Subgroups — C₂×C₁₀₇×#1846 #41046760

Order 27392.

1182 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₀₇	C₂	1	2	yes	maximal
2	C₂²×#106×C₁₀₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#382×C₁₀₇	C₂	1	2	yes	maximal
5	C₁₀₇×#1846	C₂	1	8	yes	maximal, direct product factor
6	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₀₇	—	2	8	no	—
11	C₂×#107×C₁₀₇	C₂²	1	8	yes	—
12	C₂×#106×C₁₀₇	—	2	8	no	—
13	C₂×#106×C₁₀₇	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₁₀₇	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₁₀₇	C₂²	1	4	yes	—
16	#382×C₁₀₇	—	2	4	no	—
17	#382×C₁₀₇	C₂²	1	8	yes	—
18	C₂×#95×C₁₀₇	—	2	4	no	—
19	C₂³×C₄×C₁₀₇	—	2	4	no	—
20	C₂³×C₄×C₁₀₇	D₄	1	2	yes	—
21	C₂³×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₀₇	D₄	1	4	yes	—
25	C₂²×C₈×C₁₀₇	D₄	1	4	yes	—
26	C₂²×C₈×C₁₀₇	Q₈	1	4	yes	—
27	C₂²×C₈×C₁₀₇	Q₈	1	4	yes	—
28	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₀₇	—	2	8	no	—
31	C₂×C₄⋊C₄×C₁₀₇	—	2	8	no	—
32	C₂×C₄⋊C₄×C₁₀₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	8	yes	—
34	C₂×C₄²×C₁₀₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
36	C₂×C₄²×C₁₀₇	Q₈	1	4	yes	—
37	C₂×C₄²×C₁₀₇	C₂³	1	2	yes	—
38	#107×C₁₀₇	—	2	16	no	—
39	#106×C₁₀₇	—	2	16	no	—
40	C₄×C₈×C₁₀₇	C₂×C₄	1	16	yes	complemented
41	#95×C₁₀₇	—	2	16	no	—
42	C₂⁴×C₁₀₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₇	—	2	8	no	—
44	C₂²×C₄×C₁₀₇	—	2	8	no	—
45	C₂²×C₄×C₁₀₇	—	4	8	no	—
46	C₂²×C₄×C₁₀₇	—	4	16	no	—
47	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₁₀₇	C₄⋊C₄	1	16	yes	—
58	C₂×C₈×C₁₀₇	C₄⋊C₄	1	16	yes	—
59	C₂×C₈×C₁₀₇	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₁₀₇	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₁₀₇	C₂×D₄	1	4	yes	—
62	C₂×C₈×C₁₀₇	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₁₀₇	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₁₀₇	C₂×Q₈	1	4	yes	—
65	C₄⋊C₄×C₁₀₇	—	2	8	no	—
66	C₄⋊C₄×C₁₀₇	—	2	8	no	—
67	C₄⋊C₄×C₁₀₇	—	4	8	no	—
68	C₄²×C₁₀₇	C₄⋊C₄	1	8	yes	—
69	C₄²×C₁₀₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₁₀₇	C₂×Q₈	1	4	yes	—
71	C₂³×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₁₀₇	#127	1	2	yes	—
73	C₂³×C₁₀₇	C₄⋊Q₈	1	2	yes	—
74	C₂³×C₁₀₇	#135	1	8	yes	—
75	C₂³×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₇	—	2	8	no	—
77	C₂×C₄×C₁₀₇	—	2	16	no	—
78	C₂×C₄×C₁₀₇	—	4	8	no	—
79	C₂×C₄×C₁₀₇	—	4	16	no	—
80	C₂×C₄×C₁₀₇	—	8	16	no	—
81	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	8	yes	—
82	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₁₀₇	#127	1	2	yes	—
86	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	4	yes	—
87	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	4	yes	—
88	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
95	C₈×C₁₀₇	C₂×C₄⋊C₄	1	8	yes	—
96	C₈×C₁₀₇	C₂×C₄⋊C₄	1	8	yes	—
