Subgroups — C₁₀₇×#3357 #28334040

Order 13696.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	#554×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#500×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#485×C₁₀₇	C₂	1	2	yes	maximal, complemented
5	#480×C₁₀₇	C₂	1	2	yes	maximal, complemented
6	#477×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#476×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#468×C₁₀₇	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#436×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₀₇	C₂	1	2	yes	maximal, complemented
12	C₂×SD₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
16	#125×C₁₀₇	—	2	1	no	—
17	#124×C₁₀₇	—	2	2	no	—
18	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
19	C₂²⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
20	#121×C₁₀₇	—	2	1	no	—
21	#121×C₁₀₇	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₀₇	—	2	1	no	—
24	C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
26	#117×C₁₀₇	—	2	1	no	—
27	#117×C₁₀₇	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
29	#107×C₁₀₇	C₂²	1	2	yes	—
30	#107×C₁₀₇	C₂²	1	2	yes	—
31	#107×C₁₀₇	C₂²	1	2	yes	—
32	#106×C₁₀₇	C₂²	1	1	yes	characteristic
33	#106×C₁₀₇	C₂²	1	2	yes	—
34	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
35	#103×C₁₀₇	C₂²	1	1	yes	characteristic
36	#103×C₁₀₇	C₂²	1	2	yes	—
37	#102×C₁₀₇	C₂²	1	1	yes	characteristic
38	#102×C₁₀₇	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₁₀₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₀₇	—	2	1	no	—
42	C₂×Q₈×C₁₀₇	—	2	1	no	—
43	C₂×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₀₇	—	2	2	no	—
45	C₂×D₄×C₁₀₇	D₄	1	1	yes	characteristic
46	C₂×D₄×C₁₀₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₇	—	2	2	no	—
48	C₂²×C₄×C₁₀₇	—	2	1	no	—
49	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
50	SD₁₆×C₁₀₇	—	2	2	no	—
51	M₁₆×C₁₀₇	—	2	4	no	—
52	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
54	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₇	—	2	2	no	—
56	C₄⋊C₄×C₁₀₇	—	2	1	no	—
57	C₄⋊C₄×C₁₀₇	—	2	1	no	—
58	C₄⋊C₄×C₁₀₇	—	2	1	no	—
59	C₄⋊C₄×C₁₀₇	—	2	2	no	—
60	C₄⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
63	C₄⋊C₄×C₁₀₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
66	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
68	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
69	C₂²⋊C₄×C₁₀₇	D₄	1	2	yes	—
70	C₄²×C₁₀₇	—	2	2	no	—
71	C₄²×C₁₀₇	—	2	1	no	—
72	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
73	C₂³×C₁₀₇	—	2	1	no	—
74	C₂³×C₁₀₇	C₂×D₄	1	2	yes	—
75	Q₈×C₁₀₇	—	2	1	no	—
76	Q₈×C₁₀₇	—	2	1	no	—
77	Q₈×C₁₀₇	—	8	1	no	—
78	Q₈×C₁₀₇	—	8	1	no	—
79	D₄×C₁₀₇	—	2	2	no	—
80	D₄×C₁₀₇	—	2	1	no	—
81	D₄×C₁₀₇	—	2	1	no	—
82	D₄×C₁₀₇	—	8	2	no	—
83	C₂×C₄×C₁₀₇	—	2	2	no	—
84	C₂×C₄×C₁₀₇	—	2	2	no	—
85	C₂×C₄×C₁₀₇	—	2	1	no	—
86	C₂×C₄×C₁₀₇	—	2	1	no	—
87	C₂×C₄×C₁₀₇	—	2	2	no	—
88	C₂×C₄×C₁₀₇	—	2	1	no	—
89	C₂×C₄×C₁₀₇	—	2	2	no	—
90	C₂×C₄×C₁₀₇	—	4	2	no	—
91	C₂×C₄×C₁₀₇	—	8	2	no	—
92	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₀₇	—	2	2	no	—
97	C₈×C₁₀₇	—	2	1	no	—
98	C₈×C₁₀₇	#41	1	2	yes	—
99	C₂²×C₁₀₇	—	2	2	no	—
100	C₂²×C₁₀₇	—	2	2	no	—
101	C₂²×C₁₀₇	—	2	2	no	—
102	C₂²×C₁₀₇	—	4	1	no	—
103	C₂²×C₁₀₇	—	4	1	no	—
104	C₂²×C₁₀₇	—	4	1	no	—
105	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₀₇	—	2	2	no	—
107	C₄×C₁₀₇	—	2	1	no	—
108	C₄×C₁₀₇	—	4	1	no	—
109	C₄×C₁₀₇	—	4	1	no	—
110	C₄×C₁₀₇	—	4	1	no	—
111	C₄×C₁₀₇	—	4	2	no	—
112	C₄×C₁₀₇	—	4	2	no	—
113	C₄×C₁₀₇	#137	1	2	yes	—
114	C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
115	C₄×C₁₀₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₀₇	—	4	2	no	—
117	C₂×C₁₀₇	—	8	1	no	—
118	C₂×C₁₀₇	#547	1	1	yes	characteristic
119	C₂×C₁₀₇	C₂×#137	1	1	yes	characteristic
120	C₂×C₁₀₇	#576	1	1	yes	characteristic
121	#3357	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₀₇	#3357	1	1	yes	characteristic, direct product factor, Sylow
123	#555	C₂×C₁₀₇	1	1	yes	characteristic
