Subgroups — C₁₀₇×#33853 #41059817

Order 27392.

696 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₀₇	C₂	1	2	yes	maximal, complemented
2	C₁₀₇×#3287	C₂	1	2	yes	maximal, complemented
3	C₁₀₇×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₇×#2215	C₂	1	2	yes	maximal, complemented
5	C₁₀₇×#2207	C₂	1	2	yes	maximal, complemented
6	C₁₀₇×#2186	C₂	1	4	yes	maximal, complemented
7	C₄×Q₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₀₇	C₂²	1	4	yes	—
10	#550×C₁₀₇	—	2	2	no	—
11	#508×C₁₀₇	—	2	8	no	—
12	C₂×Q₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₁₀₇	C₂²	1	4	yes	—
14	C₂×D₁₆×C₁₀₇	C₂²	1	2	yes	—
15	C₂×D₁₆×C₁₀₇	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₁₀₇	C₂²	1	2	yes	—
17	#493×C₁₀₇	C₂²	1	2	yes	—
18	#466×C₁₀₇	C₂²	1	4	yes	—
19	#460×C₁₀₇	—	2	2	no	—
20	#449×C₁₀₇	—	2	4	no	—
21	C₄×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
22	C₈×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
23	#366×C₁₀₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
25	#358×C₁₀₇	C₂²	1	2	yes	—
26	#357×C₁₀₇	C₂²	1	4	yes	—
27	C₂²⋊C₁₆×C₁₀₇	C₂²	1	2	yes	—
28	C₄×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
29	C₂×#41×C₁₀₇	—	2	4	no	—
30	#135×C₁₀₇	—	2	8	no	—
31	#135×C₁₀₇	—	4	4	no	—
32	C₂×Q₁₆×C₁₀₇	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₁₀₇	—	2	4	no	—
34	C₂×D₈×C₁₀₇	—	2	2	no	—
35	C₂×D₈×C₁₀₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
37	#127×C₁₀₇	—	2	2	no	—
38	#127×C₁₀₇	—	4	2	no	—
39	#121×C₁₀₇	—	2	4	no	—
40	#121×C₁₀₇	—	4	4	no	—
41	C₄×Q₈×C₁₀₇	—	2	2	no	—
42	C₄×D₄×C₁₀₇	—	4	2	no	—
43	C₄×D₄×C₁₀₇	D₄	1	1	yes	characteristic
44	Q₃₂×C₁₀₇	D₄	1	4	yes	complemented
45	SD₃₂×C₁₀₇	—	2	8	no	—
46	D₁₆×C₁₀₇	—	2	4	no	—
47	D₁₆×C₁₀₇	—	4	2	no	—
48	C₂×C₁₆×C₁₀₇	—	2	4	no	—
49	C₂×C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₁₀₇	C₂³	1	2	yes	—
51	C₂×C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
52	#107×C₁₀₇	C₂³	1	2	yes	—
53	C₄⋊C₈×C₁₀₇	D₄	1	1	yes	characteristic
54	#103×C₁₀₇	—	2	2	no	—
55	#102×C₁₀₇	—	2	4	no	—
56	C₂²⋊C₈×C₁₀₇	D₄	1	2	yes	—
57	C₄×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
58	#41×C₁₀₇	—	4	8	no	—
59	#41×C₁₀₇	—	8	4	no	—
60	C₂×Q₈×C₁₀₇	—	2	2	no	—
61	C₂×D₄×C₁₀₇	—	2	4	no	—
62	C₂×D₄×C₁₀₇	—	4	4	no	—
63	C₂×D₄×C₁₀₇	—	4	4	no	—
64	C₂×D₄×C₁₀₇	—	4	2	no	—
65	C₂×D₄×C₁₀₇	—	4	2	no	—
66	C₂×D₄×C₁₀₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₇	—	4	4	no	—
68	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
69	Q₁₆×C₁₀₇	—	2	2	no	—
70	Q₁₆×C₁₀₇	C₂×D₄	1	4	yes	—
71	SD₁₆×C₁₀₇	—	4	8	no	—
72	D₈×C₁₀₇	—	2	4	no	—
73	D₈×C₁₀₇	—	2	4	no	—
74	D₈×C₁₀₇	—	8	2	no	—
75	C₂×C₈×C₁₀₇	—	2	2	no	—
76	C₂×C₈×C₁₀₇	—	2	2	no	—
77	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₁₀₇	—	2	2	no	—
81	C₄⋊C₄×C₁₀₇	—	4	2	no	—
82	C₄⋊C₄×C₁₀₇	D₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
84	C₂²⋊C₄×C₁₀₇	D₈	1	2	yes	—
85	C₄²×C₁₀₇	—	4	2	no	—
