Subgroups — C₂×C₆₇×#3112 #30739795

Order 17152.

2398 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3112	C₂	1	16	yes	maximal, direct product factor
4	C₂×#480×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#477×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#460×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#448×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#440×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	—	2	1	no	—
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
13	#532×C₆₇	—	2	4	no	—
14	#532×C₆₇	C₂²	1	8	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	—	2	1	no	—
18	C₂×#121×C₆₇	—	2	1	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
21	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄×Q₈×C₆₇	—	2	4	no	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
26	C₂×C₄×D₄×C₆₇	—	2	1	no	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₆₇	—	2	1	no	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#480×C₆₇	C₂²	1	16	yes	complemented
33	#477×C₆₇	C₂²	1	16	yes	complemented
34	#460×C₆₇	C₂²	1	16	yes	complemented
35	#448×C₆₇	C₂²	1	16	yes	complemented
36	#440×C₆₇	C₂²	1	32	yes	complemented
37	#432×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#107×C₆₇	C₂²	1	4	yes	—
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₂×#103×C₆₇	C₂²	1	4	yes	—
41	C₂×#102×C₆₇	C₂²	1	4	yes	—
42	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₂×#97×C₆₇	C₂²	1	2	yes	—
44	C₂×#41×C₆₇	—	2	8	no	—
45	C₂×#41×C₆₇	—	4	8	no	—
46	C₂²×Q₈×C₆₇	—	2	8	no	—
47	C₂²×Q₈×C₆₇	—	2	2	no	—
48	C₂²×Q₈×C₆₇	—	2	1	no	—
49	C₂²×Q₈×C₆₇	—	2	1	no	—
50	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
51	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	—	2	2	no	—
59	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×SD₁₆×C₆₇	—	2	16	no	—
63	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
65	C₄⋊Q₈×C₆₇	—	2	2	no	—
66	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
67	#127×C₆₇	—	2	2	no	—
68	#127×C₆₇	C₂³	1	4	yes	—
69	#124×C₆₇	—	2	8	no	—
70	#121×C₆₇	—	2	8	no	—
71	#121×C₆₇	—	2	4	no	—
72	#121×C₆₇	C₂³	1	8	yes	—
73	C₄×Q₈×C₆₇	—	2	4	no	—
74	C₄×Q₈×C₆₇	—	2	4	no	—
75	C₄×Q₈×C₆₇	—	2	8	no	—
76	C₄×Q₈×C₆₇	—	2	8	no	—
77	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
78	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
79	C₄×D₄×C₆₇	—	2	8	no	—
80	C₄×D₄×C₆₇	—	2	4	no	—
81	C₄×D₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₄²×C₆₇	—	2	2	no	—
94	C₂×C₄²×C₆₇	—	2	2	no	—
95	C₂×C₄²×C₆₇	—	2	8	no	—
96	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	D₄	1	4	yes	—
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
102	#107×C₆₇	C₂³	1	16	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
104	#103×C₆₇	C₂³	1	16	yes	—
105	#102×C₆₇	C₂³	1	16	yes	—
106	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
107	#97×C₆₇	C₂³	1	8	yes	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
110	#41×C₆₇	—	4	32	no	—
111	C₂×Q₈×C₆₇	—	2	16	no	—
112	C₂×Q₈×C₆₇	—	2	8	no	—
113	C₂×Q₈×C₆₇	—	2	16	no	—
114	C₂×Q₈×C₆₇	—	2	4	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	4	no	—
118	C₂×Q₈×C₆₇	—	4	4	no	—
119	C₂×Q₈×C₆₇	#41	1	8	yes	complemented
120	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	8	no	—
126	C₂×D₄×C₆₇	—	4	4	no	—
127	C₂×D₄×C₆₇	—	4	8	no	—
128	C₂×D₄×C₆₇	—	4	4	no	—
129	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	8	no	—
140	C₂²×C₄×C₆₇	—	4	8	no	—
141	C₂²×C₄×C₆₇	—	4	4	no	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	SD₁₆×C₆₇	—	2	32	no	—
153	C₂×C₈×C₆₇	—	2	8	no	—
154	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	8	no	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	4	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²⋊C₄×C₆₇	—	2	8	no	—
164	C₂²⋊C₄×C₆₇	—	2	8	no	—
165	C₄²×C₆₇	—	2	8	no	—
166	C₄²×C₆₇	—	2	8	no	—
167	C₄²×C₆₇	C₂×D₄	1	2	yes	—
168	C₄²×C₆₇	C₂×D₄	1	4	yes	—
169	C₄²×C₆₇	C₂×D₄	1	4	yes	—
170	C₄²×C₆₇	C₂×D₄	1	2	yes	—
171	C₄²×C₆₇	C₂⁴	1	2	yes	—
172	C₄²×C₆₇	C₂⁴	1	2	yes	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	4	4	no	—
177	C₂³×C₆₇	—	4	4	no	—
178	C₂³×C₆₇	—	4	4	no	—
179	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
180	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	16	no	—
183	Q₈×C₆₇	—	2	8	no	—
184	Q₈×C₆₇	—	2	4	no	—
185	Q₈×C₆₇	—	4	8	no	—
186	Q₈×C₆₇	C₂×#41	1	8	yes	complemented
187	D₄×C₆₇	—	2	8	no	—
