Subgroups — C₁₀₇×#9704 #41035668

Order 27392.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#2858	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#332×C₁₀₇	C₂	1	2	yes	maximal, complemented
5	C₂×#331×C₁₀₇	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂×#127×C₁₀₇	C₂²	1	1	yes	characteristic
8	C₂×#124×C₁₀₇	—	2	1	no	—
9	C₂×#124×C₁₀₇	—	2	1	no	—
10	C₂×C₄⋊C₈×C₁₀₇	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	2	yes	—
13	C₂×#97×C₁₀₇	—	2	1	no	—
14	#390×C₁₀₇	—	2	2	no	—
15	#390×C₁₀₇	C₄	1	4	yes	—
16	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
17	#336×C₁₀₇	—	2	1	no	—
18	#332×C₁₀₇	—	2	4	no	—
19	#331×C₁₀₇	—	2	4	no	—
20	C₂²×Q₈×C₁₀₇	—	2	1	no	—
21	C₂²×D₄×C₁₀₇	—	2	1	no	—
22	C₂²×D₄×C₁₀₇	—	2	1	no	—
23	C₂²×D₄×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₀₇	—	2	2	no	—
25	C₂²×C₈×C₁₀₇	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
27	C₄⋊Q₈×C₁₀₇	—	2	2	no	—
28	#127×C₁₀₇	—	2	2	no	—
29	#127×C₁₀₇	—	2	2	no	—
30	#124×C₁₀₇	—	4	4	no	—
31	#124×C₁₀₇	—	4	2	no	—
32	#124×C₁₀₇	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₀₇	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₁₀₇	—	2	4	no	—
37	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
39	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₁₀₇	—	2	2	no	—
43	C₄⋊C₈×C₁₀₇	—	2	2	no	—
44	C₄⋊C₈×C₁₀₇	—	2	2	no	—
45	C₄⋊C₈×C₁₀₇	—	4	4	no	—
46	C₄⋊C₈×C₁₀₇	D₄	1	4	yes	—
47	#97×C₁₀₇	—	4	2	no	—
48	C₂⁴×C₁₀₇	—	2	2	no	—
49	C₂×Q₈×C₁₀₇	—	4	2	no	—
50	C₂×Q₈×C₁₀₇	—	4	1	no	—
51	C₂×Q₈×C₁₀₇	—	4	1	no	—
52	C₂×Q₈×C₁₀₇	—	4	2	no	—
53	C₂×D₄×C₁₀₇	—	2	4	no	—
54	C₂×D₄×C₁₀₇	—	4	4	no	—
55	C₂×D₄×C₁₀₇	—	4	2	no	—
56	C₂×D₄×C₁₀₇	—	4	1	no	—
57	C₂×D₄×C₁₀₇	—	4	1	no	—
58	C₂×D₄×C₁₀₇	—	4	1	no	—
59	C₂×D₄×C₁₀₇	—	4	1	no	—
60	C₂×D₄×C₁₀₇	—	4	2	no	—
61	C₂×D₄×C₁₀₇	—	4	2	no	—
62	C₂×D₄×C₁₀₇	—	4	2	no	—
63	C₂²×C₄×C₁₀₇	—	2	2	no	—
64	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₀₇	—	2	4	no	—
70	C₂×C₈×C₁₀₇	—	4	2	no	—
71	C₂×C₈×C₁₀₇	—	4	2	no	—
72	C₂×C₈×C₁₀₇	—	4	2	no	—
73	C₂×C₈×C₁₀₇	—	4	4	no	—
74	C₄⋊C₄×C₁₀₇	—	2	4	no	—
75	C₄⋊C₄×C₁₀₇	—	4	1	no	—
76	C₄⋊C₄×C₁₀₇	—	4	1	no	—
77	C₂²⋊C₄×C₁₀₇	—	4	8	no	—
78	C₂²⋊C₄×C₁₀₇	—	4	8	no	—
79	C₄²×C₁₀₇	—	2	2	no	—
80	C₄²×C₁₀₇	—	2	2	no	—
81	C₄²×C₁₀₇	—	2	2	no	—
82	C₄²×C₁₀₇	—	4	2	no	—
83	C₄²×C₁₀₇	D₈	1	2	yes	—
84	C₄²×C₁₀₇	SD₁₆	1	2	yes	—
85	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₀₇	#41	1	2	yes	—
87	C₂³×C₁₀₇	—	4	2	no	—
88	C₂³×C₁₀₇	—	4	4	no	—
89	C₂³×C₁₀₇	—	4	2	no	—
90	C₂³×C₁₀₇	—	4	4	no	—
91	C₂³×C₁₀₇	—	4	2	no	—
92	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	Q₈×C₁₀₇	—	8	2	no	—
