Subgroups — C₁₀₇×#34258 #41060222

Order 27392.

5682 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#41×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₇×#2341	C₂	1	4	yes	maximal, complemented
5	D₄×C₂²⋊C₄×C₁₀₇	C₂	1	12	yes	maximal, complemented
6	C₁₀₇×#2311	C₂	1	3	yes	maximal, complemented
7	C₁₀₇×#2309	C₂	1	3	yes	maximal, complemented
8	C₁₀₇×#2297	C₂	1	3	yes	maximal, complemented
9	C₂×#390×C₁₀₇	C₂	1	3	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₁₀₇	—	2	8	no	—
11	C₂×H(5,𝔽₂)×C₁₀₇	—	2	8	no	—
12	C₂×H(5,𝔽₂)×C₁₀₇	C₄	1	16	yes	complemented
13	C₂²×#41×C₁₀₇	C₂²	1	3	yes	—
14	C₂²×#41×C₁₀₇	C₂²	1	2	yes	—
15	C₂²×#41×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₁₀₇	C₂²	1	3	yes	—
17	C₂³×D₄×C₁₀₇	C₂²	1	6	yes	—
18	#520×C₁₀₇	—	2	8	no	—
19	C₄×#41×C₁₀₇	—	2	8	no	—
20	C₂×C₄×Q₈×C₁₀₇	C₂²	1	2	yes	complemented
21	C₂×C₄×Q₈×C₁₀₇	C₂²	1	2	yes	complemented
22	C₂×C₄×D₄×C₁₀₇	C₂²	1	6	yes	complemented
23	C₂×C₄×D₄×C₁₀₇	C₂²	1	6	yes	complemented
24	C₂×#117×C₁₀₇	C₂²	1	3	yes	complemented
25	C₂×#117×C₁₀₇	C₂²	1	3	yes	complemented
26	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	3	yes	complemented
27	C₂²×C₂²⋊C₄×C₁₀₇	C₂²	1	12	yes	complemented
28	C₂²×C₄²×C₁₀₇	C₂²	1	3	yes	complemented
29	#391×C₁₀₇	C₂²	1	1	yes	characteristic, complemented
30	#391×C₁₀₇	C₂²	1	3	yes	complemented
31	#390×C₁₀₇	C₂²	1	3	yes	complemented
32	#390×C₁₀₇	C₂²	1	6	yes	complemented
33	#390×C₁₀₇	C₂²	1	12	yes	complemented
34	#390×C₁₀₇	C₂²	1	3	yes	complemented
35	#390×C₁₀₇	C₂²	1	12	yes	complemented
36	#386×C₁₀₇	C₂²	1	12	yes	complemented
37	#386×C₁₀₇	C₂²	1	12	yes	complemented
38	#380×C₁₀₇	C₂²	1	12	yes	complemented
39	#379×C₁₀₇	C₂²	1	12	yes	complemented
40	C₄×C₂²⋊C₄×C₁₀₇	C₂²	1	12	yes	complemented
41	C₂⁵×C₁₀₇	C₂³	1	6	yes	—
42	H(5,𝔽₂)×C₁₀₇	—	2	128	no	—
43	C₂×#41×C₁₀₇	—	2	8	no	—
44	C₂×#41×C₁₀₇	—	2	8	no	—
45	C₂×#41×C₁₀₇	—	2	12	no	—
46	C₂×#41×C₁₀₇	—	2	12	no	—
47	C₂×#41×C₁₀₇	C₂×C₄	1	8	yes	—
48	C₂×#41×C₁₀₇	C₂×C₄	1	16	yes	complemented
49	C₂×#41×C₁₀₇	C₂×C₄	1	24	yes	complemented
50	C₂×#41×C₁₀₇	D₄	1	8	yes	—
51	C₂×#41×C₁₀₇	D₄	1	8	yes	—
52	C₂²×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₁₀₇	—	2	24	no	—
55	C₂²×D₄×C₁₀₇	—	2	24	no	—
56	C₂²×D₄×C₁₀₇	—	2	12	no	—
57	C₂²×D₄×C₁₀₇	—	2	12	no	—
58	C₂²×D₄×C₁₀₇	C₂×C₄	1	48	yes	complemented
