Subgroups — C₁₀₇×#8173 #41034137

Order 27392.

610 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈×C₁₀₇	C₂	1	2	yes	maximal
5	C₁₀₇×#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₀₇	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
9	C₂×#97×C₁₀₇	—	2	2	no	—
10	C₂×#97×C₁₀₇	—	2	1	no	—
11	C₂×C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
12	C₄×C₄⋊C₄×C₁₀₇	—	2	1	no	—
13	C₄×C₄⋊C₄×C₁₀₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₀₇	C₄	1	2	yes	—
15	C₄×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₄³×C₁₀₇	—	2	1	no	—
17	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
18	#336×C₁₀₇	—	2	1	no	—
19	Q₈⋊C₈×C₁₀₇	C₄	1	8	yes	complemented
20	C₂²×Q₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
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	—	2	4	no	—
25	C₄×Q₈×C₁₀₇	—	2	2	no	—
26	C₄×Q₈×C₁₀₇	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₀₇	—	2	2	no	—
32	C₂×C₄²×C₁₀₇	—	2	1	no	—
33	C₂×C₄²×C₁₀₇	—	2	2	no	—
34	C₂×C₄²×C₁₀₇	—	2	1	no	—
35	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
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	2	no	—
41	C₄⋊C₈×C₁₀₇	C₂×C₄	1	4	yes	—
42	#97×C₁₀₇	—	2	8	no	—
43	#97×C₁₀₇	—	4	2	no	—
44	C₄×C₈×C₁₀₇	—	2	2	no	—
45	C₄×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
46	#95×C₁₀₇	—	2	2	no	—
47	#95×C₁₀₇	—	2	1	no	—
48	C₂×Q₈×C₁₀₇	—	2	1	no	—
49	C₂×Q₈×C₁₀₇	—	2	2	no	—
50	C₂×Q₈×C₁₀₇	C₄²	1	2	yes	—
51	C₂×Q₈×C₁₀₇	M₁₆	1	4	yes	—
52	C₂²×C₄×C₁₀₇	—	2	1	no	—
53	C₂²×C₄×C₁₀₇	—	2	1	no	—
54	C₂²×C₄×C₁₀₇	—	2	2	no	—
55	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₀₇	—	2	8	no	—
61	C₂×C₈×C₁₀₇	—	2	4	no	—
62	C₂×C₈×C₁₀₇	—	2	4	no	—
63	C₂×C₈×C₁₀₇	—	4	2	no	—
64	C₂×C₈×C₁₀₇	—	4	2	no	—
65	C₄⋊C₄×C₁₀₇	—	2	8	no	—
66	C₄⋊C₄×C₁₀₇	—	2	1	no	—
67	C₄⋊C₄×C₁₀₇	—	2	4	no	—
68	C₄⋊C₄×C₁₀₇	—	2	2	no	—
69	C₄⋊C₄×C₁₀₇	C₄²	1	2	yes	—
70	C₄⋊C₄×C₁₀₇	M₁₆	1	4	yes	—
71	C₄²×C₁₀₇	—	2	8	no	—
72	C₄²×C₁₀₇	—	2	4	no	—
73	C₄²×C₁₀₇	—	2	4	no	—
74	C₄²×C₁₀₇	—	2	2	no	—
75	C₄²×C₁₀₇	—	2	2	no	—
76	C₄²×C₁₀₇	—	2	1	no	—
77	C₄²×C₁₀₇	—	2	2	no	—
78	C₄²×C₁₀₇	—	2	1	no	—
79	C₄²×C₁₀₇	—	4	4	no	—
80	C₄²×C₁₀₇	—	4	1	no	—
81	C₄²×C₁₀₇	—	4	1	no	—
82	C₄²×C₁₀₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₀₇	#41	1	2	yes	—
84	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₁₀₇	—	2	2	no	—
86	Q₈×C₁₀₇	—	2	4	no	—
87	Q₈×C₁₀₇	#97	1	4	yes	complemented
88	C₂×C₄×C₁₀₇	—	2	2	no	—
89	C₂×C₄×C₁₀₇	—	2	2	no	—
90	C₂×C₄×C₁₀₇	—	2	4	no	—
91	C₂×C₄×C₁₀₇	—	2	2	no	—
92	C₂×C₄×C₁₀₇	—	2	2	no	—
93	C₂×C₄×C₁₀₇	—	2	2	no	—
94	C₂×C₄×C₁₀₇	—	2	2	no	—
95	C₂×C₄×C₁₀₇	—	2	4	no	—
96	C₂×C₄×C₁₀₇	—	2	2	no	—
97	C₂×C₄×C₁₀₇	—	2	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	#104	1	4	yes	—
102	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₁₀₇	#137	1	2	yes	—
110	C₂×C₄×C₁₀₇	#137	1	1	yes	characteristic
111	C₈×C₁₀₇	—	2	8	no	—
112	C₈×C₁₀₇	—	4	4	no	—
113	C₂²×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₀₇	#407	1	2	yes	—
115	C₂²×C₁₀₇	#417	1	1	yes	characteristic
116	C₂²×C₁₀₇	#419	1	1	yes	characteristic
117	C₂²×C₁₀₇	C₂×#104	1	2	yes	—
118	C₄×C₁₀₇	—	2	4	no	—
119	C₄×C₁₀₇	—	2	4	no	—
120	C₄×C₁₀₇	—	2	2	no	—
121	C₄×C₁₀₇	—	4	2	no	—
122	C₄×C₁₀₇	—	4	4	no	—
123	C₄×C₁₀₇	—	4	2	no	—
124	C₄×C₁₀₇	C₂×#97	1	1	yes	characteristic
125	C₄×C₁₀₇	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₁₀₇	#432	1	2	yes	—
127	C₄×C₁₀₇	#440	1	2	yes	—
128	C₄×C₁₀₇	#441	1	2	yes	—
129	#8173	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₀₇	#1489	1	2	yes	—
131	C₂×C₁₀₇	#1491	1	2	yes	—
132	C₂×C₁₀₇	#1749	1	1	yes	characteristic
133	C₂×C₁₀₇	C₄×#104	1	1	yes	characteristic