188	D₄×C₆₇	—	4	8	no	—
189	D₄×C₆₇	—	4	16	no	—
190	D₄×C₆₇	—	4	4	no	—
191	D₄×C₆₇	—	4	8	no	—
192	C₂×C₄×C₆₇	—	2	16	no	—
193	C₂×C₄×C₆₇	—	2	8	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	8	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	8	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	4	16	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	#137	1	4	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
217	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
218	C₈×C₆₇	—	2	8	no	—
219	C₂²×C₆₇	—	2	8	no	—
220	C₂²×C₆₇	—	2	2	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	—	4	4	no	—
223	C₂²×C₆₇	—	4	4	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	8	8	no	—
226	C₂²×C₆₇	#525	1	2	yes	—
227	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
228	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
229	C₂²×C₆₇	C₂×#137	1	2	yes	—
230	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₆₇	—	2	8	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	—	4	4	no	—
238	C₄×C₆₇	#525	1	2	yes	—
239	C₄×C₆₇	C₂×#137	1	4	yes	—
240	C₄×C₆₇	C₂²×#41	1	2	yes	—
241	C₂×#3112	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂×C₆₇	—	2	4	no	—
243	C₂×C₆₇	—	8	4	no	—
244	C₂×C₆₇	#3112	1	4	yes	direct product factor
245	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
246	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
248	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
249	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#3112	C₂×C₆₇	1	16	yes	direct product factor
251	C₂×#480	C₂×C₆₇	1	2	yes	complemented
252	C₂×#477	C₂×C₆₇	1	2	yes	complemented
253	C₂×#460	C₂×C₆₇	1	2	yes	complemented
254	C₂×#448	C₂×C₆₇	1	2	yes	complemented
255	C₂×#440	C₂×C₆₇	1	4	yes	complemented
256	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₆₇	C₂×#3112	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²×#41	—	2	1	no	—
259	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
261	#532	—	2	4	no	—
262	#532	C₂²×C₆₇	1	8	yes	—
263	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×#124	—	2	1	no	—
266	C₂×#121	—	2	1	no	—
267	C₂×#121	C₂²×C₆₇	1	2	yes	—
268	C₂×C₄×Q₈	—	2	1	no	—
269	C₂×C₄×Q₈	—	2	1	no	—
270	C₂×C₄×Q₈	—	2	4	no	—
271	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
272	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
273	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
274	C₂×C₄×D₄	—	2	1	no	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
276	C₂²×C₄⋊C₄	—	2	1	no	—
277	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
278	C₂²×C₄²	—	2	1	no	—
279	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
280	#480	C₂²×C₆₇	1	16	yes	complemented
281	#477	C₂²×C₆₇	1	16	yes	complemented
282	#460	C₂²×C₆₇	1	16	yes	complemented
283	#448	C₂²×C₆₇	1	16	yes	complemented
284	#440	C₂²×C₆₇	1	32	yes	complemented
285	#432	C₂²×C₆₇	1	8	yes	complemented
286	C₂×#107	C₂²×C₆₇	1	4	yes	—
287	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
288	C₂×#103	C₂²×C₆₇	1	4	yes	—
289	C₂×#102	C₂²×C₆₇	1	4	yes	—
290	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
291	C₂×#97	C₂²×C₆₇	1	2	yes	—
292	C₂×#41	—	2	8	no	—
293	C₂×#41	—	4	8	no	—
294	C₂²×Q₈	—	2	8	no	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×Q₈	—	2	2	no	—
297	C₂²×Q₈	—	2	1	no	—
298	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
299	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
300	C₂²×D₄	—	2	2	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	—	2	1	no	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
308	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂×SD₁₆	—	2	16	no	—
311	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
312	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
315	#127	—	2	2	no	—
316	#127	C₂³×C₆₇	1	4	yes	—
317	#124	—	2	8	no	—
318	#121	—	2	8	no	—
319	#121	—	2	4	no	—
320	#121	C₂³×C₆₇	1	8	yes	—
321	C₄×Q₈	—	2	4	no	—
322	C₄×Q₈	—	2	4	no	—
323	C₄×Q₈	—	2	8	no	—
324	C₄×Q₈	—	2	8	no	—
325	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
326	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
327	C₄×D₄	—	2	8	no	—
328	C₄×D₄	—	2	4	no	—
329	C₄×D₄	C₂³×C₆₇	1	8	yes	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	8	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₄²	—	2	2	no	—
342	C₂×C₄²	—	2	2	no	—
343	C₂×C₄²	—	2	8	no	—
