Subgroups — C₁₀₇×#7625 #41033589

Order 27392.

462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1839	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1767	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#1271	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₀₇	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₀₇	—	2	1	no	—
7	C₂×C₄⋊C₈×C₁₀₇	—	2	1	no	—
8	C₂×#97×C₁₀₇	—	2	1	no	—
9	#391×C₁₀₇	—	2	1	no	—
10	#389×C₁₀₇	—	2	1	no	—
11	#389×C₁₀₇	C₂²	1	2	yes	—
12	#384×C₁₀₇	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₀₇	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₀₇	C₄	1	2	yes	—
15	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
16	#336×C₁₀₇	—	2	1	no	—
17	#336×C₁₀₇	—	2	1	no	—
18	#336×C₁₀₇	C₂²	1	2	yes	—
19	#336×C₁₀₇	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₀₇	—	2	1	no	—
21	C₂²×C₈×C₁₀₇	—	2	2	no	—
22	C₂²×C₈×C₁₀₇	—	2	2	no	—
23	C₄⋊Q₈×C₁₀₇	—	2	4	no	—
24	C₄×Q₈×C₁₀₇	—	4	2	no	—
25	C₄×Q₈×C₁₀₇	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₇	—	2	2	no	—
33	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₀₇	—	2	4	no	—
41	#97×C₁₀₇	—	4	2	no	—
42	#95×C₁₀₇	—	2	2	no	—
43	#95×C₁₀₇	—	2	2	no	—
44	C₂×Q₈×C₁₀₇	—	2	4	no	—
45	C₂×Q₈×C₁₀₇	—	4	2	no	—
46	C₂×Q₈×C₁₀₇	—	4	2	no	—
47	C₂²×C₄×C₁₀₇	—	2	2	no	—
48	C₂²×C₄×C₁₀₇	—	2	2	no	—
49	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₀₇	—	2	4	no	—
57	C₂×C₈×C₁₀₇	—	4	2	no	—
58	C₂×C₈×C₁₀₇	—	4	2	no	—
59	C₂×C₈×C₁₀₇	—	4	2	no	—
60	C₂×C₈×C₁₀₇	—	4	2	no	—
61	C₂×C₈×C₁₀₇	—	4	2	no	—
62	C₄⋊C₄×C₁₀₇	—	2	4	no	—
63	C₄⋊C₄×C₁₀₇	—	2	2	no	—
64	C₄⋊C₄×C₁₀₇	—	4	2	no	—
65	C₄⋊C₄×C₁₀₇	—	4	2	no	—
66	C₄⋊C₄×C₁₀₇	—	4	2	no	—
67	C₄⋊C₄×C₁₀₇	—	4	2	no	—
68	C₄⋊C₄×C₁₀₇	—	4	2	no	—
69	C₄⋊C₄×C₁₀₇	M₁₆	1	4	yes	—
70	C₄²×C₁₀₇	—	2	2	no	—
71	C₄²×C₁₀₇	—	2	2	no	—
72	C₄²×C₁₀₇	—	2	2	no	—
73	C₄²×C₁₀₇	—	2	2	no	—
74	C₄²×C₁₀₇	—	4	1	no	—
75	C₄²×C₁₀₇	—	4	1	no	—
76	C₄²×C₁₀₇	—	4	4	no	—
77	C₄²×C₁₀₇	D₈	1	2	yes	—
78	C₄²×C₁₀₇	SD₁₆	1	2	yes	—
79	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	Q₈×C₁₀₇	—	4	8	no	—
81	C₂×C₄×C₁₀₇	—	2	1	no	—
82	C₂×C₄×C₁₀₇	—	2	1	no	—
83	C₂×C₄×C₁₀₇	—	2	1	no	—
84	C₂×C₄×C₁₀₇	—	2	1	no	—
85	C₂×C₄×C₁₀₇	—	2	1	no	—
86	C₂×C₄×C₁₀₇	—	2	1	no	—
87	C₂×C₄×C₁₀₇	—	2	1	no	—
88	C₂×C₄×C₁₀₇	—	2	1	no	—
89	C₂×C₄×C₁₀₇	—	2	1	no	—
90	C₂×C₄×C₁₀₇	—	2	1	no	—
91	C₂×C₄×C₁₀₇	—	2	1	no	—
92	C₂×C₄×C₁₀₇	—	2	1	no	—
93	C₂×C₄×C₁₀₇	—	2	1	no	—
94	C₂×C₄×C₁₀₇	—	2	1	no	—
95	C₂×C₄×C₁₀₇	—	2	4	no	—
96	C₂×C₄×C₁₀₇	—	4	2	no	—
97	C₂×C₄×C₁₀₇	—	4	2	no	—
98	C₂×C₄×C₁₀₇	—	4	2	no	—
99	C₂×C₄×C₁₀₇	—	4	2	no	—
100	C₂×C₄×C₁₀₇	—	4	2	no	—
101	C₂×C₄×C₁₀₇	#101	1	2	yes	—
102	C₂×C₄×C₁₀₇	#102	1	2	yes	—
103	C₂×C₄×C₁₀₇	#102	1	2	yes	—
104	C₂×C₄×C₁₀₇	#104	1	2	yes	—
105	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₀₇	#137	1	2	yes	—
109	C₈×C₁₀₇	—	4	4	no	—
110	C₈×C₁₀₇	—	8	2	no	—
111	C₂²×C₁₀₇	#407	1	1	yes	characteristic
112	C₂²×C₁₀₇	#410	1	1	yes	characteristic
113	C₂²×C₁₀₇	C₂×#101	1	1	yes	characteristic
114	C₂²×C₁₀₇	C₂×#102	1	1	yes	characteristic
115	C₂²×C₁₀₇	#419	1	1	yes	characteristic
116	C₂²×C₁₀₇	C₂×#104	1	1	yes	characteristic
