Subgroups — C₂×#386×C₁₀₇ #28332970

Order 13696.

1576 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂²⋊C₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊C₄×C₁₀₇	C₂	1	2	yes	maximal, complemented
5	#386×C₁₀₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#95×C₁₀₇	C₂	1	2	yes	maximal, complemented
7	C₂⁵×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₂⁵×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₁₀₇	—	2	4	no	—
10	C₂²×D₄×C₁₀₇	—	2	4	no	—
11	C₂²×D₄×C₁₀₇	C₄	1	8	yes	complemented
12	C₂²×D₄×C₁₀₇	C₂²	1	4	yes	—
13	C₂³×C₄×C₁₀₇	—	2	4	no	—
14	C₂³×C₄×C₁₀₇	—	2	8	no	—
15	C₂³×C₄×C₁₀₇	C₂²	1	2	yes	—
16	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₇	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₀₇	C₂²	1	4	yes	complemented
19	C₂×C₂²⋊C₄×C₁₀₇	—	2	4	no	—
20	C₂×C₂²⋊C₄×C₁₀₇	—	2	8	no	—
21	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	4	yes	complemented
22	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	8	yes	complemented
23	#95×C₁₀₇	C₂²	1	8	yes	complemented
24	C₂⁴×C₁₀₇	—	2	4	no	—
25	C₂⁴×C₁₀₇	—	2	4	no	—
26	C₂⁴×C₁₀₇	—	2	4	no	—
27	C₂⁴×C₁₀₇	—	2	2	no	—
28	C₂⁴×C₁₀₇	—	2	2	no	—
29	C₂⁴×C₁₀₇	—	2	4	no	—
30	C₂⁴×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₂⁴×C₁₀₇	C₂×C₄	1	4	yes	complemented
32	C₂⁴×C₁₀₇	D₄	1	4	yes	—
33	C₂⁴×C₁₀₇	D₄	1	4	yes	—
34	C₂⁴×C₁₀₇	C₂³	1	2	yes	—
35	C₂⁴×C₁₀₇	C₂³	1	2	yes	—
36	C₂⁴×C₁₀₇	C₂³	1	1	yes	characteristic
37	C₂×D₄×C₁₀₇	—	2	32	no	—
38	C₂×D₄×C₁₀₇	—	2	8	no	—
39	C₂×D₄×C₁₀₇	—	2	8	no	—
40	C₂×D₄×C₁₀₇	C₂×C₄	1	16	yes	complemented
41	C₂²×C₄×C₁₀₇	—	2	32	no	—
42	C₂²×C₄×C₁₀₇	—	2	4	no	—
43	C₂²×C₄×C₁₀₇	—	2	4	no	—
44	C₂²×C₄×C₁₀₇	—	2	8	no	—
45	C₂²×C₄×C₁₀₇	—	2	2	no	—
46	C₂²×C₄×C₁₀₇	—	2	2	no	—
47	C₂²×C₄×C₁₀₇	—	2	16	no	—
48	C₂²×C₄×C₁₀₇	—	2	4	no	—
49	C₂²×C₄×C₁₀₇	—	2	4	no	—
50	C₂²×C₄×C₁₀₇	—	2	4	no	—
51	C₂²×C₄×C₁₀₇	C₂×C₄	1	4	yes	complemented
52	C₂²×C₄×C₁₀₇	D₄	1	8	yes	complemented
53	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
56	C₂²⋊C₄×C₁₀₇	—	2	8	no	—
57	C₂²⋊C₄×C₁₀₇	—	2	16	no	—
58	C₂³×C₁₀₇	—	2	16	no	—
59	C₂³×C₁₀₇	—	2	8	no	—
60	C₂³×C₁₀₇	—	2	16	no	—
61	C₂³×C₁₀₇	—	2	8	no	—
62	C₂³×C₁₀₇	—	2	4	no	—
63	C₂³×C₁₀₇	—	2	4	no	—
64	C₂³×C₁₀₇	—	2	8	no	—
65	C₂³×C₁₀₇	—	2	16	no	—
66	C₂³×C₁₀₇	—	2	4	no	—
67	C₂³×C₁₀₇	—	2	2	no	—
68	C₂³×C₁₀₇	—	2	2	no	—
69	C₂³×C₁₀₇	—	2	4	no	—
70	C₂³×C₁₀₇	—	4	4	no	—
71	C₂³×C₁₀₇	—	4	8	no	—
72	C₂³×C₁₀₇	—	4	4	no	—