344	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
346	C₂×C₄²	D₄×C₆₇	1	4	yes	—
347	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
350	#107	C₂³×C₆₇	1	16	yes	—
351	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
352	#103	C₂³×C₆₇	1	16	yes	—
353	#102	C₂³×C₆₇	1	16	yes	—
354	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
355	#97	C₂³×C₆₇	1	8	yes	—
356	C₂⁴	—	2	2	no	—
357	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
358	#41	—	4	32	no	—
359	C₂×Q₈	—	2	16	no	—
360	C₂×Q₈	—	2	8	no	—
361	C₂×Q₈	—	2	16	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	2	4	no	—
366	C₂×Q₈	—	4	4	no	—
367	C₂×Q₈	#41×C₆₇	1	8	yes	complemented
368	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	4	8	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
392	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
397	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
399	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
400	SD₁₆	—	2	32	no	—
401	C₂×C₈	—	2	8	no	—
402	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
410	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₄²	—	2	8	no	—
414	C₄²	—	2	8	no	—
415	C₄²	C₂×D₄×C₆₇	1	2	yes	—
416	C₄²	C₂×D₄×C₆₇	1	4	yes	—
417	C₄²	C₂×D₄×C₆₇	1	4	yes	—
418	C₄²	C₂×D₄×C₆₇	1	2	yes	—
419	C₄²	C₂⁴×C₆₇	1	2	yes	—
420	C₄²	C₂⁴×C₆₇	1	2	yes	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	4	4	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
428	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
429	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
430	Q₈	—	2	16	no	—
431	Q₈	—	2	8	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	4	8	no	—
434	Q₈	C₂×#41×C₆₇	1	8	yes	complemented
435	D₄	—	2	8	no	—
436	D₄	—	4	8	no	—
437	D₄	—	4	16	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	8	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	4	16	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	#137×C₆₇	1	4	yes	—
458	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
464	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
465	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
466	C₈	—	2	8	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	8	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	8	8	no	—
474	C₂²	#525×C₆₇	1	2	yes	—
475	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
476	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
477	C₂²	C₂×#137×C₆₇	1	2	yes	—
478	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	8	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	#525×C₆₇	1	2	yes	—
487	C₄	C₂×#137×C₆₇	1	4	yes	—
488	C₄	C₂²×#41×C₆₇	1	2	yes	—
489	C₂	—	2	4	no	—
490	C₂	—	8	4	no	—
491	C₂	C₆₇×#3112	1	4	yes	direct product factor
492	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
493	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
494	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3112	C₂	direct (N × H)
4. C₂×#480×C₆₇	C₂	semidirect
5. C₂×#477×C₆₇	C₂	semidirect
6. C₂×#460×C₆₇	C₂	semidirect
7. C₂×#448×C₆₇	C₂	semidirect
8. C₂×#440×C₆₇	C₂	semidirect
9. C₂×#432×C₆₇	C₂	semidirect
25. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
32. #480×C₆₇	C₂²	semidirect
33. #477×C₆₇	C₂²	semidirect
34. #460×C₆₇	C₂²	semidirect
35. #448×C₆₇	C₂²	semidirect
36. #440×C₆₇	C₂²	semidirect
37. #432×C₆₇	C₂²	semidirect
119. C₂×Q₈×C₆₇	#41	semidirect
186. Q₈×C₆₇	C₂×#41	semidirect
241. C₂×#3112	C₆₇	direct (N × H)
244. C₂×C₆₇	#3112	direct (N × H)
249. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
250. #3112	C₂×C₆₇	direct (N × H)
251. C₂×#480	C₂×C₆₇	semidirect
252. C₂×#477	C₂×C₆₇	semidirect
253. C₂×#460	C₂×C₆₇	semidirect
254. C₂×#448	C₂×C₆₇	semidirect
255. C₂×#440	C₂×C₆₇	semidirect
256. C₂×#432	C₂×C₆₇	semidirect
257. C₆₇	C₂×#3112	direct (N × H)
273. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
280. #480	C₂²×C₆₇	semidirect
281. #477	C₂²×C₆₇	semidirect
282. #460	C₂²×C₆₇	semidirect
283. #448	C₂²×C₆₇	semidirect
284. #440	C₂²×C₆₇	semidirect
285. #432	C₂²×C₆₇	semidirect
367. C₂×Q₈	#41×C₆₇	semidirect
434. Q₈	C₂×#41×C₆₇	semidirect
491. C₂	C₆₇×#3112	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#3112	2
C₂×#480×C₆₇	2
C₂×#477×C₆₇	2
C₂×#460×C₆₇	2
C₂×#448×C₆₇	2
C₂×#440×C₆₇	2
C₂×#432×C₆₇	2
C₂×#3112	67

Subgroups — C2×C67×#3112 #30739795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3112 #30739795