Subgroups — C₁₀₇×#33829 #41059793

Order 27392.

488 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#3175	C₂	1	2	yes	maximal, complemented
2	C₁₀₇×#2226	C₂	1	2	yes	maximal, complemented
3	C₁₀₇×#2225	C₂	1	2	yes	maximal, complemented
4	C₁₀₇×#2219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₇×#2218	C₂	1	1	yes	maximal, characteristic
6	C₁₀₇×#2217	C₂	1	2	yes	maximal, complemented
7	C₁₀₇×#2211	C₂	1	2	yes	maximal, complemented
8	C₁₀₇×#2210	C₂	1	2	yes	maximal, complemented
9	C₁₀₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#542×C₁₀₇	—	2	2	no	—
11	C₂×Q₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₁₀₇	C₂²	1	2	yes	—
13	C₂×D₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
16	#500×C₁₀₇	C₂²	1	2	yes	—
17	#495×C₁₀₇	C₂²	1	2	yes	—
18	#489×C₁₀₇	—	2	2	no	—
19	#468×C₁₀₇	—	2	2	no	—
20	#467×C₁₀₇	C₂²	1	2	yes	—
21	#466×C₁₀₇	C₂²	1	2	yes	—
22	#440×C₁₀₇	—	2	2	no	—
23	C₄×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
24	C₄×D₈×C₁₀₇	C₂²	1	2	yes	—
25	#405×C₁₀₇	C₂²	1	1	yes	characteristic
26	#367×C₁₀₇	C₂²	1	2	yes	—
27	#367×C₁₀₇	C₂²	1	1	yes	characteristic
28	#367×C₁₀₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆×C₁₀₇	C₂²	1	2	yes	—
30	C₄⋊C₁₆×C₁₀₇	C₂²	1	2	yes	—
31	#358×C₁₀₇	C₂²	1	2	yes	—
32	#358×C₁₀₇	C₂²	1	2	yes	—
33	#357×C₁₀₇	C₂²	1	2	yes	—
34	#357×C₁₀₇	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₁₀₇	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₁₀₇	—	2	2	no	—
37	C₂×D₈×C₁₀₇	C₂³	1	2	yes	—
38	C₂×M₁₆×C₁₀₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
41	#126×C₁₀₇	—	4	2	no	—
42	#124×C₁₀₇	—	2	2	no	—
43	#124×C₁₀₇	—	4	2	no	—
44	#123×C₁₀₇	—	4	2	no	—
45	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
46	C₂²⋊Q₈×C₁₀₇	—	4	2	no	—
47	#121×C₁₀₇	—	2	2	no	—
48	C₄×Q₈×C₁₀₇	—	2	2	no	—
49	C₄×D₄×C₁₀₇	—	2	2	no	—
50	#117×C₁₀₇	D₄	1	1	yes	characteristic
51	Q₃₂×C₁₀₇	—	4	1	no	—
52	SD₃₂×C₁₀₇	—	4	2	no	—
53	D₁₆×C₁₀₇	—	4	1	no	—
54	M₃₂×C₁₀₇	—	4	1	no	—
55	C₂×C₁₆×C₁₀₇	—	4	1	no	—
56	C₂×C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₁₀₇	C₂³	1	2	yes	—
58	C₂×C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
59	#107×C₁₀₇	—	2	2	no	—
60	#107×C₁₀₇	C₂³	1	2	yes	—
61	#107×C₁₀₇	C₂³	1	2	yes	—
62	#103×C₁₀₇	—	2	2	no	—
63	#103×C₁₀₇	—	2	2	no	—
64	#102×C₁₀₇	—	2	2	no	—
65	#102×C₁₀₇	—	2	2	no	—
66	#97×C₁₀₇	D₄	1	2	yes	—
67	C₄×C₈×C₁₀₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₀₇	—	2	2	no	—
69	C₂×Q₈×C₁₀₇	—	4	2	no	—
70	C₂×D₄×C₁₀₇	—	2	2	no	—
71	C₂×D₄×C₁₀₇	—	4	2	no	—
72	C₂²×C₄×C₁₀₇	—	4	2	no	—
73	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₁₀₇	—	2	2	no	—
75	Q₁₆×C₁₀₇	—	2	2	no	—
76	SD₁₆×C₁₀₇	—	4	4	no	—
77	D₈×C₁₀₇	—	2	2	no	—
78	D₈×C₁₀₇	—	2	2	no	—
79	M₁₆×C₁₀₇	—	4	1	no	—
80	C₂×C₈×C₁₀₇	—	2	1	no	—
81	C₂×C₈×C₁₀₇	—	2	1	no	—
82	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₁₀₇	—	2	2	no	—
86	C₄⋊C₄×C₁₀₇	—	2	2	no	—
87	C₄⋊C₄×C₁₀₇	—	4	2	no	—
88	C₄⋊C₄×C₁₀₇	—	4	2	no	—
89	C₄⋊C₄×C₁₀₇	—	4	2	no	—
90	C₄⋊C₄×C₁₀₇	D₈	1	2	yes	—
91	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