134	C₂×C₁₀₇	#1763	1	1	yes	characteristic
135	C₄²×Q₈	C₂×C₁₀₇	1	1	yes	characteristic
136	#1767	C₂×C₁₀₇	1	1	yes	characteristic
137	C₄×#97	C₂×C₁₀₇	1	1	yes	characteristic
138	C₂×Q₈⋊C₈	C₂×C₁₀₇	1	2	yes	—
139	#1279	C₂×C₁₀₇	1	2	yes	—
140	C₁₀₇	#8173	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×C₄×Q₈	C₄×C₁₀₇	1	2	yes	—
142	C₂×C₄×Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
143	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
144	C₂×#97	—	2	2	no	—
145	C₂×#97	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₁₀₇	1	2	yes	—
147	C₄×C₄⋊C₄	—	2	1	no	—
148	C₄×C₄⋊C₄	—	2	2	no	—
149	C₄×C₄⋊C₄	C₄×C₁₀₇	1	2	yes	—
150	C₄×C₄⋊C₄	C₂²×C₁₀₇	1	1	yes	characteristic
151	C₄³	—	2	1	no	—
152	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
153	#336	—	2	1	no	—
154	Q₈⋊C₈	C₄×C₁₀₇	1	8	yes	complemented
155	C₂²×Q₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₄×Q₈	—	2	4	no	—
159	C₄×Q₈	—	2	4	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	C₂×C₄×C₁₀₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
171	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
173	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₂×C₄×C₁₀₇	1	4	yes	—
177	#97	—	2	8	no	—
178	#97	—	4	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₂×C₄×C₁₀₇	1	4	yes	—
181	#95	—	2	2	no	—
182	#95	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₄²×C₁₀₇	1	2	yes	—
186	C₂×Q₈	M₁₆×C₁₀₇	1	4	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
191	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	C₄²×C₁₀₇	1	2	yes	—
205	C₄⋊C₄	M₁₆×C₁₀₇	1	4	yes	—
206	C₄²	—	2	8	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	4	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₂²×C₄×C₁₀₇	1	2	yes	—
218	C₄²	#41×C₁₀₇	1	2	yes	—
219	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	4	no	—
222	Q₈	#97×C₁₀₇	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#104×C₁₀₇	1	4	yes	—
237	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
238	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
239	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
241	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
242	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
243	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	1	yes	characteristic
244	C₂×C₄	#137×C₁₀₇	1	2	yes	—
245	C₂×C₄	#137×C₁₀₇	1	1	yes	characteristic
246	C₈	—	2	8	no	—
247	C₈	—	4	4	no	—
248	C₂²	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
249	C₂²	#407×C₁₀₇	1	2	yes	—
250	C₂²	#417×C₁₀₇	1	1	yes	characteristic
251	C₂²	#419×C₁₀₇	1	1	yes	characteristic
252	C₂²	C₂×#104×C₁₀₇	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	2	no	—
259	C₄	C₂×#97×C₁₀₇	1	1	yes	characteristic
260	C₄	C₄×M₁₆×C₁₀₇	1	1	yes	characteristic
261	C₄	#432×C₁₀₇	1	2	yes	—
262	C₄	#440×C₁₀₇	1	2	yes	—
263	C₄	#441×C₁₀₇	1	2	yes	—
264	C₂	C₁₀₇×#1489	1	2	yes	—
265	C₂	C₁₀₇×#1491	1	2	yes	—
266	C₂	C₁₀₇×#1749	1	1	yes	characteristic
267	C₂	C₄×#104×C₁₀₇	1	1	yes	characteristic
268	C₂	C₁₀₇×#1763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle, prime radical	C₂³×C₁₀₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
19. Q₈⋊C₈×C₁₀₇	C₄	semidirect
87. Q₈×C₁₀₇	#97	semidirect
129. #8173	C₁₀₇	direct (N × H)
140. C₁₀₇	#8173	direct (N × H)
154. Q₈⋊C₈	C₄×C₁₀₇	semidirect
222. Q₈	#97×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₁₀₇	2
C₁₀₇×#1767	2
C₄×#97×C₁₀₇	2
C₂×Q₈⋊C₈×C₁₀₇	2
C₁₀₇×#1279	2
#8173	107

Subgroups — C107×#8173 #41034137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8173 #41034137