73	C₂³×C₁₀₇	C₂²⋊C₄	1	16	yes	complemented
74	C₂³×C₁₀₇	C₂²×C₄	1	4	yes	—
75	C₂³×C₁₀₇	C₂²×C₄	1	2	yes	—
76	C₂³×C₁₀₇	C₂²×C₄	1	4	yes	complemented
77	C₂³×C₁₀₇	C₂×D₄	1	4	yes	—
78	C₂³×C₁₀₇	C₂×D₄	1	4	yes	—
79	C₂³×C₁₀₇	C₂×D₄	1	2	yes	—
80	C₂³×C₁₀₇	C₂×D₄	1	2	yes	—
81	C₂³×C₁₀₇	C₂×D₄	1	4	yes	—
82	C₂³×C₁₀₇	#41	1	4	yes	—
83	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
84	D₄×C₁₀₇	—	2	32	no	—
85	C₂×C₄×C₁₀₇	—	2	32	no	—
86	C₂×C₄×C₁₀₇	—	2	8	no	—
87	C₂×C₄×C₁₀₇	—	2	16	no	—
88	C₂×C₄×C₁₀₇	—	2	4	no	—
89	C₂×C₄×C₁₀₇	—	2	4	no	—
90	C₂×C₄×C₁₀₇	—	2	2	no	—
91	C₂×C₄×C₁₀₇	—	2	2	no	—
92	C₂×C₄×C₁₀₇	—	2	4	no	—
93	C₂×C₄×C₁₀₇	—	2	4	no	—
94	C₂×C₄×C₁₀₇	—	2	4	no	—
95	C₂×C₄×C₁₀₇	—	4	8	no	—
96	C₂×C₄×C₁₀₇	C₂²×C₄	1	4	yes	complemented
97	C₂×C₄×C₁₀₇	C₂×D₄	1	8	yes	complemented
98	C₂²×C₁₀₇	—	2	8	no	—
99	C₂²×C₁₀₇	—	2	4	no	—
100	C₂²×C₁₀₇	—	2	8	no	—
101	C₂²×C₁₀₇	—	2	4	no	—
102	C₂²×C₁₀₇	—	2	32	no	—
103	C₂²×C₁₀₇	—	2	4	no	—
104	C₂²×C₁₀₇	—	2	8	no	—
105	C₂²×C₁₀₇	—	4	2	no	—
106	C₂²×C₁₀₇	—	4	4	no	—
107	C₂²×C₁₀₇	—	4	2	no	—
108	C₂²×C₁₀₇	—	4	16	no	—
109	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	8	yes	complemented
111	C₂²×C₁₀₇	C₄×D₄	1	8	yes	—
112	C₂²×C₁₀₇	#120	1	4	yes	—
113	C₂²×C₁₀₇	#121	1	8	yes	—
114	C₂²×C₁₀₇	#123	1	4	yes	—
115	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₁₀₇	C₂²×D₄	1	2	yes	—
119	C₂²×C₁₀₇	C₂×#41	1	2	yes	—
120	C₄×C₁₀₇	—	2	4	no	—
121	C₄×C₁₀₇	—	2	8	no	—
122	C₄×C₁₀₇	—	4	4	no	—
123	C₂×C₁₀₇	—	2	8	no	—
124	C₂×C₁₀₇	—	4	4	no	—
125	C₂×C₁₀₇	#386	1	8	yes	direct product factor
126	C₂×C₁₀₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₁₀₇	C₂×C₄×D₄	1	2	yes	—
128	C₂×C₁₀₇	C₂×#120	1	1	yes	characteristic
129	C₂×C₁₀₇	C₂×#121	1	2	yes	—
130	C₂×C₁₀₇	C₂×#123	1	1	yes	characteristic
131	C₂×#386	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₀₇	C₂×#386	1	1	yes	characteristic, direct product factor, Sylow
133	C₂³×D₄	C₂×C₁₀₇	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×C₁₀₇	1	1	yes	characteristic, complemented
135	C₂²×C₂²⋊C₄	C₂×C₁₀₇	1	1	yes	characteristic, complemented
136	C₂²×C₂²⋊C₄	C₂×C₁₀₇	1	2	yes	complemented
137	#386	C₂×C₁₀₇	1	8	yes	direct product factor
138	C₂×#95	C₂×C₁₀₇	1	2	yes	complemented
139	C₂⁵	C₂²×C₁₀₇	1	1	yes	characteristic
140	C₂⁵	C₂²×C₁₀₇	1	1	yes	characteristic
141	C₂²×D₄	—	2	4	no	—
142	C₂²×D₄	—	2	4	no	—
143	C₂²×D₄	C₄×C₁₀₇	1	8	yes	complemented
