Subgroups — C₂×C₁₀₇×#2309 #41060005

Order 27392.

7710 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	C₂⁴×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₁₀₇	C₂	1	2	yes	maximal, complemented
3	C₁₀₇×#2309	C₂	1	16	yes	maximal, direct product factor
4	C₂×#390×C₁₀₇	C₂	1	8	yes	maximal, complemented
5	C₂×#379×C₁₀₇	C₂	1	4	yes	maximal, complemented
6	C₂⁶×C₁₀₇	C₂²	1	2	yes	—
7	C₂³×D₄×C₁₀₇	—	2	16	no	—
8	C₂³×D₄×C₁₀₇	C₄	1	16	yes	complemented
9	C₂³×D₄×C₁₀₇	C₂²	1	4	yes	—
10	C₂³×D₄×C₁₀₇	C₂²	1	8	yes	—
11	C₂⁴×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₀₇	—	2	16	no	—
13	C₂×#117×C₁₀₇	C₂²	1	16	yes	complemented
14	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	4	yes	complemented
15	C₂²×C₂²⋊C₄×C₁₀₇	C₂²	1	4	yes	—
16	C₂²×C₂²⋊C₄×C₁₀₇	C₂²	1	16	yes	complemented
17	C₂²×C₄²×C₁₀₇	C₂²	1	4	yes	complemented
18	#390×C₁₀₇	C₂²	1	64	yes	complemented
19	#379×C₁₀₇	C₂²	1	32	yes	complemented
20	C₂⁵×C₁₀₇	—	2	16	no	—
21	C₂⁵×C₁₀₇	—	4	16	no	—
22	C₂⁵×C₁₀₇	C₂×C₄	1	16	yes	—
23	C₂⁵×C₁₀₇	C₂³	1	4	yes	—
24	C₂⁵×C₁₀₇	C₂³	1	1	yes	characteristic
25	C₂⁵×C₁₀₇	C₂³	1	8	yes	—
26	C₂²×D₄×C₁₀₇	—	2	128	no	—
27	C₂²×D₄×C₁₀₇	—	2	16	no	—
28	C₂²×D₄×C₁₀₇	—	2	16	no	—
29	C₂²×D₄×C₁₀₇	—	2	64	no	—
30	C₂²×D₄×C₁₀₇	C₂×C₄	1	32	yes	—
31	C₂²×D₄×C₁₀₇	C₂×C₄	1	64	yes	complemented
32	C₂²×D₄×C₁₀₇	C₂³	1	16	yes	—
33	C₂³×C₄×C₁₀₇	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₁₀₇	D₄	1	16	yes	—
35	C₂³×C₄×C₁₀₇	C₂³	1	8	yes	—
36	C₂³×C₄×C₁₀₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₁₀₇	C₂³	1	2	yes	—
38	#117×C₁₀₇	—	2	64	no	—
39	C₂×C₄⋊C₄×C₁₀₇	—	2	16	no	—
40	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	16	yes	complemented
41	C₂×C₂²⋊C₄×C₁₀₇	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₁₀₇	—	4	32	no	—
43	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	64	yes	complemented
45	C₂×C₄²×C₁₀₇	—	2	16	no	—
46	C₂×C₄²×C₁₀₇	C₂³	1	16	yes	complemented
47	C₂⁴×C₁₀₇	—	2	8	no	—
48	C₂⁴×C₁₀₇	—	2	64	no	—
49	C₂⁴×C₁₀₇	—	2	32	no	—
50	C₂⁴×C₁₀₇	—	2	16	no	—
51	C₂⁴×C₁₀₇	—	2	32	no	—
52	C₂⁴×C₁₀₇	—	4	64	no	—
53	C₂⁴×C₁₀₇	—	4	32	no	—
54	C₂⁴×C₁₀₇	—	4	128	no	—
55	C₂⁴×C₁₀₇	C₂²×C₄	1	16	yes	—
56	C₂⁴×C₁₀₇	C₂²×C₄	1	4	yes	—
57	C₂⁴×C₁₀₇	C₂²×C₄	1	32	yes	—
