Subgroups — C₆₇×#22503 #30701754

Order 17152.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#555×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1825	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1736	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	#555×C₆₇	—	2	4	no	—
14	#555×C₆₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	—	2	2	no	—
17	C₂×#121×C₆₇	—	2	1	no	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	—	2	1	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	—	2	1	no	—
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#389×C₆₇	—	2	1	no	—
33	#385×C₆₇	—	2	2	no	—
34	#384×C₆₇	C₂²	1	1	yes	characteristic
35	#380×C₆₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	—	2	1	no	—
49	C₂³×C₄×C₆₇	—	2	2	no	—
50	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₇	—	2	4	no	—
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	—	4	2	no	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
59	#124×C₆₇	—	2	16	no	—
60	#121×C₆₇	—	2	8	no	—
61	#121×C₆₇	—	2	4	no	—
62	#121×C₆₇	C₂×C₄	1	8	yes	complemented
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	—	4	4	no	—
65	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
66	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
67	C₄×D₄×C₆₇	D₄	1	4	yes	—
68	#117×C₆₇	—	2	4	no	—
69	#117×C₆₇	—	2	8	no	—
70	#117×C₆₇	C₂×C₄	1	8	yes	complemented
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
88	C₂×C₄²×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	—	2	1	no	—
90	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₆₇	—	2	2	no	—
92	#102×C₆₇	—	2	2	no	—
93	#102×C₆₇	—	2	2	no	—
94	#102×C₆₇	—	4	2	no	—
95	C₂²⋊C₈×C₆₇	—	2	4	no	—
96	C₄×C₈×C₆₇	—	2	2	no	—
97	#95×C₆₇	—	2	1	no	—
98	#95×C₆₇	—	2	2	no	—
99	C₂⁴×C₆₇	—	2	1	no	—
100	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×Q₈×C₆₇	—	2	4	no	—
102	C₂×Q₈×C₆₇	—	8	1	no	—
103	C₂×Q₈×C₆₇	—	8	1	no	—
104	C₂×D₄×C₆₇	—	2	8	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	1	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	8	2	no	—
113	C₂×D₄×C₆₇	—	8	2	no	—
114	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
115	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	4	4	no	—
127	C₂²×C₄×C₆₇	—	4	2	no	—
128	C₂²×C₄×C₆₇	—	4	2	no	—
129	C₂²×C₄×C₆₇	—	4	4	no	—
130	C₂²×C₄×C₆₇	—	8	2	no	—
131	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	M₁₆×C₆₇	—	4	8	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	1	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	1	no	—
146	C₂×C₈×C₆₇	#41	1	2	yes	—
147	C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	—	2	1	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	4	4	no	—
156	C₄⋊C₄×C₆₇	—	4	2	no	—
157	C₄⋊C₄×C₆₇	—	8	1	no	—
158	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
160	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
161	C₄⋊C₄×C₆₇	#41	1	2	yes	—
162	C₂²⋊C₄×C₆₇	—	2	8	no	—
163	C₂²⋊C₄×C₆₇	—	2	8	no	—
164	C₂²⋊C₄×C₆₇	—	2	4	no	—
165	C₂²⋊C₄×C₆₇	—	4	4	no	—
166	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
167	C₄²×C₆₇	—	2	8	no	—
168	C₄²×C₆₇	—	4	2	no	—
169	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
170	C₄²×C₆₇	C₂×D₄	1	2	yes	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	2	2	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	4	1	no	—
179	C₂³×C₆₇	—	4	1	no	—
180	C₂³×C₆₇	—	4	1	no	—
181	C₂³×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	—	4	1	no	—
183	C₂³×C₆₇	—	4	1	no	—
184	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
185	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₆₇	—	8	4	no	—
187	D₄×C₆₇	—	2	2	no	—
188	D₄×C₆₇	—	2	2	no	—
189	D₄×C₆₇	—	2	8	no	—
190	D₄×C₆₇	—	4	1	no	—
191	D₄×C₆₇	—	4	1	no	—
192	D₄×C₆₇	—	8	8	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	8	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	1	no	—
217	C₂×C₄×C₆₇	—	8	8	no	—
218	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
219	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
222	C₂×C₄×C₆₇	#137	1	2	yes	—
223	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
229	C₈×C₆₇	—	2	2	no	—
230	C₈×C₆₇	—	4	2	no	—
231	C₈×C₆₇	—	4	1	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	2	4	no	—
234	C₂²×C₆₇	—	2	4	no	—
235	C₂²×C₆₇	—	4	2	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	8	1	no	—