59	C₂²×D₄×C₁₀₇	C₂×C₄	1	24	yes	complemented
60	C₂²×D₄×C₁₀₇	C₂³	1	6	yes	—
61	C₂²×D₄×C₁₀₇	C₂³	1	3	yes	—
62	C₂²×D₄×C₁₀₇	C₂³	1	6	yes	—
63	C₂²×D₄×C₁₀₇	C₂³	1	3	yes	—
64	C₂³×C₄×C₁₀₇	C₂³	1	6	yes	—
65	C₂³×C₄×C₁₀₇	C₂³	1	6	yes	—
66	C₂³×C₄×C₁₀₇	C₂³	1	6	yes	—
67	C₂³×C₄×C₁₀₇	C₂³	1	3	yes	—
68	C₄×Q₈×C₁₀₇	—	2	8	no	—
69	C₄×Q₈×C₁₀₇	—	2	8	no	—
70	C₄×D₄×C₁₀₇	—	2	24	no	—
71	C₄×D₄×C₁₀₇	—	2	24	no	—
72	#117×C₁₀₇	—	2	12	no	—
73	#117×C₁₀₇	—	2	12	no	—
74	C₂×C₄⋊C₄×C₁₀₇	—	2	12	no	—
75	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	6	yes	—
76	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	6	yes	—
77	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	3	yes	—
78	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₁₀₇	—	4	24	no	—
80	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	12	yes	—
81	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	6	yes	—
82	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	6	yes	—
83	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	12	yes	—
84	C₂×C₂²⋊C₄×C₁₀₇	C₂³	1	24	yes	—
85	C₂×C₄²×C₁₀₇	—	2	12	no	—
86	C₂×C₄²×C₁₀₇	C₂³	1	6	yes	—
87	C₂×C₄²×C₁₀₇	C₂³	1	6	yes	—
88	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₁₀₇	C₂³	1	3	yes	—
90	#95×C₁₀₇	C₂³	1	4	yes	—
91	#95×C₁₀₇	C₂³	1	12	yes	—
92	C₂⁴×C₁₀₇	—	2	12	no	—
93	C₂⁴×C₁₀₇	—	2	12	no	—
94	C₂⁴×C₁₀₇	—	4	12	no	—
95	C₂⁴×C₁₀₇	—	4	12	no	—
96	C₂⁴×C₁₀₇	C₂²×C₄	1	24	yes	—
97	C₂⁴×C₁₀₇	C₂⁴	1	6	yes	—
98	C₂⁴×C₁₀₇	C₂⁴	1	3	yes	—
99	#41×C₁₀₇	—	2	64	no	—
100	#41×C₁₀₇	—	2	96	no	—
101	#41×C₁₀₇	C₂²⋊C₄	1	64	yes	complemented
102	C₂×Q₈×C₁₀₇	—	2	2	no	—
103	C₂×Q₈×C₁₀₇	—	2	2	no	—
104	C₂×Q₈×C₁₀₇	C₂²×C₄	1	4	yes	—
105	C₂×Q₈×C₁₀₇	C₂²×C₄	1	4	yes	—
106	C₂×Q₈×C₁₀₇	C₂×D₄	1	4	yes	—
107	C₂×Q₈×C₁₀₇	C₂×D₄	1	4	yes	—
108	C₂×D₄×C₁₀₇	—	2	12	no	—
109	C₂×D₄×C₁₀₇	—	2	12	no	—
110	C₂×D₄×C₁₀₇	—	2	192	no	—
111	C₂×D₄×C₁₀₇	—	2	96	no	—
112	C₂×D₄×C₁₀₇	—	2	6	no	—
113	C₂×D₄×C₁₀₇	—	2	12	no	—
114	C₂×D₄×C₁₀₇	—	2	12	no	—
115	C₂×D₄×C₁₀₇	—	2	6	no	—
116	C₂×D₄×C₁₀₇	C₂²×C₄	1	24	yes	—
117	C₂×D₄×C₁₀₇	C₂²×C₄	1	24	yes	—
118	C₂×D₄×C₁₀₇	C₂²×C₄	1	12	yes	—
119	C₂×D₄×C₁₀₇	C₂²×C₄	1	12	yes	—
120	C₂×D₄×C₁₀₇	C₂×D₄	1	12	yes	—
121	C₂×D₄×C₁₀₇	C₂×D₄	1	12	yes	—
122	C₂²×C₄×C₁₀₇	—	2	12	no	—
123	C₂²×C₄×C₁₀₇	—	2	12	no	—
124	C₂²×C₄×C₁₀₇	—	2	12	no	—
125	C₂²×C₄×C₁₀₇	—	2	12	no	—
126	C₂²×C₄×C₁₀₇	—	2	12	no	—