58	C₂⁴×C₁₀₇	C₂×D₄	1	8	yes	—
59	C₂⁴×C₁₀₇	C₂⁴	1	8	yes	—
60	C₂⁴×C₁₀₇	C₂⁴	1	2	yes	—
61	C₂⁴×C₁₀₇	C₂⁴	1	1	yes	characteristic
62	C₂×D₄×C₁₀₇	—	2	128	no	—
63	C₂×D₄×C₁₀₇	—	2	256	no	—
64	C₂×D₄×C₁₀₇	—	2	32	no	—
65	C₂×D₄×C₁₀₇	—	2	32	no	—
66	C₂×D₄×C₁₀₇	C₂²×C₄	1	64	yes	—
67	C₂²×C₄×C₁₀₇	—	2	16	no	—
68	C₂²×C₄×C₁₀₇	—	2	32	no	—
69	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	64	yes	complemented
70	C₂²×C₄×C₁₀₇	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₁₀₇	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₁₀₇	C₂×D₄	1	8	yes	—
73	C₂²×C₄×C₁₀₇	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₁₀₇	C₂×D₄	1	32	yes	—
75	C₂²×C₄×C₁₀₇	C₂⁴	1	16	yes	—
76	C₂²×C₄×C₁₀₇	C₂⁴	1	4	yes	—
77	C₄⋊C₄×C₁₀₇	—	2	32	no	—
78	C₂²⋊C₄×C₁₀₇	—	2	32	no	—
79	C₂²⋊C₄×C₁₀₇	—	4	64	no	—
80	C₄²×C₁₀₇	—	2	32	no	—
81	C₂³×C₁₀₇	—	2	16	no	—
82	C₂³×C₁₀₇	—	2	32	no	—
83	C₂³×C₁₀₇	—	2	8	no	—
84	C₂³×C₁₀₇	—	2	64	no	—
85	C₂³×C₁₀₇	—	2	32	no	—
86	C₂³×C₁₀₇	—	2	64	no	—
87	C₂³×C₁₀₇	—	2	16	no	—
88	C₂³×C₁₀₇	—	2	32	no	—
89	C₂³×C₁₀₇	—	4	64	no	—
90	C₂³×C₁₀₇	—	4	256	no	—
91	C₂³×C₁₀₇	—	4	128	no	—
92	C₂³×C₁₀₇	—	4	64	no	—
93	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	16	yes	—
94	C₂³×C₁₀₇	C₂³×C₄	1	16	yes	—
95	C₂³×C₁₀₇	C₂³×C₄	1	4	yes	—
96	C₂³×C₁₀₇	C₂³×C₄	1	2	yes	—
97	C₂³×C₁₀₇	C₂²×D₄	1	8	yes	—
98	C₂³×C₁₀₇	C₂²×D₄	1	4	yes	—
99	C₂³×C₁₀₇	H(5,𝔽₂)	1	8	yes	—
100	C₂³×C₁₀₇	C₂⁵	1	1	yes	characteristic
101	D₄×C₁₀₇	—	2	128	no	—
102	C₂×C₄×C₁₀₇	—	2	16	no	—
103	C₂×C₄×C₁₀₇	—	2	32	no	—
104	C₂×C₄×C₁₀₇	—	4	32	no	—
105	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	32	yes	—
106	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	64	yes	complemented
107	C₂×C₄×C₁₀₇	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₁₀₇	C₂²×D₄	1	8	yes	—
109	C₂×C₄×C₁₀₇	C₂²×D₄	1	8	yes	—
110	C₂²×C₁₀₇	—	2	32	no	—
111	C₂²×C₁₀₇	—	2	8	no	—
112	C₂²×C₁₀₇	—	2	16	no	—
113	C₂²×C₁₀₇	—	2	16	no	—
114	C₂²×C₁₀₇	—	2	32	no	—
115	C₂²×C₁₀₇	—	4	128	no	—
116	C₂²×C₁₀₇	—	4	64	no	—
117	C₂²×C₁₀₇	—	4	32	no	—
118	C₂²×C₁₀₇	C₂²×C₂²⋊C₄	1	8	yes	—
119	C₂²×C₁₀₇	C₂²×C₂²⋊C₄	1	4	yes	—
120	C₂²×C₁₀₇	#518	1	8	yes	—
121	C₂²×C₁₀₇	#535	1	16	yes	—
122	C₂²×C₁₀₇	C₂⁴×C₄	1	1	yes	characteristic