243	C₂²×C₆₇	—	8	1	no	—
244	C₂²×C₆₇	—	8	1	no	—
245	C₂²×C₆₇	—	8	1	no	—
246	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²×C₆₇	#546	1	1	yes	characteristic
248	C₂²×C₆₇	#547	1	1	yes	characteristic
249	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
251	C₂²×C₆₇	#576	1	1	yes	characteristic
252	C₂²×C₆₇	#577	1	1	yes	characteristic
253	C₄×C₆₇	—	2	4	no	—
254	C₄×C₆₇	—	2	2	no	—
255	C₄×C₆₇	—	4	2	no	—
256	C₄×C₆₇	—	4	4	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	—	8	1	no	—
259	C₄×C₆₇	—	8	2	no	—
260	C₄×C₆₇	#417	1	4	yes	—
261	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
262	C₄×C₆₇	#520	1	1	yes	characteristic
263	C₄×C₆₇	#546	1	1	yes	characteristic
264	C₄×C₆₇	#547	1	1	yes	characteristic
265	#22503	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₆₇	—	4	4	no	—
267	C₂×C₆₇	—	8	2	no	—
268	C₂×C₆₇	#2339	1	1	yes	characteristic
269	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
270	C₂×C₆₇	#2897	1	1	yes	characteristic
271	C₂×C₆₇	#3325	1	1	yes	characteristic
272	C₂×C₆₇	#3329	1	1	yes	characteristic
273	C₂×C₆₇	#3354	1	1	yes	characteristic
274	C₂×C₆₇	#3357	1	1	yes	characteristic
275	C₂×#555	C₂×C₆₇	1	1	yes	characteristic
276	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
277	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
278	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#1895	C₂×C₆₇	1	2	yes	complemented
281	#1874	C₂×C₆₇	1	2	yes	complemented
282	#1825	C₂×C₆₇	1	2	yes	complemented
283	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
285	#1736	C₂×C₆₇	1	2	yes	complemented
286	C₆₇	#22503	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
288	#555	—	2	4	no	—
289	#555	C₄×C₆₇	1	8	yes	complemented
290	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×#124	—	2	2	no	—
292	C₂×#121	—	2	1	no	—
293	C₂×#121	C₂²×C₆₇	1	2	yes	—
294	C₂×C₄×D₄	—	2	1	no	—
295	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×#117	—	2	1	no	—
298	C₂×#117	C₂²×C₆₇	1	2	yes	—
299	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
300	#436	—	2	4	no	—
301	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#102	C₂²×C₆₇	1	2	yes	—
305	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
306	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
307	#389	—	2	1	no	—
308	#385	—	2	2	no	—
309	#384	C₂²×C₆₇	1	1	yes	characteristic
310	#380	C₂²×C₆₇	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
312	#340	C₂²×C₆₇	1	2	yes	—
313	#340	C₂²×C₆₇	1	1	yes	characteristic
314	#340	C₂²×C₆₇	1	2	yes	—
315	#340	C₂²×C₆₇	1	2	yes	—
316	#340	C₂²×C₆₇	1	2	yes	—
317	#340	C₂²×C₆₇	1	2	yes	—
318	#340	C₂²×C₆₇	1	1	yes	characteristic
319	C₂²×Q₈	—	2	1	no	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
322	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
323	C₂³×C₄	—	2	1	no	—
324	C₂³×C₄	—	2	2	no	—
325	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	—	2	4	no	—
328	C₂×M₁₆	—	4	2	no	—
329	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
334	#124	—	2	16	no	—
335	#121	—	2	8	no	—
336	#121	—	2	4	no	—
337	#121	C₂×C₄×C₆₇	1	8	yes	complemented
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	4	4	no	—
340	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
341	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
342	C₄×D₄	D₄×C₆₇	1	4	yes	—
343	#117	—	2	4	no	—
344	#117	—	2	8	no	—
345	#117	C₂×C₄×C₆₇	1	8	yes	complemented
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	4	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	4	2	no	—
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
363	C₂×C₄²	—	2	2	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
366	C₄⋊C₈	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	—	4	2	no	—
370	C₂²⋊C₈	—	2	4	no	—
371	C₄×C₈	—	2	2	no	—
372	#95	—	2	1	no	—
373	#95	—	2	2	no	—
374	C₂⁴	—	2	1	no	—
375	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	8	1	no	—
378	C₂×Q₈	—	8	1	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂×D₄	—	8	2	no	—
389	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
390	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	—	8	2	no	—
406	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
410	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
412	M₁₆	—	4	8	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	#41×C₆₇	1	2	yes	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	4	4	no	—