92	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
93	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
94	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
95	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
96	C₂²⋊C₄×C₁₀₇	D₈	1	1	yes	characteristic
97	C₂²⋊C₄×C₁₀₇	D₈	1	1	yes	characteristic
98	C₄²×C₁₀₇	—	4	2	no	—
99	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
100	C₁₆×C₁₀₇	—	2	2	no	—
101	C₁₆×C₁₀₇	—	2	1	no	—
102	C₁₆×C₁₀₇	—	2	1	no	—
103	C₂³×C₁₀₇	—	4	2	no	—
104	C₂³×C₁₀₇	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₁₀₇	—	4	2	no	—
106	Q₈×C₁₀₇	—	4	2	no	—
107	Q₈×C₁₀₇	—	16	2	no	—
108	D₄×C₁₀₇	—	4	2	no	—
109	D₄×C₁₀₇	—	4	2	no	—
110	D₄×C₁₀₇	—	16	2	no	—
111	C₂×C₄×C₁₀₇	—	2	1	no	—
112	C₂×C₄×C₁₀₇	—	2	1	no	—
113	C₂×C₄×C₁₀₇	—	4	2	no	—
114	C₂×C₄×C₁₀₇	—	4	2	no	—
115	C₂×C₄×C₁₀₇	—	4	2	no	—
116	C₂×C₄×C₁₀₇	—	16	2	no	—
117	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₀₇	—	2	1	no	—
122	C₈×C₁₀₇	—	2	2	no	—
123	C₈×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₈×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂²×C₁₀₇	—	2	2	no	—
126	C₂²×C₁₀₇	—	2	1	no	—
127	C₂²×C₁₀₇	—	8	2	no	—
128	C₂²×C₁₀₇	—	8	2	no	—
129	C₂²×C₁₀₇	—	8	2	no	—
130	C₂²×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₁₀₇	—	2	2	no	—
132	C₄×C₁₀₇	—	2	2	no	—
133	C₄×C₁₀₇	—	2	1	no	—
134	C₄×C₁₀₇	—	8	2	no	—
135	C₄×C₁₀₇	—	8	2	no	—
136	C₄×C₁₀₇	—	8	2	no	—
137	C₄×C₁₀₇	#537	1	1	yes	characteristic
138	C₄×C₁₀₇	#575	1	1	yes	characteristic
139	#33829	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₁₀₇	—	4	1	no	—
141	C₂×C₁₀₇	—	16	2	no	—
142	C₂×C₁₀₇	#3245	1	1	yes	characteristic
143	C₂×C₁₀₇	#3416	1	2	yes	—
144	#3175	C₂×C₁₀₇	1	2	yes	complemented
145	#2226	C₂×C₁₀₇	1	2	yes	complemented
146	#2225	C₂×C₁₀₇	1	2	yes	complemented
147	#2219	C₂×C₁₀₇	1	1	yes	characteristic, complemented
148	#2218	C₂×C₁₀₇	1	1	yes	characteristic
149	#2217	C₂×C₁₀₇	1	2	yes	complemented
150	#2211	C₂×C₁₀₇	1	2	yes	complemented
151	#2210	C₂×C₁₀₇	1	2	yes	complemented
152	#2151	C₂×C₁₀₇	1	1	yes	characteristic, complemented
153	C₁₀₇	#33829	1	1	yes	characteristic, direct product factor, Sylow
154	#542	—	2	2	no	—
155	C₂×Q₃₂	C₂²×C₁₀₇	1	1	yes	characteristic
156	C₂×SD₃₂	C₂²×C₁₀₇	1	2	yes	—
157	C₂×D₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₀₇	1	1	yes	characteristic
159	C₂²×C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
160	#500	C₂²×C₁₀₇	1	2	yes	—
161	#495	C₂²×C₁₀₇	1	2	yes	—
162	#489	—	2	2	no	—
163	#468	—	2	2	no	—
164	#467	C₂²×C₁₀₇	1	2	yes	—
165	#466	C₂²×C₁₀₇	1	2	yes	—
166	#440	—	2	2	no	—
167	C₄×Q₁₆	C₂²×C₁₀₇	1	2	yes	—
168	C₄×D₈	C₂²×C₁₀₇	1	2	yes	—
169	#405	C₂²×C₁₀₇	1	1	yes	characteristic
170	#367	C₂²×C₁₀₇	1	2	yes	—
171	#367	C₂²×C₁₀₇	1	1	yes	characteristic
172	#367	C₂²×C₁₀₇	1	1	yes	characteristic
173	C₄⋊C₁₆	C₂²×C₁₀₇	1	2	yes	—
174	C₄⋊C₁₆	C₂²×C₁₀₇	1	2	yes	—
175	#358	C₂²×C₁₀₇	1	2	yes	—
176	#358	C₂²×C₁₀₇	1	2	yes	—
177	#357	C₂²×C₁₀₇	1	2	yes	—
178	#357	C₂²×C₁₀₇	1	2	yes	—
179	C₂×Q₁₆	C₂³×C₁₀₇	1	2	yes	—
180	C₂×SD₁₆	—	2	2	no	—
