Subgroups — C₁₀₇×#13200 #41039164

Order 27392.

438 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#2929	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₇×#2117	C₂	1	2	yes	maximal, complemented
4	C₁₀₇×#2112	C₂	1	2	yes	maximal, complemented
5	C₁₀₇×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#131×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
8	#525×C₁₀₇	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₁₀₇	C₂²	1	2	yes	—
10	C₂×M₃₂×C₁₀₇	C₂²	1	2	yes	—
11	#435×C₁₀₇	—	2	2	no	—
12	C₈×D₄×C₁₀₇	—	2	2	no	—
13	#424×C₁₀₇	C₄	1	1	yes	characteristic
14	#408×C₁₀₇	C₄	1	1	yes	characteristic
15	#407×C₁₀₇	C₄	1	1	yes	characteristic
16	#405×C₁₀₇	C₂²	1	1	yes	characteristic
17	#364×C₁₀₇	—	2	2	no	—
18	#350×C₁₀₇	—	2	4	no	—
19	C₂²⋊C₁₆×C₁₀₇	—	2	4	no	—
20	#347×C₁₀₇	—	2	2	no	—
21	C₂×#41×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	#131×C₁₀₇	—	2	2	no	—
24	#131×C₁₀₇	—	2	2	no	—
25	C₂×M₁₆×C₁₀₇	—	2	1	no	—
26	C₂×M₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₀₇	—	2	2	no	—
29	C₂²×C₈×C₁₀₇	—	2	1	no	—
30	C₂²×C₈×C₁₀₇	—	2	1	no	—
31	C₂²×C₈×C₁₀₇	—	2	1	no	—
32	C₂²×C₈×C₁₀₇	—	2	2	no	—
33	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
35	#123×C₁₀₇	—	2	1	no	—
36	C₂²⋊Q₈×C₁₀₇	—	2	1	no	—
37	#121×C₁₀₇	—	2	1	no	—
38	#120×C₁₀₇	—	2	1	no	—
39	C₄×D₄×C₁₀₇	—	2	2	no	—
40	#117×C₁₀₇	C₂×C₄	1	1	yes	characteristic
41	M₃₂×C₁₀₇	—	2	2	no	—
42	M₃₂×C₁₀₇	—	2	2	no	—
43	M₃₂×C₁₀₇	—	2	2	no	—
44	M₃₂×C₁₀₇	—	2	2	no	—
45	C₂×C₁₆×C₁₀₇	—	2	2	no	—
46	C₂×C₁₆×C₁₀₇	D₄	1	4	yes	complemented
47	C₄⋊C₈×C₁₀₇	—	2	2	no	—
48	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
49	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
50	#97×C₁₀₇	D₄	1	2	yes	—
51	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
52	C₂⁴×C₁₀₇	M₁₆	1	1	yes	characteristic
53	#41×C₁₀₇	—	2	2	no	—
54	#41×C₁₀₇	—	2	2	no	—
55	C₂×Q₈×C₁₀₇	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₁₀₇	—	2	2	no	—
57	C₂×D₄×C₁₀₇	—	2	1	no	—
58	C₂×D₄×C₁₀₇	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₇	—	2	1	no	—
60	C₂²×C₄×C₁₀₇	—	2	1	no	—
61	C₂²×C₄×C₁₀₇	—	2	2	no	—
62	C₂²×C₄×C₁₀₇	—	2	1	no	—
63	C₂²×C₄×C₁₀₇	—	2	2	no	—
64	C₂²×C₄×C₁₀₇	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₁₀₇	—	2	2	no	—
67	M₁₆×C₁₀₇	—	2	2	no	—
68	M₁₆×C₁₀₇	—	4	2	no	—
69	C₂×C₈×C₁₀₇	—	2	1	no	—
70	C₂×C₈×C₁₀₇	—	2	1	no	—
71	C₂×C₈×C₁₀₇	—	2	2	no	—
72	C₂×C₈×C₁₀₇	—	4	2	no	—
73	C₂×C₈×C₁₀₇	—	4	1	no	—
74	C₂×C₈×C₁₀₇	—	4	1	no	—
75	C₂×C₈×C₁₀₇	—	4	1	no	—
76	C₂×C₈×C₁₀₇	—	4	1	no	—
77	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₀₇	#41	1	2	yes	—
