Subgroups — C₁₀₇×#8649 #41034613

Order 27392.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1834	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1803	C₂	1	2	yes	maximal
4	C₂×#335×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₁₀₇×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₀₇	—	2	2	no	—
8	C₈×Q₈×C₁₀₇	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈×C₁₀₇	—	2	1	no	—
10	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×#103×C₁₀₇	—	2	2	no	—
13	C₂×C₄×C₈×C₁₀₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	#384×C₁₀₇	C₂²	1	2	yes	—
16	#340×C₁₀₇	—	2	1	no	—
17	#340×C₁₀₇	—	2	1	no	—
18	#340×C₁₀₇	—	2	2	no	—
19	#335×C₁₀₇	—	2	2	no	—
20	C₂²×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₇	—	2	1	no	—
22	C₂²×C₈×C₁₀₇	—	2	2	no	—
23	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₀₇	—	2	1	no	—
26	C₄×Q₈×C₁₀₇	—	2	1	no	—
27	C₄×Q₈×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₁₀₇	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₇	—	2	1	no	—
35	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₇	—	2	4	no	—
37	C₄⋊C₈×C₁₀₇	—	2	1	no	—
38	C₄⋊C₈×C₁₀₇	—	2	2	no	—
39	C₄⋊C₈×C₁₀₇	—	2	2	no	—
40	C₄⋊C₈×C₁₀₇	C₂×C₄	1	2	yes	—
41	#103×C₁₀₇	—	4	4	no	—
42	C₄×C₈×C₁₀₇	—	2	4	no	—
43	C₄×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
45	C₂×Q₈×C₁₀₇	—	2	1	no	—
46	C₂×Q₈×C₁₀₇	—	2	1	no	—
47	C₂×Q₈×C₁₀₇	—	2	1	no	—
48	C₂×Q₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
49	C₂×Q₈×C₁₀₇	D₈	1	2	yes	—
50	C₂×Q₈×C₁₀₇	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₁₀₇	—	2	1	no	—
52	C₂²×C₄×C₁₀₇	—	2	1	no	—
53	C₂²×C₄×C₁₀₇	—	4	2	no	—
54	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₇	#41	1	1	yes	characteristic
57	C₂×C₈×C₁₀₇	—	2	2	no	—
58	C₂×C₈×C₁₀₇	—	2	2	no	—
59	C₂×C₈×C₁₀₇	—	2	2	no	—
60	C₂×C₈×C₁₀₇	—	4	2	no	—
61	C₂×C₈×C₁₀₇	—	4	2	no	—
62	C₂×C₈×C₁₀₇	—	4	2	no	—
63	C₂×C₈×C₁₀₇	—	4	2	no	—
64	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
66	C₄⋊C₄×C₁₀₇	—	2	2	no	—
67	C₄⋊C₄×C₁₀₇	—	2	1	no	—
68	C₄⋊C₄×C₁₀₇	—	2	2	no	—
69	C₄⋊C₄×C₁₀₇	—	4	2	no	—
70	C₄⋊C₄×C₁₀₇	—	4	2	no	—
71	C₄⋊C₄×C₁₀₇	—	4	2	no	—
72	C₄⋊C₄×C₁₀₇	—	4	2	no	—
73	C₄⋊C₄×C₁₀₇	—	8	2	no	—
74	C₄⋊C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₀₇	—	2	2	no	—
76	C₄²×C₁₀₇	—	2	2	no	—
77	C₄²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₁₀₇	#41	1	1	yes	characteristic
81	C₂³×C₁₀₇	#121	1	1	yes	characteristic
82	Q₈×C₁₀₇	—	2	2	no	—
83	Q₈×C₁₀₇	—	2	2	no	—
84	Q₈×C₁₀₇	—	2	2	no	—
85	Q₈×C₁₀₇	#107	1	4	yes	—
86	C₂×C₄×C₁₀₇	—	2	2	no	—
87	C₂×C₄×C₁₀₇	—	2	2	no	—
88	C₂×C₄×C₁₀₇	—	2	2	no	—
89	C₂×C₄×C₁₀₇	—	2	2	no	—
90	C₂×C₄×C₁₀₇	—	2	2	no	—
91	C₂×C₄×C₁₀₇	—	4	2	no	—
92	C₂×C₄×C₁₀₇	—	8	2	no	—
93	C₂×C₄×C₁₀₇	—	8	2	no	—
94	C₂×C₄×C₁₀₇	—	8	2	no	—
95	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₀₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₀₇	#135	1	2	yes	—
104	C₂×C₄×C₁₀₇	#135	1	1	yes	characteristic
105	C₂×C₄×C₁₀₇	#135	1	1	yes	characteristic
106	C₂×C₄×C₁₀₇	#137	1	2	yes	—
107	C₈×C₁₀₇	—	2	2	no	—
108	C₈×C₁₀₇	—	2	2	no	—
109	C₈×C₁₀₇	—	4	2	no	—
110	C₈×C₁₀₇	—	8	2	no	—
111	C₂²×C₁₀₇	#380	1	1	yes	characteristic
112	C₂²×C₁₀₇	#464	1	2	yes	—
113	C₂²×C₁₀₇	#466	1	1	yes	characteristic
114	C₂²×C₁₀₇	#467	1	1	yes	characteristic
115	C₂²×C₁₀₇	#471	1	2	yes	—
116	C₄×C₁₀₇	—	2	4	no	—
117	C₄×C₁₀₇	—	2	1	no	—
118	C₄×C₁₀₇	—	2	1	no	—
119	C₄×C₁₀₇	—	4	2	no	—
120	C₄×C₁₀₇	—	16	2	no	—
121	C₄×C₁₀₇	#416	1	1	yes	characteristic
122	C₄×C₁₀₇	#419	1	1	yes	characteristic
123	C₄×C₁₀₇	C₂×#107	1	1	yes	characteristic
124	C₄×C₁₀₇	#427	1	1	yes	characteristic
125	C₄×C₁₀₇	#466	1	1	yes	characteristic
