Subgroups — C₁₀₇×#29209 #41055173

Order 27392.

686 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#2058	C₂	1	2	yes	maximal
4	C₁₀₇×#2057	C₂	1	2	yes	maximal
5	C₁₀₇×#1991	C₂	1	2	yes	maximal
6	C₁₀₇×#1949	C₂	1	4	yes	maximal
7	C₁₀₇×#1846	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
10	C₂×#125×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₀₇	C₂²	1	4	yes	—
12	C₂×#106×C₁₀₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
15	#391×C₁₀₇	—	2	2	no	—
16	#391×C₁₀₇	—	2	2	no	—
17	#389×C₁₀₇	C₂²	1	2	yes	—
18	#387×C₁₀₇	—	2	2	no	—
19	#382×C₁₀₇	C₂²	1	2	yes	—
20	#382×C₁₀₇	C₂²	1	2	yes	—
21	#382×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₀₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
24	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
25	#340×C₁₀₇	C₂²	1	4	yes	—
26	#340×C₁₀₇	C₂²	1	4	yes	—
27	#336×C₁₀₇	C₂²	1	1	yes	characteristic
28	#336×C₁₀₇	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₀₇	—	2	2	no	—
30	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₁₀₇	—	4	4	no	—
33	#125×C₁₀₇	—	2	2	no	—
34	#125×C₁₀₇	Q₈	1	4	yes	complemented
35	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₀₇	—	2	2	no	—
42	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
43	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
47	#107×C₁₀₇	—	4	4	no	—
48	#106×C₁₀₇	—	4	4	no	—
49	C₄⋊C₈×C₁₀₇	Q₈	1	8	yes	complemented
50	C₄×C₈×C₁₀₇	—	2	4	no	—
51	#95×C₁₀₇	—	2	2	no	—
52	#95×C₁₀₇	—	2	2	no	—
53	#95×C₁₀₇	—	2	2	no	—
54	#95×C₁₀₇	—	2	4	no	—
55	#95×C₁₀₇	—	2	2	no	—
56	#95×C₁₀₇	—	2	2	no	—
57	C₂×Q₈×C₁₀₇	—	4	4	no	—
58	C₂×Q₈×C₁₀₇	—	4	4	no	—
59	C₂×Q₈×C₁₀₇	—	4	4	no	—
60	C₂²×C₄×C₁₀₇	—	2	2	no	—
61	C₂²×C₄×C₁₀₇	—	2	2	no	—
62	C₂²×C₄×C₁₀₇	—	2	4	no	—
63	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₁₀₇	—	2	2	no	—
69	C₂×C₈×C₁₀₇	—	2	2	no	—
70	C₂×C₈×C₁₀₇	—	2	2	no	—
71	C₂×C₈×C₁₀₇	—	2	2	no	—
72	C₂×C₈×C₁₀₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₁₀₇	C₂×Q₈	1	4	yes	—
74	C₄⋊C₄×C₁₀₇	—	2	8	no	—
75	C₄⋊C₄×C₁₀₇	—	2	2	no	—
76	C₄⋊C₄×C₁₀₇	—	2	4	no	—
77	C₄⋊C₄×C₁₀₇	—	2	4	no	—
78	C₄⋊C₄×C₁₀₇	—	2	4	no	—
79	C₄⋊C₄×C₁₀₇	—	4	4	no	—
80	C₄⋊C₄×C₁₀₇	—	4	4	no	—
81	C₄⋊C₄×C₁₀₇	C₂×Q₈	1	4	yes	—
82	C₄²×C₁₀₇	—	2	4	no	—
83	C₄²×C₁₀₇	—	2	2	no	—
84	C₄²×C₁₀₇	—	4	2	no	—
85	C₄²×C₁₀₇	—	4	4	no	—
86	C₄²×C₁₀₇	C₂×Q₈	1	2	yes	—
87	C₄²×C₁₀₇	C₂×Q₈	1	4	yes	—
88	C₄²×C₁₀₇	#41	1	2	yes	—
89	C₂³×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₁₀₇	—	8	8	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	2	no	—
96	C₂×C₄×C₁₀₇	—	2	2	no	—
97	C₂×C₄×C₁₀₇	—	2	4	no	—
98	C₂×C₄×C₁₀₇	—	4	2	no	—
99	C₂×C₄×C₁₀₇	—	4	2	no	—
100	C₂×C₄×C₁₀₇	—	4	4	no	—
101	C₂×C₄×C₁₀₇	—	4	2	no	—
102	C₂×C₄×C₁₀₇	—	4	2	no	—
103	C₂×C₄×C₁₀₇	—	4	4	no	—
104	C₂×C₄×C₁₀₇	—	4	2	no	—
105	C₂×C₄×C₁₀₇	—	4	4	no	—
106	C₂×C₄×C₁₀₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₀₇	#135	1	4	yes	—
110	C₂×C₄×C₁₀₇	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₀₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₀₇	H(5,𝔽₂)	1	2	yes	—
114	C₈×C₁₀₇	—	2	4	no	—
115	C₈×C₁₀₇	—	2	4	no	—
116	C₂²×C₁₀₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
117	C₂²×C₁₀₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₂²×C₁₀₇	#535	1	1	yes	characteristic
119	C₂²×C₁₀₇	C₂×#135	1	2	yes	—
120	C₂²×C₁₀₇	#575	1	2	yes	—
121	C₄×C₁₀₇	—	2	4	no	—
122	C₄×C₁₀₇	—	2	2	no	—
123	C₄×C₁₀₇	—	2	2	no	—
124	C₄×C₁₀₇	—	2	4	no	—
125	C₄×C₁₀₇	—	4	4	no	—
126	C₄×C₁₀₇	—	8	2	no	—
127	C₄×C₁₀₇	—	8	4	no	—
128	C₄×C₁₀₇	#554	1	2	yes	—
129	C₄×C₁₀₇	Q₈²	1	2	yes	—
130	#29209	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₁₀₇	#2567	1	1	yes	characteristic