144	C₂²×D₄	C₂²×C₁₀₇	1	4	yes	—
145	C₂³×C₄	—	2	4	no	—
146	C₂³×C₄	—	2	8	no	—
147	C₂³×C₄	C₂²×C₁₀₇	1	2	yes	—
148	C₂³×C₄	C₂²×C₁₀₇	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₁₀₇	1	2	yes	—
150	C₂³×C₄	C₂²×C₁₀₇	1	4	yes	complemented
151	C₂×C₂²⋊C₄	—	2	4	no	—
152	C₂×C₂²⋊C₄	—	2	8	no	—
153	C₂×C₂²⋊C₄	C₂²×C₁₀₇	1	4	yes	complemented
154	C₂×C₂²⋊C₄	C₂²×C₁₀₇	1	8	yes	complemented
155	#95	C₂²×C₁₀₇	1	8	yes	complemented
156	C₂⁴	—	2	4	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	—	2	4	no	—
159	C₂⁴	—	2	2	no	—
160	C₂⁴	—	2	2	no	—
161	C₂⁴	—	2	4	no	—
162	C₂⁴	C₂×C₄×C₁₀₇	1	4	yes	—
163	C₂⁴	C₂×C₄×C₁₀₇	1	4	yes	complemented
164	C₂⁴	D₄×C₁₀₇	1	4	yes	—
165	C₂⁴	D₄×C₁₀₇	1	4	yes	—
166	C₂⁴	C₂³×C₁₀₇	1	2	yes	—
167	C₂⁴	C₂³×C₁₀₇	1	2	yes	—
168	C₂⁴	C₂³×C₁₀₇	1	1	yes	characteristic
169	C₂×D₄	—	2	32	no	—
170	C₂×D₄	—	2	8	no	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	C₂×C₄×C₁₀₇	1	16	yes	complemented
173	C₂²×C₄	—	2	32	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	C₂×C₄×C₁₀₇	1	4	yes	complemented
184	C₂²×C₄	D₄×C₁₀₇	1	8	yes	complemented
185	C₂²×C₄	C₂³×C₁₀₇	1	4	yes	—
186	C₂²×C₄	C₂³×C₁₀₇	1	2	yes	—
187	C₂²×C₄	C₂³×C₁₀₇	1	4	yes	—
188	C₂²⋊C₄	—	2	8	no	—
189	C₂²⋊C₄	—	2	16	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	8	no	—
192	C₂³	—	2	16	no	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	8	no	—
197	C₂³	—	2	16	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	8	no	—
204	C₂³	—	4	4	no	—
205	C₂³	C₂²⋊C₄×C₁₀₇	1	16	yes	complemented
206	C₂³	C₂²×C₄×C₁₀₇	1	4	yes	—
207	C₂³	C₂²×C₄×C₁₀₇	1	2	yes	—
208	C₂³	C₂²×C₄×C₁₀₇	1	4	yes	complemented
209	C₂³	C₂×D₄×C₁₀₇	1	4	yes	—
210	C₂³	C₂×D₄×C₁₀₇	1	4	yes	—
211	C₂³	C₂×D₄×C₁₀₇	1	2	yes	—
212	C₂³	C₂×D₄×C₁₀₇	1	2	yes	—
213	C₂³	C₂×D₄×C₁₀₇	1	4	yes	—
214	C₂³	#41×C₁₀₇	1	4	yes	—
215	C₂³	C₂⁴×C₁₀₇	1	1	yes	characteristic
216	D₄	—	2	32	no	—
217	C₂×C₄	—	2	32	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	C₂²×C₄×C₁₀₇	1	4	yes	complemented
229	C₂×C₄	C₂×D₄×C₁₀₇	1	8	yes	complemented
230	C₂²	—	2	32	no	—
231	C₂²	—	2	8	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	4	16	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	C₂×C₂²⋊C₄×C₁₀₇	1	4	yes	—
242	C₂²	C₂×C₂²⋊C₄×C₁₀₇	1	8	yes	complemented
243	C₂²	C₄×D₄×C₁₀₇	1	8	yes	—
244	C₂²	#120×C₁₀₇	1	4	yes	—
245	C₂²	#121×C₁₀₇	1	8	yes	—
246	C₂²	#123×C₁₀₇	1	4	yes	—
247	C₂²	C₂³×C₄×C₁₀₇	1	1	yes	characteristic