123	C₂²×C₁₀₇	C₂³×D₄	1	2	yes	—
124	C₂²×C₁₀₇	C₂×H(5,𝔽₂)	1	4	yes	—
125	C₄×C₁₀₇	—	4	16	no	—
126	C₄×C₁₀₇	C₂²×C₂²⋊C₄	1	16	yes	—
127	C₂×#2309	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₁₀₇	—	2	8	no	—
129	C₂×C₁₀₇	—	4	16	no	—
130	C₂×C₁₀₇	#2309	1	8	yes	direct product factor
131	C₂×C₁₀₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₁₀₇	C₂×#518	1	2	yes	—
133	C₂×C₁₀₇	C₂×#535	1	4	yes	—
134	C₂⁴×D₄	C₂×C₁₀₇	1	1	yes	characteristic
135	C₂²×#117	C₂×C₁₀₇	1	2	yes	complemented
136	#2309	C₂×C₁₀₇	1	16	yes	direct product factor
137	C₂×#390	C₂×C₁₀₇	1	8	yes	complemented
138	C₂×#379	C₂×C₁₀₇	1	4	yes	complemented
139	C₁₀₇	C₂×#2309	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁶	C₂²×C₁₀₇	1	2	yes	—
141	C₂³×D₄	—	2	16	no	—
142	C₂³×D₄	C₄×C₁₀₇	1	16	yes	complemented
143	C₂³×D₄	C₂²×C₁₀₇	1	4	yes	—
144	C₂³×D₄	C₂²×C₁₀₇	1	8	yes	—
145	C₂⁴×C₄	C₂²×C₁₀₇	1	1	yes	characteristic
146	C₂×#117	—	2	16	no	—
147	C₂×#117	C₂²×C₁₀₇	1	16	yes	complemented
148	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	4	yes	complemented
149	C₂²×C₂²⋊C₄	C₂²×C₁₀₇	1	4	yes	—
150	C₂²×C₂²⋊C₄	C₂²×C₁₀₇	1	16	yes	complemented
151	C₂²×C₄²	C₂²×C₁₀₇	1	4	yes	complemented
152	#390	C₂²×C₁₀₇	1	64	yes	complemented
153	#379	C₂²×C₁₀₇	1	32	yes	complemented
154	C₂⁵	—	2	16	no	—
155	C₂⁵	—	4	16	no	—
156	C₂⁵	C₂×C₄×C₁₀₇	1	16	yes	—
157	C₂⁵	C₂³×C₁₀₇	1	4	yes	—
158	C₂⁵	C₂³×C₁₀₇	1	1	yes	characteristic
159	C₂⁵	C₂³×C₁₀₇	1	8	yes	—
160	C₂²×D₄	—	2	128	no	—
161	C₂²×D₄	—	2	16	no	—
162	C₂²×D₄	—	2	16	no	—
163	C₂²×D₄	—	2	64	no	—
164	C₂²×D₄	C₂×C₄×C₁₀₇	1	32	yes	—
165	C₂²×D₄	C₂×C₄×C₁₀₇	1	64	yes	complemented
166	C₂²×D₄	C₂³×C₁₀₇	1	16	yes	—
167	C₂³×C₄	C₂×C₄×C₁₀₇	1	8	yes	—
168	C₂³×C₄	D₄×C₁₀₇	1	16	yes	—
169	C₂³×C₄	C₂³×C₁₀₇	1	8	yes	—
170	C₂³×C₄	C₂³×C₁₀₇	1	2	yes	—
171	C₂³×C₄	C₂³×C₁₀₇	1	4	yes	—
172	#117	—	2	64	no	—
173	C₂×C₄⋊C₄	—	2	16	no	—
174	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	16	yes	complemented
175	C₂×C₂²⋊C₄	—	2	16	no	—
176	C₂×C₂²⋊C₄	—	4	32	no	—
177	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	16	yes	—
178	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	64	yes	complemented
179	C₂×C₄²	—	2	16	no	—
180	C₂×C₄²	C₂³×C₁₀₇	1	16	yes	complemented