124	#554	C₂×C₁₀₇	1	1	yes	characteristic, complemented
125	#500	C₂×C₁₀₇	1	1	yes	characteristic, complemented
126	#485	C₂×C₁₀₇	1	2	yes	complemented
127	#480	C₂×C₁₀₇	1	2	yes	complemented
128	#477	C₂×C₁₀₇	1	1	yes	characteristic, complemented
129	#476	C₂×C₁₀₇	1	1	yes	characteristic, complemented
130	#468	C₂×C₁₀₇	1	2	yes	complemented
131	C₄×SD₁₆	C₂×C₁₀₇	1	1	yes	characteristic, complemented
132	#436	C₂×C₁₀₇	1	1	yes	characteristic, complemented
133	#428	C₂×C₁₀₇	1	2	yes	complemented
134	C₂×SD₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
135	C₂×M₁₆	C₂²×C₁₀₇	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
138	#125	—	2	1	no	—
139	#124	—	2	2	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	C₂²×C₁₀₇	1	2	yes	—
142	#121	—	2	1	no	—
143	#121	C₂²×C₁₀₇	1	2	yes	—
144	C₄×Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	C₂²×C₁₀₇	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₁₀₇	1	1	yes	characteristic
148	#117	—	2	1	no	—
149	#117	C₂²×C₁₀₇	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₁₀₇	1	2	yes	—
151	#107	C₂²×C₁₀₇	1	2	yes	—
152	#107	C₂²×C₁₀₇	1	2	yes	—
153	#107	C₂²×C₁₀₇	1	2	yes	—
154	#106	C₂²×C₁₀₇	1	1	yes	characteristic
155	#106	C₂²×C₁₀₇	1	2	yes	—
156	C₄⋊C₈	C₂²×C₁₀₇	1	1	yes	characteristic
157	#103	C₂²×C₁₀₇	1	1	yes	characteristic
158	#103	C₂²×C₁₀₇	1	2	yes	—
159	#102	C₂²×C₁₀₇	1	1	yes	characteristic
160	#102	C₂²×C₁₀₇	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₁₀₇	1	2	yes	—
162	C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂³×C₁₀₇	1	1	yes	characteristic
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	D₄×C₁₀₇	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₁₀₇	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₂³×C₁₀₇	1	2	yes	—
172	SD₁₆	—	2	2	no	—
173	M₁₆	—	2	4	no	—
174	C₂×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₁₀₇	1	2	yes	—
176	C₂×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	D₄×C₁₀₇	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₁₀₇	1	1	yes	characteristic
184	C₄⋊C₄	C₂³×C₁₀₇	1	2	yes	—
185	C₄⋊C₄	C₂³×C₁₀₇	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₁₀₇	1	2	yes	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	D₄×C₁₀₇	1	2	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	C₂×D₄×C₁₀₇	1	2	yes	—
197	Q₈	—	2	1	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	8	1	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	1	no	—
203	D₄	—	2	1	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₁₀₇	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	#41×C₁₀₇	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	#137×C₁₀₇	1	2	yes	—
236	C₄	C₂×#41×C₁₀₇	1	1	yes	characteristic
237	C₄	2₋¹⁺⁴×C₁₀₇	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	8	1	no	—
240	C₂	#547×C₁₀₇	1	1	yes	characteristic
241	C₂	C₂×#137×C₁₀₇	1	1	yes	characteristic
242	C₂	#576×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₇
prime radical	#555×C₁₀₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₄×D₄×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #554×C₁₀₇	C₂	semidirect
3. #500×C₁₀₇	C₂	semidirect
4. #485×C₁₀₇	C₂	semidirect
5. #480×C₁₀₇	C₂	semidirect
6. #477×C₁₀₇	C₂	semidirect
7. #476×C₁₀₇	C₂	semidirect
8. #468×C₁₀₇	C₂	semidirect
9. C₄×SD₁₆×C₁₀₇	C₂	semidirect
10. #436×C₁₀₇	C₂	semidirect
11. #428×C₁₀₇	C₂	semidirect
121. #3357	C₁₀₇	direct (N × H)
122. C₁₀₇	#3357	direct (N × H)
124. #554	C₂×C₁₀₇	semidirect
125. #500	C₂×C₁₀₇	semidirect
126. #485	C₂×C₁₀₇	semidirect
127. #480	C₂×C₁₀₇	semidirect
128. #477	C₂×C₁₀₇	semidirect
129. #476	C₂×C₁₀₇	semidirect
130. #468	C₂×C₁₀₇	semidirect
131. C₄×SD₁₆	C₂×C₁₀₇	semidirect
132. #436	C₂×C₁₀₇	semidirect
133. #428	C₂×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#555×C₁₀₇	2
#554×C₁₀₇	2
#500×C₁₀₇	2
#485×C₁₀₇	2
#480×C₁₀₇	2
#477×C₁₀₇	2
#476×C₁₀₇	2
#468×C₁₀₇	2
C₄×SD₁₆×C₁₀₇	2
#436×C₁₀₇	2
#428×C₁₀₇	2
#3357	107

Subgroups — C107×#3357 #28334040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#3357 #28334040