181	C₂×D₈	C₂³×C₁₀₇	1	2	yes	—
182	C₂×M₁₆	D₄×C₁₀₇	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	2	no	—
185	#126	—	4	2	no	—
186	#124	—	2	2	no	—
187	#124	—	4	2	no	—
188	#123	—	4	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₂²⋊Q₈	—	4	2	no	—
191	#121	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	#117	D₄×C₁₀₇	1	1	yes	characteristic
195	Q₃₂	—	4	1	no	—
196	SD₃₂	—	4	2	no	—
197	D₁₆	—	4	1	no	—
198	M₃₂	—	4	1	no	—
199	C₂×C₁₆	—	4	1	no	—
200	C₂×C₁₆	C₂³×C₁₀₇	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₁₀₇	1	2	yes	—
202	C₂×C₁₆	C₂³×C₁₀₇	1	1	yes	characteristic
203	#107	—	2	2	no	—
204	#107	C₂³×C₁₀₇	1	2	yes	—
205	#107	C₂³×C₁₀₇	1	2	yes	—
206	#103	—	2	2	no	—
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	#102	—	2	2	no	—
210	#97	D₄×C₁₀₇	1	2	yes	—
211	C₄×C₈	C₂³×C₁₀₇	1	2	yes	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	SD₁₆	—	4	4	no	—
221	D₈	—	2	2	no	—
222	D₈	—	2	2	no	—
223	M₁₆	—	4	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
228	C₂×C₈	C₂⁴×C₁₀₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	D₈×C₁₀₇	1	2	yes	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	D₈×C₁₀₇	1	1	yes	characteristic
241	C₂²⋊C₄	D₈×C₁₀₇	1	1	yes	characteristic
242	C₄²	—	4	2	no	—
243	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	2	1	no	—
246	C₁₆	—	2	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×D₈×C₁₀₇	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	16	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	16	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	16	2	no	—
261	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	H(5,𝔽₂)×C₁₀₇	1	1	yes	characteristic
268	C₈	2₋¹⁺⁴×C₁₀₇	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	C₂²×D₈×C₁₀₇	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	#537×C₁₀₇	1	1	yes	characteristic
282	C₄	#575×C₁₀₇	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	16	2	no	—
285	C₂	C₁₀₇×#3245	1	1	yes	characteristic
286	C₂	C₁₀₇×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₇
prime radical	C₁₀₇×#2218	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ #117×C₁₀₇ ◁ #405×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₇×#3175	C₂	semidirect
2. C₁₀₇×#2226	C₂	semidirect
3. C₁₀₇×#2225	C₂	semidirect
4. C₁₀₇×#2219	C₂	semidirect
6. C₁₀₇×#2217	C₂	semidirect
7. C₁₀₇×#2211	C₂	semidirect
8. C₁₀₇×#2210	C₂	semidirect
9. C₁₀₇×#2151	C₂	semidirect
139. #33829	C₁₀₇	direct (N × H)
144. #3175	C₂×C₁₀₇	semidirect
145. #2226	C₂×C₁₀₇	semidirect
146. #2225	C₂×C₁₀₇	semidirect
147. #2219	C₂×C₁₀₇	semidirect
149. #2217	C₂×C₁₀₇	semidirect
150. #2211	C₂×C₁₀₇	semidirect
151. #2210	C₂×C₁₀₇	semidirect
152. #2151	C₂×C₁₀₇	semidirect
153. C₁₀₇	#33829	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×#3175	2
C₁₀₇×#2226	2
C₁₀₇×#2225	2
C₁₀₇×#2219	2
C₁₀₇×#2218	2
C₁₀₇×#2217	2
C₁₀₇×#2211	2
C₁₀₇×#2210	2
C₁₀₇×#2151	2
#33829	107

Subgroups — C107×#33829 #41059793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#33829 #41059793