Subgroups — C₁₀₇×#8178 #41034142

Order 27392.

486 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#2295	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1767	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1744	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	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
8	C₂×#97×C₁₀₇	—	2	1	no	—
9	C₂×C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
10	#391×C₁₀₇	C₄	1	2	yes	—
11	#389×C₁₀₇	C₄	1	2	yes	—
12	#387×C₁₀₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₁₀₇	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
15	#376×C₁₀₇	—	2	1	no	—
16	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
17	#336×C₁₀₇	—	2	2	no	—
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	2	no	—
24	C₄×Q₈×C₁₀₇	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₇	—	2	1	no	—
30	C₂×C₄²×C₁₀₇	—	2	1	no	—
31	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
35	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₇	—	2	2	no	—
37	C₄⋊C₈×C₁₀₇	C₂×C₄	1	4	yes	—
38	#97×C₁₀₇	—	4	2	no	—
39	C₄×C₈×C₁₀₇	—	2	2	no	—
40	C₄×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
41	#95×C₁₀₇	—	2	2	no	—
42	#95×C₁₀₇	—	2	2	no	—
43	#95×C₁₀₇	—	2	2	no	—
44	#95×C₁₀₇	—	2	1	no	—
45	C₂×Q₈×C₁₀₇	—	2	1	no	—
46	C₂×Q₈×C₁₀₇	—	2	4	no	—
47	C₂×Q₈×C₁₀₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₀₇	—	2	1	no	—
49	C₂²×C₄×C₁₀₇	—	2	1	no	—
50	C₂²×C₄×C₁₀₇	—	2	2	no	—
51	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
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	—	2	4	no	—
58	C₂×C₈×C₁₀₇	—	4	4	no	—
59	C₂×C₈×C₁₀₇	—	4	2	no	—
60	C₂×C₈×C₁₀₇	—	4	2	no	—
61	C₄⋊C₄×C₁₀₇	—	2	1	no	—
62	C₄⋊C₄×C₁₀₇	—	2	4	no	—
63	C₄⋊C₄×C₁₀₇	—	2	4	no	—
64	C₄⋊C₄×C₁₀₇	—	4	4	no	—
65	C₄⋊C₄×C₁₀₇	C₄²	1	2	yes	—
66	C₄²×C₁₀₇	—	2	4	no	—
67	C₄²×C₁₀₇	—	2	4	no	—
68	C₄²×C₁₀₇	—	2	1	no	—
69	C₄²×C₁₀₇	—	2	1	no	—
70	C₄²×C₁₀₇	—	2	2	no	—
71	C₄²×C₁₀₇	—	4	2	no	—
72	C₄²×C₁₀₇	—	4	1	no	—
73	C₄²×C₁₀₇	—	4	1	no	—
74	C₄²×C₁₀₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₀₇	#41	1	2	yes	—
76	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₁₀₇	—	2	4	no	—
78	Q₈×C₁₀₇	—	4	2	no	—
79	C₂×C₄×C₁₀₇	—	2	2	no	—
80	C₂×C₄×C₁₀₇	—	2	2	no	—
81	C₂×C₄×C₁₀₇	—	2	2	no	—
82	C₂×C₄×C₁₀₇	—	2	2	no	—
83	C₂×C₄×C₁₀₇	—	2	2	no	—
84	C₂×C₄×C₁₀₇	—	2	2	no	—
85	C₂×C₄×C₁₀₇	—	2	2	no	—
86	C₂×C₄×C₁₀₇	—	2	2	no	—
87	C₂×C₄×C₁₀₇	—	2	2	no	—
88	C₂×C₄×C₁₀₇	—	4	2	no	—
89	C₂×C₄×C₁₀₇	—	4	2	no	—
90	C₂×C₄×C₁₀₇	—	4	2	no	—
91	C₂×C₄×C₁₀₇	—	4	2	no	—
92	C₂×C₄×C₁₀₇	—	4	2	no	—
93	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₀₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₇	#131	1	4	yes	—
98	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₁₀₇	#137	1	2	yes	—
100	C₂×C₄×C₁₀₇	#137	1	1	yes	characteristic
101	C₈×C₁₀₇	—	4	4	no	—
102	C₈×C₁₀₇	—	4	4	no	—
103	C₂²×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₁₀₇	#408	1	2	yes	—
105	C₂²×C₁₀₇	#417	1	1	yes	characteristic
106	C₂²×C₁₀₇	#419	1	1	yes	characteristic
107	C₂²×C₁₀₇	#421	1	2	yes	—
108	C₄×C₁₀₇	—	2	2	no	—
109	C₄×C₁₀₇	—	4	2	no	—
110	C₄×C₁₀₇	—	4	2	no	—
111	C₄×C₁₀₇	—	4	2	no	—
112	C₄×C₁₀₇	—	4	2	no	—
113	C₄×C₁₀₇	—	8	2	no	—