82	C₄⋊C₄×C₁₀₇	—	2	2	no	—
83	C₄⋊C₄×C₁₀₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
85	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
86	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
87	C₄²×C₁₀₇	C₂²⋊C₄	1	2	yes	—
88	C₁₆×C₁₀₇	—	2	4	no	—
89	C₁₆×C₁₀₇	—	4	2	no	—
90	C₂³×C₁₀₇	—	2	1	no	—
91	C₂³×C₁₀₇	—	2	2	no	—
92	C₂³×C₁₀₇	—	2	1	no	—
93	C₂³×C₁₀₇	—	4	2	no	—
94	C₂³×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
95	Q₈×C₁₀₇	—	2	2	no	—
96	D₄×C₁₀₇	—	2	2	no	—
97	D₄×C₁₀₇	—	4	4	no	—
98	D₄×C₁₀₇	—	4	2	no	—
99	C₂×C₄×C₁₀₇	—	2	2	no	—
100	C₂×C₄×C₁₀₇	—	2	1	no	—
101	C₂×C₄×C₁₀₇	—	2	1	no	—
102	C₂×C₄×C₁₀₇	—	2	2	no	—
103	C₂×C₄×C₁₀₇	—	4	1	no	—
104	C₂×C₄×C₁₀₇	—	4	1	no	—
105	C₂×C₄×C₁₀₇	—	4	1	no	—
106	C₂×C₄×C₁₀₇	—	4	1	no	—
107	C₂×C₄×C₁₀₇	—	4	2	no	—
108	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₀₇	#131	1	2	yes	—
112	C₈×C₁₀₇	—	2	2	no	—
113	C₈×C₁₀₇	—	4	1	no	—
114	C₈×C₁₀₇	—	4	1	no	—
115	C₈×C₁₀₇	#121	1	2	yes	—
116	C₈×C₁₀₇	#124	1	1	yes	characteristic
117	C₈×C₁₀₇	#127	1	1	yes	characteristic
118	C₂²×C₁₀₇	—	2	2	no	—
119	C₂²×C₁₀₇	—	2	4	no	—
120	C₂²×C₁₀₇	—	4	1	no	—
121	C₂²×C₁₀₇	—	4	1	no	—
122	C₂²×C₁₀₇	—	4	2	no	—
123	C₂²×C₁₀₇	—	4	2	no	—
124	C₂²×C₁₀₇	—	4	1	no	—
125	C₂²×C₁₀₇	—	4	1	no	—
126	C₂²×C₁₀₇	#407	1	1	yes	characteristic
127	C₂²×C₁₀₇	#435	1	2	yes	—
128	C₄×C₁₀₇	—	4	2	no	—
129	C₄×C₁₀₇	—	4	1	no	—
130	C₄×C₁₀₇	—	4	1	no	—
131	C₄×C₁₀₇	#390	1	1	yes	characteristic
132	C₄×C₁₀₇	#408	1	1	yes	characteristic
133	C₄×C₁₀₇	#435	1	2	yes	—
134	#13200	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₁₀₇	—	2	1	no	—
136	C₂×C₁₀₇	—	2	2	no	—
137	C₂×C₁₀₇	—	4	2	no	—
138	C₂×C₁₀₇	—	8	1	no	—
139	C₂×C₁₀₇	#1955	1	1	yes	characteristic
140	#2929	C₂×C₁₀₇	1	1	yes	characteristic
141	#2153	C₂×C₁₀₇	1	1	yes	characteristic, complemented
142	#2117	C₂×C₁₀₇	1	2	yes	complemented
143	#2112	C₂×C₁₀₇	1	2	yes	complemented
144	#2110	C₂×C₁₀₇	1	1	yes	characteristic, complemented
145	C₁₀₇	#13200	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#131	C₂²×C₁₀₇	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
148	#525	C₄×C₁₀₇	1	1	yes	characteristic
149	C₂×M₃₂	C₂²×C₁₀₇	1	2	yes	—
150	C₂×M₃₂	C₂²×C₁₀₇	1	2	yes	—
151	#435	—	2	2	no	—
152	C₈×D₄	—	2	2	no	—
153	#424	C₄×C₁₀₇	1	1	yes	characteristic
154	#408	C₄×C₁₀₇	1	1	yes	characteristic
155	#407	C₄×C₁₀₇	1	1	yes	characteristic
156	#405	C₂²×C₁₀₇	1	1	yes	characteristic
157	#364	—	2	2	no	—
158	#350	—	2	4	no	—
159	C₂²⋊C₁₆	—	2	4	no	—
160	#347	—	2	2	no	—
161	C₂×#41	C₂×C₄×C₁₀₇	1	1	yes	characteristic
162	C₂³×C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
163	#131	—	2	2	no	—