126	C₄×C₁₀₇	#467	1	1	yes	characteristic
127	C₄×C₁₀₇	#479	1	2	yes	—
128	#8649	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₇	#1682	1	2	yes	—
130	C₂×C₁₀₇	#1684	1	2	yes	—
131	C₂×C₁₀₇	#1805	1	1	yes	characteristic
132	C₂×C₁₀₇	#1808	1	1	yes	characteristic
133	C₂×C₁₀₇	#1815	1	1	yes	characteristic
134	C₂×C₈×Q₈	C₂×C₁₀₇	1	1	yes	characteristic
135	#1834	C₂×C₁₀₇	1	1	yes	characteristic
136	#1803	C₂×C₁₀₇	1	2	yes	—
137	C₂×#335	C₂×C₁₀₇	1	1	yes	characteristic
138	#1296	C₂×C₁₀₇	1	2	yes	—
139	C₁₀₇	#8649	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×C₄×Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
141	C₈×Q₈	—	2	2	no	—
142	C₈×Q₈	C₄×C₁₀₇	1	4	yes	complemented
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	1	yes	characteristic
146	C₂×#103	—	2	2	no	—
147	C₂×C₄×C₈	—	2	1	no	—
148	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
149	#384	C₂²×C₁₀₇	1	2	yes	—
150	#340	—	2	1	no	—
151	#340	—	2	1	no	—
152	#340	—	2	2	no	—
153	#335	—	2	2	no	—
154	C₂²×Q₈	D₄×C₁₀₇	1	1	yes	characteristic
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₂×C₄×C₁₀₇	1	2	yes	—
162	C₄×Q₈	Q₈×C₁₀₇	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	D₄×C₁₀₇	1	1	yes	characteristic
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	C₂×C₄×C₁₀₇	1	2	yes	—
175	#103	—	4	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
178	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
183	C₂×Q₈	D₈×C₁₀₇	1	2	yes	—
184	C₂×Q₈	Q₁₆×C₁₀₇	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
190	C₂²×C₄	#41×C₁₀₇	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
199	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	C₄⋊C₄×C₁₀₇	1	2	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
213	C₄²	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
214	C₄²	#41×C₁₀₇	1	1	yes	characteristic
215	C₂³	#121×C₁₀₇	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	#107×C₁₀₇	1	4	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
231	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
232	C₂×C₄	#121×C₁₀₇	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
235	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₁₆×C₁₀₇	1	1	yes	characteristic
237	C₂×C₄	#135×C₁₀₇	1	2	yes	—
238	C₂×C₄	#135×C₁₀₇	1	1	yes	characteristic
239	C₂×C₄	#135×C₁₀₇	1	1	yes	characteristic
240	C₂×C₄	#137×C₁₀₇	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₂²	#380×C₁₀₇	1	1	yes	characteristic
246	C₂²	#464×C₁₀₇	1	2	yes	—
247	C₂²	#466×C₁₀₇	1	1	yes	characteristic
248	C₂²	#467×C₁₀₇	1	1	yes	characteristic
249	C₂²	#471×C₁₀₇	1	2	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	16	2	no	—
255	C₄	#416×C₁₀₇	1	1	yes	characteristic
256	C₄	#419×C₁₀₇	1	1	yes	characteristic
257	C₄	C₂×#107×C₁₀₇	1	1	yes	characteristic
258	C₄	#427×C₁₀₇	1	1	yes	characteristic
259	C₄	#466×C₁₀₇	1	1	yes	characteristic
260	C₄	#467×C₁₀₇	1	1	yes	characteristic
261	C₄	#479×C₁₀₇	1	2	yes	—
262	C₂	C₁₀₇×#1682	1	2	yes	—
263	C₂	C₁₀₇×#1684	1	2	yes	—
264	C₂	C₁₀₇×#1805	1	1	yes	characteristic
265	C₂	C₁₀₇×#1808	1	1	yes	characteristic
266	C₂	C₁₀₇×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	#121
commutator	C₄²	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
LCS	G ▷ 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
8. C₈×Q₈×C₁₀₇	C₄	semidirect
128. #8649	C₁₀₇	direct (N × H)
139. C₁₀₇	#8649	direct (N × H)
142. C₈×Q₈	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×C₈×Q₈×C₁₀₇	2
C₁₀₇×#1834	2
C₁₀₇×#1803	2
C₂×#335×C₁₀₇	2
C₁₀₇×#1296	2
#8649	107

Subgroups — C107×#8649 #41034613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8649 #41034613