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	64	no	—
183	C₂⁴	—	2	32	no	—
184	C₂⁴	—	2	16	no	—
185	C₂⁴	—	2	32	no	—
186	C₂⁴	—	4	64	no	—
187	C₂⁴	—	4	32	no	—
188	C₂⁴	—	4	128	no	—
189	C₂⁴	C₂²×C₄×C₁₀₇	1	16	yes	—
190	C₂⁴	C₂²×C₄×C₁₀₇	1	4	yes	—
191	C₂⁴	C₂²×C₄×C₁₀₇	1	32	yes	—
192	C₂⁴	C₂×D₄×C₁₀₇	1	8	yes	—
193	C₂⁴	C₂⁴×C₁₀₇	1	8	yes	—
194	C₂⁴	C₂⁴×C₁₀₇	1	2	yes	—
195	C₂⁴	C₂⁴×C₁₀₇	1	1	yes	characteristic
196	C₂×D₄	—	2	128	no	—
197	C₂×D₄	—	2	256	no	—
198	C₂×D₄	—	2	32	no	—
199	C₂×D₄	—	2	32	no	—
200	C₂×D₄	C₂²×C₄×C₁₀₇	1	64	yes	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	64	yes	complemented
204	C₂²×C₄	C₂²×C₄×C₁₀₇	1	8	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₀₇	1	16	yes	—
206	C₂²×C₄	C₂×D₄×C₁₀₇	1	8	yes	—
207	C₂²×C₄	C₂×D₄×C₁₀₇	1	8	yes	—
208	C₂²×C₄	C₂×D₄×C₁₀₇	1	32	yes	—
209	C₂²×C₄	C₂⁴×C₁₀₇	1	16	yes	—
210	C₂²×C₄	C₂⁴×C₁₀₇	1	4	yes	—
211	C₄⋊C₄	—	2	32	no	—
212	C₂²⋊C₄	—	2	32	no	—
213	C₂²⋊C₄	—	4	64	no	—
214	C₄²	—	2	32	no	—
215	C₂³	—	2	16	no	—
216	C₂³	—	2	32	no	—
217	C₂³	—	2	8	no	—
218	C₂³	—	2	64	no	—
219	C₂³	—	2	32	no	—
220	C₂³	—	2	64	no	—
221	C₂³	—	2	16	no	—
222	C₂³	—	2	32	no	—
223	C₂³	—	4	64	no	—
224	C₂³	—	4	128	no	—
225	C₂³	—	4	256	no	—
226	C₂³	—	4	64	no	—
227	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	16	yes	—
228	C₂³	C₂³×C₄×C₁₀₇	1	16	yes	—
229	C₂³	C₂³×C₄×C₁₀₇	1	4	yes	—
230	C₂³	C₂³×C₄×C₁₀₇	1	2	yes	—
231	C₂³	C₂²×D₄×C₁₀₇	1	8	yes	—
232	C₂³	C₂²×D₄×C₁₀₇	1	4	yes	—
233	C₂³	H(5,𝔽₂)×C₁₀₇	1	8	yes	—
234	C₂³	C₂⁵×C₁₀₇	1	1	yes	characteristic
235	D₄	—	2	128	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	32	no	—
238	C₂×C₄	—	4	32	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	32	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	64	yes	complemented
241	C₂×C₄	C₂³×C₄×C₁₀₇	1	8	yes	—
242	C₂×C₄	C₂²×D₄×C₁₀₇	1	8	yes	—
243	C₂×C₄	C₂²×D₄×C₁₀₇	1	8	yes	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	16	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	16	no	—
248	C₂²	—	2	32	no	—
249	C₂²	—	4	128	no	—
250	C₂²	—	4	32	no	—
251	C₂²	—	4	64	no	—
252	C₂²	C₂²×C₂²⋊C₄×C₁₀₇	1	8	yes	—
253	C₂²	C₂²×C₂²⋊C₄×C₁₀₇	1	4	yes	—
254	C₂²	#518×C₁₀₇	1	8	yes	—
255	C₂²	#535×C₁₀₇	1	16	yes	—
