Subgroups — C₁₀₇×#9726 #41035690

Order 27392.

436 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	C₁₀₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₇×#1525	C₂	1	2	yes	maximal, complemented
4	C₁₀₇×#1484	C₂	1	2	yes	maximal, complemented
5	C₁₀₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
6	#532×C₁₀₇	C₂²	1	2	yes	—
7	#500×C₁₀₇	C₄	1	1	yes	characteristic
8	#493×C₁₀₇	C₄	1	1	yes	characteristic
9	#486×C₁₀₇	C₄	1	2	yes	—
10	#466×C₁₀₇	—	2	1	no	—
11	#466×C₁₀₇	—	2	1	no	—
12	C₄×D₈×C₁₀₇	—	2	1	no	—
13	C₄×D₈×C₁₀₇	—	2	1	no	—
14	#431×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₁₀₇	—	2	1	no	—
16	C₄×M₁₆×C₁₀₇	C₂²	1	2	yes	—
17	C₄×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
19	#341×C₁₀₇	—	2	1	no	—
20	#329×C₁₀₇	—	2	4	no	—
21	D₄⋊C₈×C₁₀₇	—	2	4	no	—
22	#322×C₁₀₇	—	2	1	no	—
23	#322×C₁₀₇	—	2	1	no	—
24	C₂×#41×C₁₀₇	—	2	2	no	—
25	C₂×D₈×C₁₀₇	—	2	1	no	—
26	C₂×D₈×C₁₀₇	—	2	1	no	—
27	C₂×M₁₆×C₁₀₇	—	2	1	no	—
28	C₂×M₁₆×C₁₀₇	—	2	2	no	—
29	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₀₇	C₂×C₄	1	2	yes	—
32	#127×C₁₀₇	C₂×C₄	1	2	yes	—
33	#124×C₁₀₇	—	2	2	no	—
34	#121×C₁₀₇	—	2	2	no	—
35	#121×C₁₀₇	—	2	2	no	—
36	C₄×D₄×C₁₀₇	—	2	2	no	—
37	C₄×D₄×C₁₀₇	—	2	2	no	—
38	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
39	#108×C₁₀₇	—	2	4	no	—
40	#107×C₁₀₇	—	2	1	no	—
41	#107×C₁₀₇	—	2	1	no	—
42	C₄⋊C₈×C₁₀₇	D₄	1	2	yes	—
43	#102×C₁₀₇	—	2	2	no	—
44	#102×C₁₀₇	—	2	2	no	—
45	C₂²⋊C₈×C₁₀₇	—	2	1	no	—
46	#97×C₁₀₇	—	2	2	no	—
47	#97×C₁₀₇	D₄	1	2	yes	—
48	C₄×C₈×C₁₀₇	—	2	1	no	—
49	C₄×C₈×C₁₀₇	—	2	1	no	—
50	C₄×C₈×C₁₀₇	—	2	2	no	—
51	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
55	#41×C₁₀₇	—	8	4	no	—
56	C₂×Q₈×C₁₀₇	—	2	2	no	—
57	C₂×D₄×C₁₀₇	—	2	2	no	—
58	C₂×D₄×C₁₀₇	—	2	2	no	—
59	C₂×D₄×C₁₀₇	—	2	2	no	—
60	C₂×D₄×C₁₀₇	—	4	2	no	—
61	C₂²×C₄×C₁₀₇	—	4	2	no	—
62	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₇	#41	1	1	yes	characteristic
65	D₈×C₁₀₇	—	4	2	no	—
66	D₈×C₁₀₇	—	4	2	no	—
67	M₁₆×C₁₀₇	—	2	4	no	—
68	M₁₆×C₁₀₇	—	4	2	no	—
69	M₁₆×C₁₀₇	—	4	2	no	—
70	C₂×C₈×C₁₀₇	—	2	2	no	—
71	C₂×C₈×C₁₀₇	—	2	2	no	—
72	C₂×C₈×C₁₀₇	—	2	2	no	—
73	C₂×C₈×C₁₀₇	—	4	1	no	—
74	C₂×C₈×C₁₀₇	—	4	2	no	—
75	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₀₇	—	2	2	no	—
80	C₄⋊C₄×C₁₀₇	—	2	2	no	—
81	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
82	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
83	C₄²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₁₀₇	—	4	2	no	—
88	C₂³×C₁₀₇	#124	1	1	yes	characteristic
89	Q₈×C₁₀₇	—	8	2	no	—
90	D₄×C₁₀₇	—	4	2	no	—
91	D₄×C₁₀₇	—	4	2	no	—
92	D₄×C₁₀₇	—	4	4	no	—
93	D₄×C₁₀₇	—	8	2	no	—
94	D₄×C₁₀₇	—	8	4	no	—
95	C₂×C₄×C₁₀₇	—	2	1	no	—
96	C₂×C₄×C₁₀₇	—	2	1	no	—
97	C₂×C₄×C₁₀₇	—	2	2	no	—
98	C₂×C₄×C₁₀₇	—	2	2	no	—
99	C₂×C₄×C₁₀₇	—	4	2	no	—
100	C₂×C₄×C₁₀₇	—	8	4	no	—
101	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₀₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₀₇	#127	1	1	yes	characteristic
107	C₈×C₁₀₇	—	2	1	no	—
108	C₈×C₁₀₇	—	2	1	no	—
109	C₈×C₁₀₇	—	2	1	no	—
110	C₈×C₁₀₇	—	4	2	no	—
111	C₈×C₁₀₇	—	4	4	no	—
112	C₈×C₁₀₇	—	4	2	no	—
113	C₈×C₁₀₇	#104	1	2	yes	—
114	C₂²×C₁₀₇	—	2	1	no	—
115	C₂²×C₁₀₇	—	2	1	no	—
116	C₂²×C₁₀₇	—	2	1	no	—
117	C₂²×C₁₀₇	—	8	2	no	—
118	C₂²×C₁₀₇	—	8	2	no	—
119	C₂²×C₁₀₇	—	8	2	no	—
120	C₂²×C₁₀₇	#390	1	1	yes	characteristic
121	C₄×C₁₀₇	—	2	1	no	—
122	C₄×C₁₀₇	—	2	1	no	—
123	C₄×C₁₀₇	—	2	1	no	—
124	C₄×C₁₀₇	—	8	2	no	—
125	C₄×C₁₀₇	C₂×#104	1	1	yes	characteristic
126	C₄×C₁₀₇	#421	1	1	yes	characteristic
127	C₄×C₁₀₇	#444	1	2	yes	—
128	C₄×C₁₀₇	#472	1	2	yes	—