117	C₂²×C₁₀₇	#421	1	1	yes	characteristic
118	C₄×C₁₀₇	—	2	2	no	—
119	C₄×C₁₀₇	—	2	2	no	—
120	C₄×C₁₀₇	—	2	2	no	—
121	C₄×C₁₀₇	—	2	2	no	—
122	C₄×C₁₀₇	—	2	1	no	—
123	C₄×C₁₀₇	—	2	1	no	—
124	C₄×C₁₀₇	—	2	2	no	—
125	C₄×C₁₀₇	—	2	2	no	—
126	C₄×C₁₀₇	—	4	4	no	—
127	C₄×C₁₀₇	—	8	2	no	—
128	#7625	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₇	#1467	1	1	yes	characteristic
130	C₂×C₁₀₇	#1486	1	1	yes	characteristic
131	C₂×C₁₀₇	#1491	1	1	yes	characteristic
132	C₂×C₁₀₇	#1502	1	1	yes	characteristic
133	C₂×C₁₀₇	#1515	1	1	yes	characteristic
134	C₂×C₁₀₇	#1531	1	1	yes	characteristic
135	C₂×C₁₀₇	#1546	1	1	yes	characteristic
136	C₄×C₄⋊Q₈	C₂×C₁₀₇	1	1	yes	characteristic
137	#1839	C₂×C₁₀₇	1	1	yes	characteristic
138	#1767	C₂×C₁₀₇	1	1	yes	characteristic
139	#1271	C₂×C₁₀₇	1	4	yes	—
140	C₁₀₇	#7625	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₄⋊Q₈	C₄×C₁₀₇	1	1	yes	characteristic
142	C₂×C₄×Q₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	2	1	no	—
144	C₂×#97	—	2	1	no	—
145	#391	—	2	1	no	—
146	#389	—	2	1	no	—
147	#389	C₂²×C₁₀₇	1	2	yes	—
148	#384	C₄×C₁₀₇	1	1	yes	characteristic
149	C₄×C₄⋊C₄	—	2	1	no	—
150	C₄×C₄⋊C₄	C₄×C₁₀₇	1	2	yes	—
151	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	#336	—	2	1	no	—
154	#336	C₂²×C₁₀₇	1	2	yes	—
155	#336	C₂²×C₁₀₇	1	2	yes	—
156	C₂²×Q₈	—	2	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	4	no	—
160	C₄×Q₈	—	4	2	no	—
161	C₄×Q₈	—	4	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
173	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	#97	—	4	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	M₁₆×C₁₀₇	1	4	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	4	no	—
213	C₄²	D₈×C₁₀₇	1	2	yes	—
214	C₄²	SD₁₆×C₁₀₇	1	2	yes	—
215	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
216	Q₈	—	4	8	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#101×C₁₀₇	1	2	yes	—
238	C₂×C₄	#102×C₁₀₇	1	2	yes	—
239	C₂×C₄	#102×C₁₀₇	1	2	yes	—
240	C₂×C₄	#104×C₁₀₇	1	2	yes	—
241	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	1	yes	characteristic
244	C₂×C₄	#137×C₁₀₇	1	2	yes	—
245	C₈	—	4	4	no	—
246	C₈	—	8	2	no	—
247	C₂²	#407×C₁₀₇	1	1	yes	characteristic
248	C₂²	#410×C₁₀₇	1	1	yes	characteristic
249	C₂²	C₂×#101×C₁₀₇	1	1	yes	characteristic
250	C₂²	C₂×#102×C₁₀₇	1	1	yes	characteristic
251	C₂²	#419×C₁₀₇	1	1	yes	characteristic
252	C₂²	C₂×#104×C₁₀₇	1	1	yes	characteristic
253	C₂²	#421×C₁₀₇	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₂	C₁₀₇×#1467	1	1	yes	characteristic
265	C₂	C₁₀₇×#1486	1	1	yes	characteristic
266	C₂	C₁₀₇×#1491	1	1	yes	characteristic
267	C₂	C₁₀₇×#1502	1	1	yes	characteristic
268	C₂	C₁₀₇×#1515	1	1	yes	characteristic
269	C₂	C₁₀₇×#1531	1	1	yes	characteristic
270	C₂	C₁₀₇×#1546	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
128. #7625	C₁₀₇	direct (N × H)
140. C₁₀₇	#7625	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₀₇	2
C₁₀₇×#1839	2
C₁₀₇×#1767	2
C₁₀₇×#1271	2
#7625	107

Subgroups — C107×#7625 #41033589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7625 #41033589