94	Q₈×C₁₀₇	—	8	2	no	—
95	D₄×C₁₀₇	—	8	4	no	—
96	D₄×C₁₀₇	—	8	2	no	—
97	D₄×C₁₀₇	—	8	2	no	—
98	D₄×C₁₀₇	—	8	2	no	—
99	D₄×C₁₀₇	—	8	1	no	—
100	D₄×C₁₀₇	—	8	1	no	—
101	C₂×C₄×C₁₀₇	—	2	2	no	—
102	C₂×C₄×C₁₀₇	—	2	2	no	—
103	C₂×C₄×C₁₀₇	—	2	2	no	—
104	C₂×C₄×C₁₀₇	—	2	2	no	—
105	C₂×C₄×C₁₀₇	—	2	2	no	—
106	C₂×C₄×C₁₀₇	—	2	2	no	—
107	C₂×C₄×C₁₀₇	—	4	2	no	—
108	C₂×C₄×C₁₀₇	—	4	2	no	—
109	C₂×C₄×C₁₀₇	—	4	2	no	—
110	C₂×C₄×C₁₀₇	#102	1	4	yes	—
111	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₇	#121	1	2	yes	—
113	C₂×C₄×C₁₀₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₀₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₀₇	#137	1	2	yes	—
120	C₈×C₁₀₇	—	4	4	no	—
121	C₈×C₁₀₇	—	8	2	no	—
122	C₂²×C₁₀₇	—	8	4	no	—
123	C₂²×C₁₀₇	—	8	4	no	—
124	C₂²×C₁₀₇	—	8	2	no	—
125	C₂²×C₁₀₇	—	8	2	no	—
126	C₂²×C₁₀₇	—	8	2	no	—
127	C₂²×C₁₀₇	#390	1	1	yes	characteristic
128	C₂²×C₁₀₇	#413	1	2	yes	—
129	C₂²×C₁₀₇	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₀₇	#417	1	2	yes	—
131	C₂²×C₁₀₇	#418	1	1	yes	characteristic
132	C₄×C₁₀₇	—	2	4	no	—
133	C₄×C₁₀₇	—	2	2	no	—
134	C₄×C₁₀₇	—	4	2	no	—
135	C₄×C₁₀₇	—	8	2	no	—
136	C₄×C₁₀₇	#459	1	2	yes	—
137	C₄×C₁₀₇	#460	1	2	yes	—
138	C₄×C₁₀₇	#489	1	2	yes	—
139	C₄×C₁₀₇	#496	1	1	yes	characteristic
140	C₄×C₁₀₇	#497	1	1	yes	characteristic
141	#9704	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₁₀₇	—	16	2	no	—
143	C₂×C₁₀₇	#1547	1	2	yes	—
144	C₂×C₁₀₇	#1557	1	2	yes	—
145	C₂×C₁₀₇	#1959	1	1	yes	characteristic
146	C₂×C₁₀₇	#1967	1	1	yes	characteristic
147	C₂×C₁₀₇	#1982	1	1	yes	characteristic
148	#2858	C₂×C₁₀₇	1	1	yes	characteristic
149	#1842	C₂×C₁₀₇	1	1	yes	characteristic, complemented
150	#1767	C₂×C₁₀₇	1	1	yes	characteristic, complemented
151	C₂×#332	C₂×C₁₀₇	1	2	yes	complemented
152	C₂×#331	C₂×C₁₀₇	1	2	yes	—
153	C₁₀₇	#9704	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×C₄⋊Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
155	C₂×#127	C₂²×C₁₀₇	1	1	yes	characteristic
156	C₂×#124	—	2	1	no	—
157	C₂×#124	—	2	1	no	—
158	C₂×C₄⋊C₈	—	2	2	no	—
159	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	2	yes	—
161	C₂×#97	—	2	1	no	—
162	#390	—	2	2	no	—
163	#390	C₄×C₁₀₇	1	4	yes	—
164	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
165	#336	—	2	1	no	—
166	#332	—	2	4	no	—
167	#331	—	2	4	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	C₂×C₄×C₁₀₇	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	#127	—	2	2	no	—
177	#127	—	2	2	no	—
178	#124	—	4	4	no	—
179	#124	—	4	2	no	—
180	#124	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