127	C₂²×C₄×C₁₀₇	—	4	6	no	—
128	C₂²×C₄×C₁₀₇	—	4	12	no	—
129	C₂²×C₄×C₁₀₇	C₂²×C₄	1	24	yes	—
130	C₂²×C₄×C₁₀₇	C₂²×C₄	1	12	yes	—
131	C₂²×C₄×C₁₀₇	C₂×D₄	1	12	yes	—
132	C₂²×C₄×C₁₀₇	C₂×D₄	1	12	yes	—
133	C₂²×C₄×C₁₀₇	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₁₀₇	C₂⁴	1	6	yes	—
135	C₂²×C₄×C₁₀₇	C₂⁴	1	6	yes	—
136	C₂²×C₄×C₁₀₇	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₁₀₇	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
139	C₂²×C₄×C₁₀₇	C₂⁴	1	3	yes	—
140	C₄⋊C₄×C₁₀₇	—	2	6	no	—
141	C₄⋊C₄×C₁₀₇	—	2	12	no	—
142	C₄⋊C₄×C₁₀₇	—	2	2	no	—
143	C₄⋊C₄×C₁₀₇	—	2	12	no	—
144	C₂²⋊C₄×C₁₀₇	—	2	12	no	—
145	C₂²⋊C₄×C₁₀₇	—	2	12	no	—
146	C₂²⋊C₄×C₁₀₇	—	2	24	no	—
147	C₂²⋊C₄×C₁₀₇	—	4	24	no	—
148	C₂²⋊C₄×C₁₀₇	—	4	12	no	—
149	C₄²×C₁₀₇	—	2	2	no	—
150	C₄²×C₁₀₇	—	2	6	no	—
151	C₄²×C₁₀₇	—	2	12	no	—
152	C₄²×C₁₀₇	—	2	12	no	—
153	C₂³×C₁₀₇	—	2	6	no	—
154	C₂³×C₁₀₇	—	2	6	no	—
155	C₂³×C₁₀₇	—	2	6	no	—
156	C₂³×C₁₀₇	—	2	6	no	—
157	C₂³×C₁₀₇	—	2	12	no	—
158	C₂³×C₁₀₇	—	2	12	no	—
159	C₂³×C₁₀₇	—	2	48	no	—
160	C₂³×C₁₀₇	—	4	48	no	—
161	C₂³×C₁₀₇	—	8	24	no	—
162	C₂³×C₁₀₇	—	8	12	no	—
163	C₂³×C₁₀₇	—	8	12	no	—
164	C₂³×C₁₀₇	C₂³×C₄	1	12	yes	—
165	C₂³×C₁₀₇	C₂³×C₄	1	6	yes	—
166	C₂³×C₁₀₇	C₂²×D₄	1	6	yes	—
167	C₂³×C₁₀₇	C₂²×D₄	1	6	yes	—
168	C₂³×C₁₀₇	C₂⁵	1	1	yes	characteristic
169	Q₈×C₁₀₇	—	2	8	no	—
170	Q₈×C₁₀₇	C₂×C₂²⋊C₄	1	16	yes	complemented
171	D₄×C₁₀₇	—	2	48	no	—
172	D₄×C₁₀₇	—	2	48	no	—
173	D₄×C₁₀₇	—	2	24	no	—
174	D₄×C₁₀₇	C₂×C₂²⋊C₄	1	48	yes	complemented
175	C₂×C₄×C₁₀₇	—	2	2	no	—
176	C₂×C₄×C₁₀₇	—	2	2	no	—
177	C₂×C₄×C₁₀₇	—	2	2	no	—
178	C₂×C₄×C₁₀₇	—	2	2	no	—
179	C₂×C₄×C₁₀₇	—	2	48	no	—
180	C₂×C₄×C₁₀₇	—	2	2	no	—
181	C₂×C₄×C₁₀₇	—	2	6	no	—
182	C₂×C₄×C₁₀₇	—	2	6	no	—
183	C₂×C₄×C₁₀₇	—	2	12	no	—
184	C₂×C₄×C₁₀₇	—	2	6	no	—
185	C₂×C₄×C₁₀₇	—	2	4	no	—
186	C₂×C₄×C₁₀₇	—	2	6	no	—
187	C₂×C₄×C₁₀₇	—	4	24	no	—
188	C₂×C₄×C₁₀₇	—	8	12	no	—
189	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	48	yes	complemented
190	C₂×C₄×C₁₀₇	C₂³×C₄	1	4	yes	—
191	C₂×C₄×C₁₀₇	C₂³×C₄	1	6	yes	—
192	C₂×C₄×C₁₀₇	C₂³×C₄	1	2	yes	—
193	C₂×C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₁₀₇	C₂²×D₄	1	6	yes	—
195	C₂×C₄×C₁₀₇	C₂²×D₄	1	6	yes	—
196	C₂×C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₁₀₇	H(5,𝔽₂)	1	4	yes	complemented
198	C₂²×C₁₀₇	—	2	12	no	—
199	C₂²×C₁₀₇	—	2	12	no	—
