₈	—	8	4	no	—
243	C₂²	—	4	4	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	#380×C₁₀₇	1	1	yes	characteristic
249	C₂²	C₂×#100×C₁₀₇	1	1	yes	characteristic
250	C₂²	#413×C₁₀₇	1	1	yes	characteristic
251	C₂²	C₂×#102×C₁₀₇	1	2	yes	—
252	C₂²	#417×C₁₀₇	1	2	yes	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	4	no	—
256	C₄	—	4	1	no	—
257	C₄	—	8	2	no	—
258	C₄	#331×C₁₀₇	1	4	yes	complemented
259	C₄	#459×C₁₀₇	1	2	yes	—
260	C₄	#461×C₁₀₇	1	2	yes	—
261	C₄	#474×C₁₀₇	1	2	yes	—
262	C₄	#476×C₁₀₇	1	2	yes	—
263	C₂	—	8	4	no	—
264	C₂	C₂×#331×C₁₀₇	1	1	yes	characteristic
265	C₂	C₁₀₇×#1547	1	2	yes	—
266	C₂	C₁₀₇×#1560	1	1	yes	characteristic
267	C₂	C₁₀₇×#1800	1	1	yes	characteristic
268	C₂	C₁₀₇×#1802	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
prime radical	C₂×#127×C₁₀₇	C₂²
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
2. C₁₀₇×#1842	C₂	semidirect
4. C₁₀₇×#1296	C₂	semidirect
123. C₄×C₁₀₇	#331	semidirect
128. #8585	C₁₀₇	direct (N × H)
136. #1842	C₂×C₁₀₇	semidirect
138. #1296	C₂×C₁₀₇	semidirect
139. C₁₀₇	#8585	direct (N × H)
258. C₄	#331×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#127×C₁₀₇	2
C₁₀₇×#1842	2
C₂×#331×C₁₀₇	2
C₁₀₇×#1296	2
#8585	107

Subgroups — C107×#8585 #41034549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8585 #41034549