186	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
188	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	4	4	no	—
194	C₄⋊C₈	D₄×C₁₀₇	1	4	yes	—
195	#97	—	4	2	no	—
196	C₂⁴	—	2	2	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂×Q₈	—	4	1	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	D₈×C₁₀₇	1	2	yes	—
232	C₄²	SD₁₆×C₁₀₇	1	2	yes	—
233	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
234	C₄²	#41×C₁₀₇	1	2	yes	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	2	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	2	no	—
240	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
241	Q₈	—	8	2	no	—
242	Q₈	—	8	2	no	—
243	D₄	—	8	4	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	1	no	—
248	D₄	—	8	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	#102×C₁₀₇	1	4	yes	—
259	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
260	C₂×C₄	#121×C₁₀₇	1	2	yes	—
261	C₂×C₄	#124×C₁₀₇	1	1	yes	characteristic
262	C₂×C₄	#127×C₁₀₇	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
264	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	1	yes	characteristic
265	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
266	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
267	C₂×C₄	#137×C₁₀₇	1	2	yes	—
268	C₈	—	4	4	no	—
269	C₈	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	#390×C₁₀₇	1	1	yes	characteristic
276	C₂²	#413×C₁₀₇	1	2	yes	—
277	C₂²	C₂×#102×C₁₀₇	1	1	yes	characteristic
278	C₂²	#417×C₁₀₇	1	2	yes	—
279	C₂²	#418×C₁₀₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	8	2	no	—
284	C₄	#459×C₁₀₇	1	2	yes	—
285	C₄	#460×C₁₀₇	1	2	yes	—
286	C₄	#489×C₁₀₇	1	2	yes	—
287	C₄	#496×C₁₀₇	1	1	yes	characteristic
288	C₄	#497×C₁₀₇	1	1	yes	characteristic
289	C₂	—	16	2	no	—
290	C₂	C₁₀₇×#1547	1	2	yes	—
291	C₂	C₁₀₇×#1557	1	2	yes	—
292	C₂	C₁₀₇×#1959	1	1	yes	characteristic
293	C₂	C₁₀₇×#1967	1	1	yes	characteristic
294	C₂	C₁₀₇×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
prime radical	C₂×#127×C₁₀₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₇×#1842	C₂	semidirect
3. C₁₀₇×#1767	C₂	semidirect
4. C₂×#332×C₁₀₇	C₂	semidirect
141. #9704	C₁₀₇	direct (N × H)
149. #1842	C₂×C₁₀₇	semidirect
150. #1767	C₂×C₁₀₇	semidirect
151. C₂×#332	C₂×C₁₀₇	semidirect
153. C₁₀₇	#9704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₇×#2858	2
C₁₀₇×#1842	2
C₁₀₇×#1767	2
C₂×#332×C₁₀₇	2
C₂×#331×C₁₀₇	2
#9704	107

Subgroups — C107×#9704 #41035668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#9704 #41035668