164	#131	—	2	2	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	C₂×C₄×C₁₀₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₁₀₇	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
175	#123	—	2	1	no	—
176	C₂²⋊Q₈	—	2	1	no	—
177	#121	—	2	1	no	—
178	#120	—	2	1	no	—
179	C₄×D₄	—	2	2	no	—
180	#117	C₂×C₄×C₁₀₇	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	D₄×C₁₀₇	1	4	yes	complemented
187	C₄⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	#97	D₄×C₁₀₇	1	2	yes	—
191	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
192	C₂⁴	M₁₆×C₁₀₇	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	M₁₆×C₁₀₇	1	1	yes	characteristic
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	M₁₆×C₁₀₇	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	M₁₆×C₁₀₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	4	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
221	C₂×C₈	#41×C₁₀₇	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₄²	C₂²⋊C₄×C₁₀₇	1	2	yes	—
228	C₁₆	—	2	4	no	—
229	C₁₆	—	4	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
249	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
250	C₂×C₄	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
251	C₂×C₄	#131×C₁₀₇	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	#121×C₁₀₇	1	2	yes	—
256	C₈	#124×C₁₀₇	1	1	yes	characteristic
257	C₈	#127×C₁₀₇	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	#407×C₁₀₇	1	1	yes	characteristic
267	C₂²	#435×C₁₀₇	1	2	yes	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	#390×C₁₀₇	1	1	yes	characteristic
272	C₄	#408×C₁₀₇	1	1	yes	characteristic
273	C₄	#435×C₁₀₇	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₁₀₇×#1955	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₇	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₇
socle	C₂×C₁₀₇	#1955
prime radical	#525×C₁₀₇	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₇ ◁ C₂²×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₇×#2153	C₂	semidirect
3. C₁₀₇×#2117	C₂	semidirect
4. C₁₀₇×#2112	C₂	semidirect
5. C₁₀₇×#2110	C₂	semidirect
46. C₂×C₁₆×C₁₀₇	D₄	semidirect
134. #13200	C₁₀₇	direct (N × H)
141. #2153	C₂×C₁₀₇	semidirect
142. #2117	C₂×C₁₀₇	semidirect
143. #2112	C₂×C₁₀₇	semidirect
144. #2110	C₂×C₁₀₇	semidirect
145. C₁₀₇	#13200	direct (N × H)
186. C₂×C₁₆	D₄×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₇×#2929	2
C₁₀₇×#2153	2
C₁₀₇×#2117	2
C₁₀₇×#2112	2
C₁₀₇×#2110	2
#13200	107

Subgroups — C107×#13200 #41039164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#13200 #41039164