200	C₂²×C₁₀₇	—	2	6	no	—
201	C₂²×C₁₀₇	—	2	6	no	—
202	C₂²×C₁₀₇	—	2	12	no	—
203	C₂²×C₁₀₇	—	2	24	no	—
204	C₂²×C₁₀₇	—	4	12	no	—
205	C₂²×C₁₀₇	—	4	6	no	—
206	C₂²×C₁₀₇	—	4	6	no	—
207	C₂²×C₁₀₇	—	8	48	no	—
208	C₂²×C₁₀₇	C₂²×C₂²⋊C₄	1	12	yes	—
209	C₂²×C₁₀₇	C₂⁴×C₄	1	1	yes	characteristic
210	C₂²×C₁₀₇	C₂³×D₄	1	1	yes	characteristic
211	C₂²×C₁₀₇	C₂³×D₄	1	1	yes	characteristic
212	C₂²×C₁₀₇	C₂×H(5,𝔽₂)	1	2	yes	—
213	C₂²×C₁₀₇	#585	1	2	yes	—
214	C₄×C₁₀₇	—	2	4	no	—
215	C₄×C₁₀₇	—	2	4	no	—
216	C₄×C₁₀₇	—	4	6	no	—
217	C₄×C₁₀₇	—	4	6	no	—
218	C₄×C₁₀₇	—	4	2	no	—
219	C₄×C₁₀₇	C₂²×C₂²⋊C₄	1	12	yes	—
220	C₄×C₁₀₇	C₂²×C₂²⋊C₄	1	4	yes	—
221	#34258	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂×C₁₀₇	—	2	12	no	—
223	C₂×C₁₀₇	—	4	12	no	—
224	C₂×C₁₀₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₁₀₇	C₄×H(5,𝔽₂)	1	2	yes	—
226	C₂×C₁₀₇	#3488	1	2	yes	—
227	C₂×C₁₀₇	#3498	1	1	yes	characteristic
228	C₂×C₁₀₇	#3500	1	1	yes	characteristic
229	C₂²×H(5,𝔽₂)	C₂×C₁₀₇	1	1	yes	characteristic
230	C₂×#520	C₂×C₁₀₇	1	1	yes	characteristic, complemented
231	C₂×C₄×#41	C₂×C₁₀₇	1	1	yes	characteristic, complemented
232	#2341	C₂×C₁₀₇	1	4	yes	complemented
233	D₄×C₂²⋊C₄	C₂×C₁₀₇	1	12	yes	complemented
234	#2311	C₂×C₁₀₇	1	3	yes	complemented
235	#2309	C₂×C₁₀₇	1	3	yes	complemented
236	#2297	C₂×C₁₀₇	1	3	yes	complemented
237	C₂×#390	C₂×C₁₀₇	1	3	yes	complemented
238	C₁₀₇	#34258	1	1	yes	characteristic, direct product factor, Sylow
239	C₂×H(5,𝔽₂)	—	2	8	no	—
240	C₂×H(5,𝔽₂)	—	2	8	no	—
241	C₂×H(5,𝔽₂)	C₄×C₁₀₇	1	16	yes	complemented
242	C₂²×#41	C₂²×C₁₀₇	1	3	yes	—
243	C₂²×#41	C₂²×C₁₀₇	1	2	yes	—
244	C₂²×#41	C₂²×C₁₀₇	1	1	yes	characteristic
245	C₂³×D₄	C₂²×C₁₀₇	1	3	yes	—
246	C₂³×D₄	C₂²×C₁₀₇	1	6	yes	—
247	#520	—	2	8	no	—
248	C₄×#41	—	2	8	no	—
249	C₂×C₄×Q₈	C₂²×C₁₀₇	1	2	yes	complemented
250	C₂×C₄×Q₈	C₂²×C₁₀₇	1	2	yes	complemented
251	C₂×C₄×D₄	C₂²×C₁₀₇	1	6	yes	complemented
252	C₂×C₄×D₄	C₂²×C₁₀₇	1	6	yes	complemented
253	C₂×#117	C₂²×C₁₀₇	1	3	yes	complemented
254	C₂×#117	C₂²×C₁₀₇	1	3	yes	complemented
255	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	3	yes	complemented
256	C₂²×C₂²⋊C₄	C₂²×C₁₀₇	1	12	yes	complemented
257	C₂²×C₄²	C₂²×C₁₀₇	1	3	yes	complemented
258	#391	C₂²×C₁₀₇	1	3	yes	complemented
259	#391	C₂²×C₁₀₇	1	1	yes	characteristic, complemented