248	C₂²	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
249	C₂²	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
250	C₂²	C₂²×D₄×C₁₀₇	1	2	yes	—
251	C₂²	C₂×#41×C₁₀₇	1	2	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	8	no	—
254	C₄	—	4	4	no	—
255	C₂	—	2	8	no	—
256	C₂	—	4	4	no	—
257	C₂	#386×C₁₀₇	1	8	yes	direct product factor
258	C₂	C₂²×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
259	C₂	C₂×C₄×D₄×C₁₀₇	1	2	yes	—
260	C₂	C₂×#120×C₁₀₇	1	1	yes	characteristic
261	C₂	C₂×#121×C₁₀₇	1	2	yes	—
262	C₂	C₂×#123×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₀₇	C₂³
commutator	C₂²	C₂³×C₄×C₁₀₇
Frattini, cosocle	C₂³	C₂⁴×C₁₀₇
prime radical	C₂³×D₄×C₁₀₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×C₁₀₇	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₁₀₇	C₂	semidirect
4. C₂²×C₂²⋊C₄×C₁₀₇	C₂	semidirect
5. #386×C₁₀₇	C₂	direct (N × H)
6. C₂×#95×C₁₀₇	C₂	semidirect
11. C₂²×D₄×C₁₀₇	C₄	semidirect
18. C₂³×C₄×C₁₀₇	C₂²	semidirect
21. C₂×C₂²⋊C₄×C₁₀₇	C₂²	semidirect
22. C₂×C₂²⋊C₄×C₁₀₇	C₂²	semidirect
23. #95×C₁₀₇	C₂²	semidirect
31. C₂⁴×C₁₀₇	C₂×C₄	semidirect
40. C₂×D₄×C₁₀₇	C₂×C₄	semidirect
51. C₂²×C₄×C₁₀₇	C₂×C₄	semidirect
52. C₂²×C₄×C₁₀₇	D₄	semidirect
73. C₂³×C₁₀₇	C₂²⋊C₄	semidirect
76. C₂³×C₁₀₇	C₂²×C₄	semidirect
96. C₂×C₄×C₁₀₇	C₂²×C₄	semidirect
97. C₂×C₄×C₁₀₇	C₂×D₄	semidirect
110. C₂²×C₁₀₇	C₂×C₂²⋊C₄	semidirect
125. C₂×C₁₀₇	#386	direct (N × H)
131. C₂×#386	C₁₀₇	direct (N × H)
132. C₁₀₇	C₂×#386	direct (N × H)
134. C₂⁴×C₄	C₂×C₁₀₇	semidirect
135. C₂²×C₂²⋊C₄	C₂×C₁₀₇	semidirect
136. C₂²×C₂²⋊C₄	C₂×C₁₀₇	semidirect
137. #386	C₂×C₁₀₇	direct (N × H)
138. C₂×#95	C₂×C₁₀₇	semidirect
143. C₂²×D₄	C₄×C₁₀₇	semidirect
150. C₂³×C₄	C₂²×C₁₀₇	semidirect
153. C₂×C₂²⋊C₄	C₂²×C₁₀₇	semidirect
154. C₂×C₂²⋊C₄	C₂²×C₁₀₇	semidirect
155. #95	C₂²×C₁₀₇	semidirect
163. C₂⁴	C₂×C₄×C₁₀₇	semidirect
172. C₂×D₄	C₂×C₄×C₁₀₇	semidirect
183. C₂²×C₄	C₂×C₄×C₁₀₇	semidirect
184. C₂²×C₄	D₄×C₁₀₇	semidirect
205. C₂³	C₂²⋊C₄×C₁₀₇	semidirect
208. C₂³	C₂²×C₄×C₁₀₇	semidirect
228. C₂×C₄	C₂²×C₄×C₁₀₇	semidirect
229. C₂×C₄	C₂×D₄×C₁₀₇	semidirect
242. C₂²	C₂×C₂²⋊C₄×C₁₀₇	semidirect
257. C₂	#386×C₁₀₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₁₀₇	2
C₂⁴×C₄×C₁₀₇	2
C₂²×C₂²⋊C₄×C₁₀₇	2
#386×C₁₀₇	2
C₂×#95×C₁₀₇	2
C₂×#386	107

Subgroups — C2×#386×C107 #28332970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#386×C₁₀₇ #28332970