256	C₂²	C₂⁴×C₄×C₁₀₇	1	1	yes	characteristic
257	C₂²	C₂³×D₄×C₁₀₇	1	2	yes	—
258	C₂²	C₂×H(5,𝔽₂)×C₁₀₇	1	4	yes	—
259	C₄	—	4	16	no	—
260	C₄	C₂²×C₂²⋊C₄×C₁₀₇	1	16	yes	—
261	C₂	—	2	8	no	—
262	C₂	—	4	16	no	—
263	C₂	C₁₀₇×#2309	1	8	yes	direct product factor
264	C₂	C₂³×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
265	C₂	C₂×#518×C₁₀₇	1	2	yes	—
266	C₂	C₂×#535×C₁₀₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₀₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₀₇
Frattini, cosocle	C₂³	C₂⁵×C₁₀₇
prime radical	C₂⁴×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. C₂²×#117×C₁₀₇	C₂	semidirect
3. C₁₀₇×#2309	C₂	direct (N × H)
4. C₂×#390×C₁₀₇	C₂	semidirect
5. C₂×#379×C₁₀₇	C₂	semidirect
8. C₂³×D₄×C₁₀₇	C₄	semidirect
13. C₂×#117×C₁₀₇	C₂²	semidirect
14. C₂²×C₄⋊C₄×C₁₀₇	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₁₀₇	C₂²	semidirect
17. C₂²×C₄²×C₁₀₇	C₂²	semidirect
18. #390×C₁₀₇	C₂²	semidirect
19. #379×C₁₀₇	C₂²	semidirect
31. C₂²×D₄×C₁₀₇	C₂×C₄	semidirect
40. C₂×C₄⋊C₄×C₁₀₇	C₂³	semidirect
44. C₂×C₂²⋊C₄×C₁₀₇	C₂³	semidirect
46. C₂×C₄²×C₁₀₇	C₂³	semidirect
69. C₂²×C₄×C₁₀₇	C₂²⋊C₄	semidirect
106. C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	semidirect
127. C₂×#2309	C₁₀₇	direct (N × H)
130. C₂×C₁₀₇	#2309	direct (N × H)
135. C₂²×#117	C₂×C₁₀₇	semidirect
136. #2309	C₂×C₁₀₇	direct (N × H)
137. C₂×#390	C₂×C₁₀₇	semidirect
138. C₂×#379	C₂×C₁₀₇	semidirect
139. C₁₀₇	C₂×#2309	direct (N × H)
142. C₂³×D₄	C₄×C₁₀₇	semidirect
147. C₂×#117	C₂²×C₁₀₇	semidirect
148. C₂²×C₄⋊C₄	C₂²×C₁₀₇	semidirect
150. C₂²×C₂²⋊C₄	C₂²×C₁₀₇	semidirect
151. C₂²×C₄²	C₂²×C₁₀₇	semidirect
152. #390	C₂²×C₁₀₇	semidirect
153. #379	C₂²×C₁₀₇	semidirect
165. C₂²×D₄	C₂×C₄×C₁₀₇	semidirect
174. C₂×C₄⋊C₄	C₂³×C₁₀₇	semidirect
178. C₂×C₂²⋊C₄	C₂³×C₁₀₇	semidirect
180. C₂×C₄²	C₂³×C₁₀₇	semidirect
203. C₂²×C₄	C₂²⋊C₄×C₁₀₇	semidirect
240. C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	semidirect
263. C₂	C₁₀₇×#2309	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₁₀₇	2
C₂²×#117×C₁₀₇	2
C₁₀₇×#2309	2
C₂×#390×C₁₀₇	2
C₂×#379×C₁₀₇	2
C₂×#2309	107

Subgroups — C2×C107×#2309 #41060005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#2309 #41060005