260	#390	C₂²×C₁₀₇	1	3	yes	complemented
261	#390	C₂²×C₁₀₇	1	6	yes	complemented
262	#390	C₂²×C₁₀₇	1	12	yes	complemented
263	#390	C₂²×C₁₀₇	1	12	yes	complemented
264	#390	C₂²×C₁₀₇	1	3	yes	complemented
265	#386	C₂²×C₁₀₇	1	12	yes	complemented
266	#386	C₂²×C₁₀₇	1	12	yes	complemented
267	#380	C₂²×C₁₀₇	1	12	yes	complemented
268	#379	C₂²×C₁₀₇	1	12	yes	complemented
269	C₄×C₂²⋊C₄	C₂²×C₁₀₇	1	12	yes	complemented
270	C₂⁵	C₂³×C₁₀₇	1	6	yes	—
271	H(5,𝔽₂)	—	2	128	no	—
272	C₂×#41	—	2	8	no	—
273	C₂×#41	—	2	8	no	—
274	C₂×#41	—	2	12	no	—
275	C₂×#41	—	2	12	no	—
276	C₂×#41	C₂×C₄×C₁₀₇	1	8	yes	—
277	C₂×#41	C₂×C₄×C₁₀₇	1	16	yes	complemented
278	C₂×#41	C₂×C₄×C₁₀₇	1	24	yes	complemented
279	C₂×#41	D₄×C₁₀₇	1	8	yes	—
280	C₂×#41	D₄×C₁₀₇	1	8	yes	—
281	C₂²×Q₈	C₂³×C₁₀₇	1	1	yes	characteristic
282	C₂²×Q₈	C₂³×C₁₀₇	1	1	yes	characteristic
283	C₂²×D₄	—	2	24	no	—
284	C₂²×D₄	—	2	24	no	—
285	C₂²×D₄	—	2	12	no	—
286	C₂²×D₄	—	2	12	no	—
287	C₂²×D₄	C₂×C₄×C₁₀₇	1	48	yes	complemented
288	C₂²×D₄	C₂×C₄×C₁₀₇	1	24	yes	complemented
289	C₂²×D₄	C₂³×C₁₀₇	1	6	yes	—
290	C₂²×D₄	C₂³×C₁₀₇	1	3	yes	—
291	C₂²×D₄	C₂³×C₁₀₇	1	6	yes	—
292	C₂²×D₄	C₂³×C₁₀₇	1	3	yes	—
293	C₂³×C₄	C₂³×C₁₀₇	1	6	yes	—
294	C₂³×C₄	C₂³×C₁₀₇	1	6	yes	—
295	C₂³×C₄	C₂³×C₁₀₇	1	6	yes	—
296	C₂³×C₄	C₂³×C₁₀₇	1	3	yes	—
297	C₄×Q₈	—	2	8	no	—
298	C₄×Q₈	—	2	8	no	—
299	C₄×D₄	—	2	24	no	—
300	C₄×D₄	—	2	24	no	—
301	#117	—	2	12	no	—
302	#117	—	2	12	no	—
303	C₂×C₄⋊C₄	—	2	12	no	—
304	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	3	yes	—
305	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	6	yes	—
306	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	6	yes	—
308	C₂×C₂²⋊C₄	—	4	24	no	—
309	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	12	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	6	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	6	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	24	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₁₀₇	1	12	yes	—
314	C₂×C₄²	—	2	12	no	—
315	C₂×C₄²	C₂³×C₁₀₇	1	3	yes	—
316	C₂×C₄²	C₂³×C₁₀₇	1	6	yes	—
317	C₂×C₄²	C₂³×C₁₀₇	1	6	yes	—
318	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
319	#95	C₂³×C₁₀₇	1	4	yes	—
320	#95	C₂³×C₁₀₇	1	12	yes	—
321	C₂⁴	—	2	12	no	—
322	C₂⁴	—	2	12	no	—
323	C₂⁴	—	4	12	no	—
324	C₂⁴	—	4	12	no	—
325	C₂⁴	C₂²×C₄×C₁₀₇	1	24	yes	—