97	C₂²×C₁₀₇	#384	1	4	yes	—
98	C₂²×C₁₀₇	#427	1	8	yes	—
99	C₂²×C₁₀₇	#495	1	8	yes	—
100	C₂²×C₁₀₇	#499	1	8	yes	—
101	C₂²×C₁₀₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₁₀₇	C₂×#127	1	1	yes	characteristic
103	C₂²×C₁₀₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₁₀₇	C₂×#135	1	4	yes	—
105	C₄×C₁₀₇	—	2	8	no	—
106	C₄×C₁₀₇	—	8	8	no	—
107	C₄×C₁₀₇	#384	1	4	yes	—
108	C₄×C₁₀₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₀₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₁₀₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₁₀₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₂×#1846	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₇	#1846	1	8	yes	direct product factor
114	C₂×C₁₀₇	C₂×#384	1	1	yes	characteristic
115	C₂×C₁₀₇	C₂×#427	1	2	yes	—
116	C₂×C₁₀₇	C₂×#495	1	2	yes	—
117	C₂×C₁₀₇	C₂×#499	1	2	yes	—
118	C₂²×#107	C₂×C₁₀₇	1	2	yes	—
119	C₂²×#106	C₂×C₁₀₇	1	2	yes	—
120	C₂²×C₄×C₈	C₂×C₁₀₇	1	1	yes	characteristic
121	C₂×#382	C₂×C₁₀₇	1	2	yes	—
122	#1846	C₂×C₁₀₇	1	8	yes	direct product factor
123	C₁₀₇	C₂×#1846	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	4	yes	—
127	C₂²×C₄²	C₂²×C₁₀₇	1	1	yes	characteristic
128	C₂×#107	—	2	8	no	—
129	C₂×#107	C₂²×C₁₀₇	1	8	yes	—
130	C₂×#106	—	2	8	no	—
131	C₂×#106	C₂²×C₁₀₇	1	8	yes	—
132	C₂×C₄×C₈	C₄×C₁₀₇	1	8	yes	complemented
133	C₂×C₄×C₈	C₂²×C₁₀₇	1	4	yes	—
134	#382	—	2	4	no	—
135	#382	C₂²×C₁₀₇	1	8	yes	—
136	C₂×#95	—	2	4	no	—
137	C₂³×C₄	—	2	4	no	—
138	C₂³×C₄	D₄×C₁₀₇	1	2	yes	—
139	C₂³×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₁₀₇	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₁₀₇	1	4	yes	—
142	C₂²×C₈	D₄×C₁₀₇	1	4	yes	—
143	C₂²×C₈	D₄×C₁₀₇	1	4	yes	—
144	C₂²×C₈	Q₈×C₁₀₇	1	4	yes	—
145	C₂²×C₈	Q₈×C₁₀₇	1	4	yes	—
146	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
147	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	8	no	—
149	C₂×C₄⋊C₄	—	2	8	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	8	yes	—
152	C₂×C₄²	C₂×C₄×C₁₀₇	1	4	yes	—
153	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
154	C₂×C₄²	Q₈×C₁₀₇	1	4	yes	—
155	C₂×C₄²	C₂³×C₁₀₇	1	2	yes	—
156	#107	—	2	16	no	—
157	#106	—	2	16	no	—
158	C₄×C₈	C₂×C₄×C₁₀₇	1	16	yes	complemented
159	#95	—	2	16	no	—
160	C₂⁴	C₂×D₄×C₁₀₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	4	8	no	—
164	C₂²×C₄	—	4	16	no	—
165	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	8	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₀₇	1	2	yes	—
167	C₂²×C₄	C₂×D₄×C₁₀₇	1	4	yes	—
168	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
171	C₂²×C₄	C₂×Q₈×C₁₀₇	1	2	yes	—
172	C₂²×C₄	C₂×Q₈×C₁₀₇	1	2	yes	—
173	C₂²×C₄	C₂×Q₈×C₁₀₇	1	2	yes	—
174	C₂²×C₄	C₂⁴×C₁₀₇	1	1	yes	characteristic
175	C₂×C₈	C₄⋊C₄×C₁₀₇	1	16	yes	—
176	C₂×C₈	C₄⋊C₄×C₁₀₇	1	16	yes	—
177	C₂×C₈	C₂²×C₄×C₁₀₇	1	4	yes	—