132	C₂×C₁₀₇	#3076	1	2	yes	—
133	C₂×C₁₀₇	#3160	1	1	yes	characteristic
134	C₂×C₁₀₇	#3166	1	1	yes	characteristic
135	C₂×C₁₀₇	#3211	1	2	yes	—
136	#2862	C₂×C₁₀₇	1	1	yes	characteristic
137	C₄×#125	C₂×C₁₀₇	1	1	yes	characteristic
138	#2058	C₂×C₁₀₇	1	2	yes	—
139	#2057	C₂×C₁₀₇	1	2	yes	—
140	#1991	C₂×C₁₀₇	1	2	yes	—
141	#1949	C₂×C₁₀₇	1	4	yes	—
142	#1846	C₂×C₁₀₇	1	2	yes	—
143	C₄×C₄⋊C₈	C₂×C₁₀₇	1	1	yes	characteristic
144	C₁₀₇	#29209	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×C₄⋊Q₈	C₂²×C₁₀₇	1	2	yes	—
146	C₂×#125	C₂²×C₁₀₇	1	1	yes	characteristic
147	C₂×#107	C₂²×C₁₀₇	1	4	yes	—
148	C₂×#106	C₂²×C₁₀₇	1	4	yes	—
149	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₁₀₇	1	2	yes	—
151	#391	—	2	2	no	—
152	#391	—	2	2	no	—
153	#389	C₂²×C₁₀₇	1	2	yes	—
154	#387	—	2	2	no	—
155	#382	C₂²×C₁₀₇	1	1	yes	characteristic
156	#382	C₂²×C₁₀₇	1	2	yes	—
157	#382	C₂²×C₁₀₇	1	2	yes	—
158	C₄×C₄⋊C₄	—	2	2	no	—
159	C₄×C₄⋊C₄	C₂²×C₁₀₇	1	2	yes	—
160	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
161	#340	C₂²×C₁₀₇	1	4	yes	—
162	#340	C₂²×C₁₀₇	1	4	yes	—
163	#336	C₂²×C₁₀₇	1	1	yes	characteristic
164	#336	C₂²×C₁₀₇	1	1	yes	characteristic
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
168	C₄⋊Q₈	—	4	4	no	—
169	#125	—	2	2	no	—
170	#125	Q₈×C₁₀₇	1	4	yes	complemented
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	4	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
179	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
180	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
183	#107	—	4	4	no	—
184	#106	—	4	4	no	—
185	C₄⋊C₈	Q₈×C₁₀₇	1	8	yes	complemented
186	C₄×C₈	—	2	4	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	4	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
203	C₂²×C₄	C₂⁴×C₁₀₇	1	1	yes	characteristic
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂×Q₈×C₁₀₇	1	4	yes	—
209	C₂×C₈	C₂×Q₈×C₁₀₇	1	4	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂×Q₈×C₁₀₇	1	4	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	C₂×Q₈×C₁₀₇	1	2	yes	—
223	C₄²	C₂×Q₈×C₁₀₇	1	4	yes	—
224	C₄²	#41×C₁₀₇	1	2	yes	—
225	C₂³	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
226	Q₈	—	8	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₂²⋊Q₈×C₁₀₇	1	4	yes	—
243	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	2	yes	—
244	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	2	yes	—
245	C₂×C₄	#135×C₁₀₇	1	4	yes	—
246	C₂×C₄	C₂²×Q₈×C₁₀₇	1	2	yes	—
247	C₂×C₄	C₂²×Q₈×C₁₀₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₁₀₇	1	1	yes	characteristic
249	C₂×C₄	H(5,𝔽₂)×C₁₀₇	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂×C₂²⋊Q₈×C₁₀₇	1	1	yes	characteristic
253	C₂²	C₂×C₄⋊Q₈×C₁₀₇	1	1	yes	characteristic
254	C₂²	#535×C₁₀₇	1	1	yes	characteristic
255	C₂²	C₂×#135×C₁₀₇	1	2	yes	—
256	C₂²	#575×C₁₀₇	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	#554×C₁₀₇	1	2	yes	—
265	C₄	Q₈²×C₁₀₇	1	2	yes	—
266	C₂	C₁₀₇×#2567	1	1	yes	characteristic
267	C₂	C₁₀₇×#3076	1	2	yes	—
268	C₂	C₁₀₇×#3160	1	1	yes	characteristic
269	C₂	C₁₀₇×#3166	1	1	yes	characteristic
270	C₂	C₁₀₇×#3211	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
34. #125×C₁₀₇	Q₈	semidirect
49. C₄⋊C₈×C₁₀₇	Q₈	semidirect
130. #29209	C₁₀₇	direct (N × H)
144. C₁₀₇	#29209	direct (N × H)
170. #125	Q₈×C₁₀₇	semidirect
185. C₄⋊C₈	Q₈×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₇×#2862	2
C₄×#125×C₁₀₇	2
C₁₀₇×#2058	2
C₁₀₇×#2057	2
C₁₀₇×#1991	2
C₁₀₇×#1949	2
C₁₀₇×#1846	2
C₄×C₄⋊C₈×C₁₀₇	2
#29209	107

Subgroups — C107×#29209 #41055173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#29209 #41055173