326	C₂⁴	C₂⁴×C₁₀₇	1	6	yes	—
327	C₂⁴	C₂⁴×C₁₀₇	1	3	yes	—
328	#41	—	2	64	no	—
329	#41	—	2	96	no	—
330	#41	C₂²⋊C₄×C₁₀₇	1	64	yes	complemented
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	C₂²×C₄×C₁₀₇	1	4	yes	—
334	C₂×Q₈	C₂²×C₄×C₁₀₇	1	4	yes	—
335	C₂×Q₈	C₂×D₄×C₁₀₇	1	4	yes	—
336	C₂×Q₈	C₂×D₄×C₁₀₇	1	4	yes	—
337	C₂×D₄	—	2	12	no	—
338	C₂×D₄	—	2	12	no	—
339	C₂×D₄	—	2	192	no	—
340	C₂×D₄	—	2	96	no	—
341	C₂×D₄	—	2	6	no	—
342	C₂×D₄	—	2	12	no	—
343	C₂×D₄	—	2	12	no	—
344	C₂×D₄	—	2	6	no	—
345	C₂×D₄	C₂²×C₄×C₁₀₇	1	24	yes	—
346	C₂×D₄	C₂²×C₄×C₁₀₇	1	24	yes	—
347	C₂×D₄	C₂²×C₄×C₁₀₇	1	12	yes	—
348	C₂×D₄	C₂²×C₄×C₁₀₇	1	12	yes	—
349	C₂×D₄	C₂×D₄×C₁₀₇	1	12	yes	—
350	C₂×D₄	C₂×D₄×C₁₀₇	1	12	yes	—
351	C₂²×C₄	—	2	12	no	—
352	C₂²×C₄	—	2	12	no	—
353	C₂²×C₄	—	2	12	no	—
354	C₂²×C₄	—	2	12	no	—
355	C₂²×C₄	—	2	12	no	—
356	C₂²×C₄	—	4	6	no	—
357	C₂²×C₄	—	4	12	no	—
358	C₂²×C₄	C₂²×C₄×C₁₀₇	1	24	yes	—
359	C₂²×C₄	C₂²×C₄×C₁₀₇	1	12	yes	—
360	C₂²×C₄	C₂×D₄×C₁₀₇	1	12	yes	—
361	C₂²×C₄	C₂×D₄×C₁₀₇	1	12	yes	—
362	C₂²×C₄	C₂⁴×C₁₀₇	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₁₀₇	1	6	yes	—
364	C₂²×C₄	C₂⁴×C₁₀₇	1	6	yes	—
365	C₂²×C₄	C₂⁴×C₁₀₇	1	2	yes	—
366	C₂²×C₄	C₂⁴×C₁₀₇	1	2	yes	—
367	C₂²×C₄	C₂⁴×C₁₀₇	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₁₀₇	1	3	yes	—
369	C₄⋊C₄	—	2	6	no	—
370	C₄⋊C₄	—	2	12	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	12	no	—
373	C₂²⋊C₄	—	2	12	no	—
374	C₂²⋊C₄	—	2	12	no	—
375	C₂²⋊C₄	—	2	24	no	—
376	C₂²⋊C₄	—	4	24	no	—
377	C₂²⋊C₄	—	4	12	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	6	no	—
380	C₄²	—	2	12	no	—
381	C₄²	—	2	12	no	—
382	C₂³	—	2	6	no	—
383	C₂³	—	2	6	no	—
384	C₂³	—	2	6	no	—
385	C₂³	—	2	6	no	—
386	C₂³	—	2	12	no	—
387	C₂³	—	2	12	no	—
388	C₂³	—	2	48	no	—
389	C₂³	—	4	48	no	—
390	C₂³	—	8	24	no	—
391	C₂³	—	8	12	no	—
392	C₂³	—	8	12	no	—
393	C₂³	C₂³×C₄×C₁₀₇	1	12	yes	—
394	C₂³	C₂³×C₄×C₁₀₇	1	6	yes	—
395	C₂³	C₂²×D₄×C₁₀₇	1	6	yes	—
396	C₂³	C₂²×D₄×C₁₀₇	1	6	yes	—
397	C₂³	C₂⁵×C₁₀₇	1	1	yes	characteristic
398	Q₈	—	2	8	no	—
399	Q₈	C₂×C₂²⋊C₄×C₁₀₇	1	16	yes	complemented
400	D₄	—	2	48	no	—
401	D₄	—	2	48	no	—
402	D₄	—	2	24	no	—
403	D₄	C₂×C₂²⋊C₄×C₁₀₇	1	48	yes	complemented
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	48	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	6	no	—