86	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₁₀₇	—	2	2	no	—
88	C₁₆×C₁₀₇	—	2	1	no	—
89	C₁₆×C₁₀₇	C₂×D₄	1	2	yes	—
90	C₂³×C₁₀₇	—	4	4	no	—
91	C₂³×C₁₀₇	C₂×D₈	1	2	yes	—
92	Q₈×C₁₀₇	—	2	4	no	—
93	Q₈×C₁₀₇	—	4	2	no	—
94	D₄×C₁₀₇	—	2	2	no	—
95	D₄×C₁₀₇	—	4	4	no	—
96	D₄×C₁₀₇	—	4	4	no	—
97	D₄×C₁₀₇	—	8	8	no	—
98	D₄×C₁₀₇	—	8	4	no	—
99	D₄×C₁₀₇	—	16	2	no	—
100	D₄×C₁₀₇	—	16	4	no	—
101	C₂×C₄×C₁₀₇	—	2	2	no	—
102	C₂×C₄×C₁₀₇	—	2	2	no	—
103	C₂×C₄×C₁₀₇	—	4	2	no	—
104	C₂×C₄×C₁₀₇	—	4	2	no	—
105	C₂×C₄×C₁₀₇	—	8	8	no	—
106	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₁₀₇	—	2	2	no	—
111	C₈×C₁₀₇	—	2	1	no	—
112	C₈×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₁₀₇	—	2	2	no	—
115	C₂²×C₁₀₇	—	2	2	no	—
116	C₂²×C₁₀₇	—	2	2	no	—
117	C₂²×C₁₀₇	—	8	4	no	—
118	C₂²×C₁₀₇	—	8	4	no	—
119	C₂²×C₁₀₇	—	8	4	no	—
120	C₂²×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₁₀₇	—	2	1	no	—
122	C₄×C₁₀₇	—	2	2	no	—
123	C₄×C₁₀₇	—	4	4	no	—
124	C₄×C₁₀₇	—	8	2	no	—
125	C₄×C₁₀₇	#508	1	2	yes	complemented
126	C₄×C₁₀₇	D₄²	1	1	yes	characteristic
127	C₄×C₁₀₇	#576	1	1	yes	characteristic
128	#33853	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₇	—	4	2	no	—
130	C₂×C₁₀₇	—	16	4	no	—
131	C₂×C₁₀₇	D₄×D₈	1	1	yes	characteristic
132	C₂×C₁₀₇	C₂×#508	1	1	yes	characteristic
133	C₂×C₁₀₇	#3415	1	1	yes	characteristic
134	C₂×#508	C₂×C₁₀₇	1	2	yes	complemented
135	#3287	C₂×C₁₀₇	1	2	yes	complemented
136	#2246	C₂×C₁₀₇	1	1	yes	characteristic, complemented
137	#2215	C₂×C₁₀₇	1	2	yes	complemented
138	#2207	C₂×C₁₀₇	1	2	yes	complemented
139	#2186	C₂×C₁₀₇	1	4	yes	complemented
140	C₄×Q₃₂	C₂×C₁₀₇	1	1	yes	characteristic, complemented
141	D₄×C₁₆	C₂×C₁₀₇	1	1	yes	characteristic, complemented
142	C₁₀₇	#33853	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#135	C₂²×C₁₀₇	1	4	yes	—
144	#550	—	2	2	no	—
145	#508	—	2	8	no	—
146	C₂×Q₃₂	C₂²×C₁₀₇	1	1	yes	characteristic
147	C₂×SD₃₂	C₂²×C₁₀₇	1	4	yes	—
148	C₂×D₁₆	C₂²×C₁₀₇	1	2	yes	—
149	C₂×D₁₆	C₂²×C₁₀₇	1	2	yes	—
150	C₂²×C₁₆	C₂²×C₁₀₇	1	2	yes	—
151	#493	C₂²×C₁₀₇	1	2	yes	—
152	#466	C₂²×C₁₀₇	1	4	yes	—
153	#460	—	2	2	no	—
154	#449	—	2	4	no	—
155	C₄×Q₁₆	C₂²×C₁₀₇	1	2	yes	—
156	C₈×D₄	C₂²×C₁₀₇	1	1	yes	characteristic
157	#366	C₂²×C₁₀₇	1	1	yes	characteristic
158	C₄⋊C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
159	#358	C₂²×C₁₀₇	1	2	yes	—
160	#357	C₂²×C₁₀₇	1	4	yes	—
161	C₂²⋊C₁₆	C₂²×C₁₀₇	1	2	yes	—
162	C₄×C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
163	C₂×#41	—	2	4	no	—
164	#135	—	2	8	no	—
165	#135	—	4	4	no	—
166	C₂×Q₁₆	C₂³×C₁₀₇	1	2	yes	—
167	C₂×SD₁₆	—	2	4	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	C₂³×C₁₀₇	1	4	yes	—
170	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
171	#127	—	2	2	no	—
172	#127	—	4	2	no	—
173	#121	—	2	4	no	—
174	#121	—	4	4	no	—
175	C₄×Q₈	—	2	2	no	—
176	C₄×D₄	—	4	2	no	—
177	C₄×D₄	D₄×C₁₀₇	1	1	yes	characteristic