129	#9726	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₀₇	—	4	1	no	—
131	C₂×C₁₀₇	—	16	2	no	—
132	C₂×C₁₀₇	#1975	1	1	yes	characteristic
133	C₂×C₁₀₇	#1978	1	1	yes	characteristic
134	C₂×C₁₀₇	#1980	1	1	yes	characteristic
135	#3169	C₂×C₁₀₇	1	1	yes	characteristic
136	C₄×#108	C₂×C₁₀₇	1	1	yes	characteristic, complemented
137	#1525	C₂×C₁₀₇	1	2	yes	complemented
138	#1484	C₂×C₁₀₇	1	2	yes	complemented
139	#1451	C₂×C₁₀₇	1	1	yes	characteristic, complemented
140	C₁₀₇	#9726	1	1	yes	characteristic, direct product factor, Sylow
141	#532	C₂²×C₁₀₇	1	2	yes	—
142	#500	C₄×C₁₀₇	1	1	yes	characteristic
143	#493	C₄×C₁₀₇	1	1	yes	characteristic
144	#486	C₄×C₁₀₇	1	2	yes	—
145	#466	—	2	1	no	—
146	#466	—	2	1	no	—
147	C₄×D₈	—	2	1	no	—
148	C₄×D₈	—	2	1	no	—
149	#431	C₂²×C₁₀₇	1	1	yes	characteristic
150	C₂×#108	—	2	1	no	—
151	C₄×M₁₆	C₂²×C₁₀₇	1	2	yes	—
152	C₄×M₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
154	#341	—	2	1	no	—
155	#329	—	2	4	no	—
156	D₄⋊C₈	—	2	4	no	—
157	#322	—	2	1	no	—
158	#322	—	2	1	no	—
159	C₂×#41	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
166	C₄⋊Q₈	C₂×C₄×C₁₀₇	1	2	yes	—
167	#127	C₂×C₄×C₁₀₇	1	2	yes	—
168	#124	—	2	2	no	—
169	#121	—	2	2	no	—
170	#121	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
174	#108	—	2	4	no	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	C₄⋊C₈	D₄×C₁₀₇	1	2	yes	—
178	#102	—	2	2	no	—
179	#102	—	2	2	no	—
180	C₂²⋊C₈	—	2	1	no	—
181	#97	—	2	2	no	—
182	#97	D₄×C₁₀₇	1	2	yes	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
188	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
189	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
190	#41	—	8	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
198	C₂²×C₄	#41×C₁₀₇	1	1	yes	characteristic
199	C₂²×C₄	#41×C₁₀₇	1	1	yes	characteristic
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	1	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	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	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	C₂×D₄×C₁₀₇	1	1	yes	characteristic
222	C₂³	—	4	2	no	—
223	C₂³	#124×C₁₀₇	1	1	yes	characteristic
224	Q₈	—	8	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₀₇	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₁₀₇	1	1	yes	characteristic
239	C₂×C₄	#121×C₁₀₇	1	1	yes	characteristic
240	C₂×C₄	#121×C₁₀₇	1	1	yes	characteristic
241	C₂×C₄	#127×C₁₀₇	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	—	4	4	no	—
247	C₈	—	4	2	no	—
248	C₈	#104×C₁₀₇	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	#390×C₁₀₇	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	2	no	—
260	C₄	C₂×#104×C₁₀₇	1	1	yes	characteristic
261	C₄	#421×C₁₀₇	1	1	yes	characteristic
262	C₄	#444×C₁₀₇	1	2	yes	—
263	C₄	#472×C₁₀₇	1	2	yes	—
264	C₂	—	4	1	no	—
265	C₂	—	16	2	no	—
266	C₂	C₁₀₇×#1975	1	1	yes	characteristic
267	C₂	C₁₀₇×#1978	1	1	yes	characteristic
268	C₂	C₁₀₇×#1980	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	#390
prime radical	C₁₀₇×#3169	C₂
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. C₄×#108×C₁₀₇	C₂	semidirect
3. C₁₀₇×#1525	C₂	semidirect
4. C₁₀₇×#1484	C₂	semidirect
5. C₁₀₇×#1451	C₂	semidirect
129. #9726	C₁₀₇	direct (N × H)
136. C₄×#108	C₂×C₁₀₇	semidirect
137. #1525	C₂×C₁₀₇	semidirect
138. #1484	C₂×C₁₀₇	semidirect
139. #1451	C₂×C₁₀₇	semidirect
140. C₁₀₇	#9726	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₇×#3169	2
C₄×#108×C₁₀₇	2
C₁₀₇×#1525	2
C₁₀₇×#1484	2
C₁₀₇×#1451	2
#9726	107

Subgroups — C107×#9726 #41035690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#9726 #41035690