411	C₂×C₄	—	2	6	no	—
412	C₂×C₄	—	2	12	no	—
413	C₂×C₄	—	2	6	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	6	no	—
416	C₂×C₄	—	4	24	no	—
417	C₂×C₄	—	8	12	no	—
418	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	48	yes	complemented
419	C₂×C₄	C₂³×C₄×C₁₀₇	1	4	yes	—
420	C₂×C₄	C₂³×C₄×C₁₀₇	1	6	yes	—
421	C₂×C₄	C₂³×C₄×C₁₀₇	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₁₀₇	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₁₀₇	1	6	yes	—
424	C₂×C₄	C₂²×D₄×C₁₀₇	1	6	yes	—
425	C₂×C₄	C₂²×D₄×C₁₀₇	1	2	yes	—
426	C₂×C₄	H(5,𝔽₂)×C₁₀₇	1	4	yes	complemented
427	C₂²	—	2	12	no	—
428	C₂²	—	2	12	no	—
429	C₂²	—	2	6	no	—
430	C₂²	—	2	6	no	—
431	C₂²	—	2	12	no	—
432	C₂²	—	2	24	no	—
433	C₂²	—	4	12	no	—
434	C₂²	—	4	6	no	—
435	C₂²	—	4	6	no	—
436	C₂²	—	8	48	no	—
437	C₂²	C₂²×C₂²⋊C₄×C₁₀₇	1	12	yes	—
438	C₂²	C₂⁴×C₄×C₁₀₇	1	1	yes	characteristic
439	C₂²	C₂³×D₄×C₁₀₇	1	1	yes	characteristic
440	C₂²	C₂³×D₄×C₁₀₇	1	1	yes	characteristic
441	C₂²	C₂×H(5,𝔽₂)×C₁₀₇	1	2	yes	—
442	C₂²	#585×C₁₀₇	1	2	yes	—
443	C₄	—	2	4	no	—
444	C₄	—	2	4	no	—
445	C₄	—	4	6	no	—
446	C₄	—	4	6	no	—
447	C₄	—	4	2	no	—
448	C₄	C₂²×C₂²⋊C₄×C₁₀₇	1	12	yes	—
449	C₄	C₂²×C₂²⋊C₄×C₁₀₇	1	4	yes	—
450	C₂	—	2	12	no	—
451	C₂	—	4	12	no	—
452	C₂	C₂³×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
453	C₂	C₄×H(5,𝔽₂)×C₁₀₇	1	2	yes	—
454	C₂	C₁₀₇×#3488	1	2	yes	—
455	C₂	C₁₀₇×#3498	1	1	yes	characteristic
456	C₂	C₁₀₇×#3500	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₀₇
Frattini, cosocle	C₂³	C₂⁵×C₁₀₇
prime radical	C₂²×H(5,𝔽₂)×C₁₀₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#520×C₁₀₇	C₂	semidirect
3. C₂×C₄×#41×C₁₀₇	C₂	semidirect
4. C₁₀₇×#2341	C₂	semidirect
5. D₄×C₂²⋊C₄×C₁₀₇	C₂	semidirect
6. C₁₀₇×#2311	C₂	semidirect
7. C₁₀₇×#2309	C₂	semidirect
8. C₁₀₇×#2297	C₂	semidirect
9. C₂×#390×C₁₀₇	C₂	semidirect
12. C₂×H(5,𝔽₂)×C₁₀₇	C₄	semidirect
20. C₂×C₄×Q₈×C₁₀₇	C₂²	semidirect
21. C₂×C₄×Q₈×C₁₀₇	C₂²	semidirect
22. C₂×C₄×D₄×C₁₀₇	C₂²	semidirect
23. C₂×C₄×D₄×C₁₀₇	C₂²	semidirect
24. C₂×#117×C₁₀₇	C₂²	semidirect
25. C₂×#117×C₁₀₇	C₂²	semidirect
26. C₂²×C₄⋊C₄×C₁₀₇	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₁₀₇	C₂²	semidirect
28. C₂²×C₄²×C₁₀₇	C₂²	semidirect
29. #391×C₁₀₇	C₂²	semidirect
30. #391×C₁₀₇	C₂²	semidirect
31. #390×C₁₀₇	C₂²	semidirect
32. #390×C₁₀₇	C₂²	semidirect