114	C₄×C₁₀₇	#405	1	2	yes	—
115	C₄×C₁₀₇	#433	1	2	yes	—
116	C₄×C₁₀₇	#440	1	2	yes	—
117	C₄×C₁₀₇	#441	1	2	yes	—
118	#8178	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₁₀₇	#1495	1	2	yes	—
120	C₂×C₁₀₇	#1497	1	2	yes	—
121	C₂×C₁₀₇	#1750	1	1	yes	characteristic
122	C₂×C₁₀₇	#1759	1	1	yes	characteristic
123	C₂×C₁₀₇	#1763	1	1	yes	characteristic
124	#2295	C₂×C₁₀₇	1	1	yes	characteristic
125	#1767	C₂×C₁₀₇	1	1	yes	characteristic
126	#1744	C₂×C₁₀₇	1	1	yes	characteristic
127	C₂×Q₈⋊C₈	C₂×C₁₀₇	1	2	yes	—
128	#1279	C₂×C₁₀₇	1	2	yes	—
129	C₁₀₇	#8178	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×C₄×Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
132	C₂×#97	—	2	1	no	—
133	C₂×C₄×C₈	C₂²×C₁₀₇	1	2	yes	—
134	#391	C₄×C₁₀₇	1	2	yes	—
135	#389	C₄×C₁₀₇	1	2	yes	—
136	#387	—	2	2	no	—
137	C₄×C₄⋊C₄	—	2	1	no	—
138	C₄×C₄⋊C₄	C₂²×C₁₀₇	1	1	yes	characteristic
139	#376	—	2	1	no	—
140	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
141	#336	—	2	2	no	—
142	#336	—	2	1	no	—
143	Q₈⋊C₈	C₄×C₁₀₇	1	8	yes	complemented
144	C₂²×Q₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	C₂×C₄×C₁₀₇	1	4	yes	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
156	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
157	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
158	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
159	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	C₂×C₄×C₁₀₇	1	4	yes	—
162	#97	—	4	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	C₂×C₄×C₁₀₇	1	4	yes	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	1	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	C₄²×C₁₀₇	1	2	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	C₄²×C₁₀₇	1	2	yes	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	C₂²×C₄×C₁₀₇	1	2	yes	—
199	C₄²	#41×C₁₀₇	1	2	yes	—
200	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
218	C₂×C₄	#117×C₁₀₇	1	1	yes	characteristic
219	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
220	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
221	C₂×C₄	#131×C₁₀₇	1	4	yes	—
222	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	1	yes	characteristic
223	C₂×C₄	#137×C₁₀₇	1	2	yes	—
224	C₂×C₄	#137×C₁₀₇	1	1	yes	characteristic
225	C₈	—	4	4	no	—
226	C₈	—	4	4	no	—
227	C₂²	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
228	C₂²	#408×C₁₀₇	1	2	yes	—
229	C₂²	#417×C₁₀₇	1	1	yes	characteristic
230	C₂²	#419×C₁₀₇	1	1	yes	characteristic
231	C₂²	#421×C₁₀₇	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	8	2	no	—
238	C₄	#405×C₁₀₇	1	2	yes	—
239	C₄	#433×C₁₀₇	1	2	yes	—
240	C₄	#440×C₁₀₇	1	2	yes	—
241	C₄	#441×C₁₀₇	1	2	yes	—
242	C₂	C₁₀₇×#1495	1	2	yes	—
243	C₂	C₁₀₇×#1497	1	2	yes	—
244	C₂	C₁₀₇×#1750	1	1	yes	characteristic
245	C₂	C₁₀₇×#1759	1	1	yes	characteristic
246	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
118. #8178	C₁₀₇	direct (N × H)
129. C₁₀₇	#8178	direct (N × H)
143. Q₈⋊C₈	C₄×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₇×#2295	2
C₁₀₇×#1767	2
C₁₀₇×#1744	2
C₂×Q₈⋊C₈×C₁₀₇	2
C₁₀₇×#1279	2
#8178	107

Subgroups — C107×#8178 #41034142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8178 #41034142