178	Q₃₂	D₄×C₁₀₇	1	4	yes	complemented
179	SD₃₂	—	2	8	no	—
180	D₁₆	—	2	4	no	—
181	D₁₆	—	4	2	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂³×C₁₀₇	1	1	yes	characteristic
184	C₂×C₁₆	C₂³×C₁₀₇	1	2	yes	—
185	C₂×C₁₆	C₂³×C₁₀₇	1	1	yes	characteristic
186	#107	C₂³×C₁₀₇	1	2	yes	—
187	C₄⋊C₈	D₄×C₁₀₇	1	1	yes	characteristic
188	#103	—	2	2	no	—
189	#102	—	2	4	no	—
190	C₂²⋊C₈	D₄×C₁₀₇	1	2	yes	—
191	C₄×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
192	#41	—	4	8	no	—
193	#41	—	8	4	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	D₈×C₁₀₇	1	1	yes	characteristic
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
203	Q₁₆	—	2	2	no	—
204	Q₁₆	C₂×D₄×C₁₀₇	1	4	yes	—
205	SD₁₆	—	4	8	no	—
206	D₈	—	2	4	no	—
207	D₈	—	2	4	no	—
208	D₈	—	8	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
213	C₂×C₈	C₂⁴×C₁₀₇	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	D₈×C₁₀₇	1	1	yes	characteristic
217	C₂²⋊C₄	—	4	4	no	—
218	C₂²⋊C₄	D₈×C₁₀₇	1	2	yes	—
219	C₄²	—	4	2	no	—
220	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	C₂×D₄×C₁₀₇	1	2	yes	—
224	C₂³	—	4	4	no	—
225	C₂³	C₂×D₈×C₁₀₇	1	2	yes	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	8	8	no	—
232	D₄	—	8	4	no	—
233	D₄	—	16	2	no	—
234	D₄	—	16	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	8	no	—
240	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
241	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
247	C₈	H(5,𝔽₂)×C₁₀₇	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	C₂²×D₈×C₁₀₇	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	8	2	no	—
259	C₄	#508×C₁₀₇	1	2	yes	complemented
260	C₄	D₄²×C₁₀₇	1	1	yes	characteristic
261	C₄	#576×C₁₀₇	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	16	4	no	—
264	C₂	D₄×D₈×C₁₀₇	1	1	yes	characteristic
265	C₂	C₂×#508×C₁₀₇	1	1	yes	characteristic
266	C₂	C₁₀₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₄×D₄×C₁₀₇ ◁ C₈×D₄×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₁₀₇	C₂	semidirect
2. C₁₀₇×#3287	C₂	semidirect
3. C₁₀₇×#2246	C₂	semidirect
4. C₁₀₇×#2215	C₂	semidirect
5. C₁₀₇×#2207	C₂	semidirect
6. C₁₀₇×#2186	C₂	semidirect
7. C₄×Q₃₂×C₁₀₇	C₂	semidirect
8. D₄×C₁₆×C₁₀₇	C₂	semidirect
44. Q₃₂×C₁₀₇	D₄	semidirect
125. C₄×C₁₀₇	#508	semidirect
128. #33853	C₁₀₇	direct (N × H)
134. C₂×#508	C₂×C₁₀₇	semidirect
135. #3287	C₂×C₁₀₇	semidirect
136. #2246	C₂×C₁₀₇	semidirect
137. #2215	C₂×C₁₀₇	semidirect
138. #2207	C₂×C₁₀₇	semidirect
139. #2186	C₂×C₁₀₇	semidirect
140. C₄×Q₃₂	C₂×C₁₀₇	semidirect
141. D₄×C₁₆	C₂×C₁₀₇	semidirect
142. C₁₀₇	#33853	direct (N × H)
178. Q₃₂	D₄×C₁₀₇	semidirect
259. C₄	#508×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₀₇	2
C₁₀₇×#3287	2
C₁₀₇×#2246	2
C₁₀₇×#2215	2
C₁₀₇×#2207	2
C₁₀₇×#2186	2
C₄×Q₃₂×C₁₀₇	2
D₄×C₁₆×C₁₀₇	2
#33853	107

Subgroups — C107×#33853 #41059817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#33853 #41059817