33. #390×C₁₀₇	C₂²	semidirect
34. #390×C₁₀₇	C₂²	semidirect
35. #390×C₁₀₇	C₂²	semidirect
36. #386×C₁₀₇	C₂²	semidirect
37. #386×C₁₀₇	C₂²	semidirect
38. #380×C₁₀₇	C₂²	semidirect
39. #379×C₁₀₇	C₂²	semidirect
40. C₄×C₂²⋊C₄×C₁₀₇	C₂²	semidirect
48. C₂×#41×C₁₀₇	C₂×C₄	semidirect
49. C₂×#41×C₁₀₇	C₂×C₄	semidirect
58. C₂²×D₄×C₁₀₇	C₂×C₄	semidirect
59. C₂²×D₄×C₁₀₇	C₂×C₄	semidirect
101. #41×C₁₀₇	C₂²⋊C₄	semidirect
170. Q₈×C₁₀₇	C₂×C₂²⋊C₄	semidirect
174. D₄×C₁₀₇	C₂×C₂²⋊C₄	semidirect
189. C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	semidirect
197. C₂×C₄×C₁₀₇	H(5,𝔽₂)	semidirect
221. #34258	C₁₀₇	direct (N × H)
230. C₂×#520	C₂×C₁₀₇	semidirect
231. C₂×C₄×#41	C₂×C₁₀₇	semidirect
232. #2341	C₂×C₁₀₇	semidirect
233. D₄×C₂²⋊C₄	C₂×C₁₀₇	semidirect
234. #2311	C₂×C₁₀₇	semidirect
235. #2309	C₂×C₁₀₇	semidirect
236. #2297	C₂×C₁₀₇	semidirect
237. C₂×#390	C₂×C₁₀₇	semidirect
238. C₁₀₇	#34258	direct (N × H)
241. C₂×H(5,𝔽₂)	C₄×C₁₀₇	semidirect
249. C₂×C₄×Q₈	C₂²×C₁₀₇	semidirect
250. C₂×C₄×Q₈	C₂²×C₁₀₇	semidirect
251. C₂×C₄×D₄	C₂²×C₁₀₇	semidirect
252. C₂×C₄×D₄	C₂²×C₁₀₇	semidirect
253. C₂×#117	C₂²×C₁₀₇	semidirect
254. C₂×#117	C₂²×C₁₀₇	semidirect
255. C₂²×C₄⋊C₄	C₂²×C₁₀₇	semidirect
256. C₂²×C₂²⋊C₄	C₂²×C₁₀₇	semidirect
257. C₂²×C₄²	C₂²×C₁₀₇	semidirect
258. #391	C₂²×C₁₀₇	semidirect
259. #391	C₂²×C₁₀₇	semidirect
260. #390	C₂²×C₁₀₇	semidirect
261. #390	C₂²×C₁₀₇	semidirect
262. #390	C₂²×C₁₀₇	semidirect
263. #390	C₂²×C₁₀₇	semidirect
264. #390	C₂²×C₁₀₇	semidirect
265. #386	C₂²×C₁₀₇	semidirect
266. #386	C₂²×C₁₀₇	semidirect
267. #380	C₂²×C₁₀₇	semidirect
268. #379	C₂²×C₁₀₇	semidirect
269. C₄×C₂²⋊C₄	C₂²×C₁₀₇	semidirect
277. C₂×#41	C₂×C₄×C₁₀₇	semidirect
278. C₂×#41	C₂×C₄×C₁₀₇	semidirect
287. C₂²×D₄	C₂×C₄×C₁₀₇	semidirect
288. C₂²×D₄	C₂×C₄×C₁₀₇	semidirect
330. #41	C₂²⋊C₄×C₁₀₇	semidirect
399. Q₈	C₂×C₂²⋊C₄×C₁₀₇	semidirect
403. D₄	C₂×C₂²⋊C₄×C₁₀₇	semidirect
418. C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	semidirect
426. C₂×C₄	H(5,𝔽₂)×C₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₁₀₇	2
C₂×#520×C₁₀₇	2
C₂×C₄×#41×C₁₀₇	2
C₁₀₇×#2341	2
D₄×C₂²⋊C₄×C₁₀₇	2
C₁₀₇×#2311	2
C₁₀₇×#2309	2
C₁₀₇×#2297	2
C₂×#390×C₁₀₇	2
#34258	107

Subgroups — C107×#34258 #41060222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#34258 #41060222