178	C₂×C₈	C₂²×C₄×C₁₀₇	1	4	yes	—
179	C₂×C₈	C₂×D₄×C₁₀₇	1	4	yes	—
180	C₂×C₈	C₂×D₄×C₁₀₇	1	4	yes	—
181	C₂×C₈	C₂×Q₈×C₁₀₇	1	4	yes	—
182	C₂×C₈	C₂×Q₈×C₁₀₇	1	4	yes	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₄⋊C₄	—	4	8	no	—
186	C₄²	C₄⋊C₄×C₁₀₇	1	8	yes	—
187	C₄²	C₂²×C₄×C₁₀₇	1	4	yes	—
188	C₄²	C₂×Q₈×C₁₀₇	1	4	yes	—
189	C₂³	C₂×C₄⋊C₄×C₁₀₇	1	2	yes	—
190	C₂³	#127×C₁₀₇	1	2	yes	—
191	C₂³	C₄⋊Q₈×C₁₀₇	1	2	yes	—
192	C₂³	#135×C₁₀₇	1	8	yes	—
193	C₂³	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	—	8	16	no	—
199	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	8	yes	—
200	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	2	yes	—
201	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	4	yes	—
203	C₂×C₄	#127×C₁₀₇	1	2	yes	—
204	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	2	yes	—
205	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	4	yes	—
206	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	4	yes	—
207	C₂×C₄	C₂³×C₄×C₁₀₇	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
209	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₁₀₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×Q₈×C₁₀₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×Q₈×C₁₀₇	1	1	yes	characteristic
213	C₈	C₂×C₄⋊C₄×C₁₀₇	1	8	yes	—
214	C₈	C₂×C₄⋊C₄×C₁₀₇	1	8	yes	—
215	C₂²	#384×C₁₀₇	1	4	yes	—
216	C₂²	#427×C₁₀₇	1	8	yes	—
217	C₂²	#495×C₁₀₇	1	8	yes	—
218	C₂²	#499×C₁₀₇	1	8	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
220	C₂²	C₂×#127×C₁₀₇	1	1	yes	characteristic
221	C₂²	C₂×C₄⋊Q₈×C₁₀₇	1	1	yes	characteristic
222	C₂²	C₂×#135×C₁₀₇	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	8	8	no	—
225	C₄	#384×C₁₀₇	1	4	yes	—
226	C₄	C₂²×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
227	C₄	C₂²×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
228	C₄	C₂×C₄⋊Q₈×C₁₀₇	1	1	yes	characteristic
229	C₄	C₂×C₄⋊Q₈×C₁₀₇	1	1	yes	characteristic
230	C₂	C₁₀₇×#1846	1	8	yes	direct product factor
231	C₂	C₂×#384×C₁₀₇	1	1	yes	characteristic
232	C₂	C₂×#427×C₁₀₇	1	2	yes	—
233	C₂	C₂×#495×C₁₀₇	1	2	yes	—
234	C₂	C₂×#499×C₁₀₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₇ ◁ C₂²×C₄²×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₀₇×#1846	C₂	direct (N × H)
14. C₂×C₄×C₈×C₁₀₇	C₄	semidirect
40. C₄×C₈×C₁₀₇	C₂×C₄	semidirect
112. C₂×#1846	C₁₀₇	direct (N × H)
113. C₂×C₁₀₇	#1846	direct (N × H)
122. #1846	C₂×C₁₀₇	direct (N × H)
123. C₁₀₇	C₂×#1846	direct (N × H)
132. C₂×C₄×C₈	C₄×C₁₀₇	semidirect
158. C₄×C₈	C₂×C₄×C₁₀₇	semidirect
230. C₂	C₁₀₇×#1846	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₀₇	2
C₂²×#106×C₁₀₇	2
C₂²×C₄×C₈×C₁₀₇	2
C₂×#382×C₁₀₇	2
C₁₀₇×#1846	2
C₂×#1846	107

Subgroups — C2×C107×#1846 #41046760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#1846 #41046760