431	C₄⋊C₄	—	4	2	no	—
432	C₄⋊C₄	—	8	1	no	—
433	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
434	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
435	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
436	C₄⋊C₄	#41×C₆₇	1	2	yes	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	4	4	no	—
441	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
442	C₄²	—	2	8	no	—
443	C₄²	—	4	2	no	—
444	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
445	C₄²	C₂×D₄×C₆₇	1	2	yes	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	4	1	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	1	no	—
456	C₂³	—	4	1	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	4	1	no	—
459	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
460	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
461	Q₈	—	8	4	no	—
462	D₄	—	2	2	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	8	no	—
465	D₄	—	4	1	no	—
466	D₄	—	4	1	no	—
467	D₄	—	8	8	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	8	8	no	—
493	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
494	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
495	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
496	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
497	C₂×C₄	#137×C₆₇	1	2	yes	—
498	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
500	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
501	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
502	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
503	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
504	C₈	—	2	2	no	—
505	C₈	—	4	2	no	—
506	C₈	—	4	1	no	—
507	C₂²	—	2	4	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	8	1	no	—
521	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
522	C₂²	#546×C₆₇	1	1	yes	characteristic
523	C₂²	#547×C₆₇	1	1	yes	characteristic
524	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
525	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
526	C₂²	#576×C₆₇	1	1	yes	characteristic
527	C₂²	#577×C₆₇	1	1	yes	characteristic
528	C₄	—	2	4	no	—
529	C₄	—	2	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	4	no	—
533	C₄	—	8	2	no	—
534	C₄	—	8	1	no	—
535	C₄	#417×C₆₇	1	4	yes	—
536	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
537	C₄	#520×C₆₇	1	1	yes	characteristic
538	C₄	#546×C₆₇	1	1	yes	characteristic
539	C₄	#547×C₆₇	1	1	yes	characteristic
540	C₂	—	4	4	no	—
541	C₂	—	8	2	no	—
542	C₂	C₆₇×#2339	1	1	yes	characteristic
543	C₂	C₂×#417×C₆₇	1	1	yes	characteristic
544	C₂	C₆₇×#2897	1	1	yes	characteristic
545	C₂	C₆₇×#3325	1	1	yes	characteristic
546	C₂	C₆₇×#3329	1	1	yes	characteristic
547	C₂	C₆₇×#3354	1	1	yes	characteristic
548	C₂	C₆₇×#3357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#555×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#436×C₆₇	C₂	semidirect
3. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1987	C₂	semidirect
6. C₆₇×#1895	C₂	semidirect
7. C₆₇×#1874	C₂	semidirect
8. C₆₇×#1825	C₂	semidirect
9. C₆₇×#1802	C₂	semidirect
10. C₄×#102×C₆₇	C₂	semidirect
11. C₆₇×#1736	C₂	semidirect
14. #555×C₆₇	C₄	semidirect
58. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
62. #121×C₆₇	C₂×C₄	semidirect
66. C₄×D₄×C₆₇	C₂×C₄	semidirect
70. #117×C₆₇	C₂×C₄	semidirect
265. #22503	C₆₇	direct (N × H)
276. C₂×#436	C₂×C₆₇	semidirect
277. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
278. #1990	C₂×C₆₇	semidirect
279. #1987	C₂×C₆₇	semidirect
280. #1895	C₂×C₆₇	semidirect
281. #1874	C₂×C₆₇	semidirect
282. #1825	C₂×C₆₇	semidirect
283. #1802	C₂×C₆₇	semidirect
284. C₄×#102	C₂×C₆₇	semidirect
285. #1736	C₂×C₆₇	semidirect
286. C₆₇	#22503	direct (N × H)
289. #555	C₄×C₆₇	semidirect
333. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
337. #121	C₂×C₄×C₆₇	semidirect
341. C₄×D₄	C₂×C₄×C₆₇	semidirect
345. #117	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#555×C₆₇	2
C₂×#436×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1990	2
C₆₇×#1987	2
C₆₇×#1895	2
C₆₇×#1874	2
C₆₇×#1825	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
C₆₇×#1736	2
#22503	67

Subgroups — C67×#22503 #30701754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22503 #30701754