Subgroups — C₆₇×#21859 #30701110

Order 17152.

1502 conjugacy classes of subgroups.

The subgroup lattice has 472 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	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1957	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1874	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1802	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	—	2	1	no	—
9	#532×C₆₇	—	2	4	no	—
10	#532×C₆₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	—	2	1	no	—
14	C₂×#121×C₆₇	—	2	1	no	—
15	C₂×#121×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#431×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	8	yes	—
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	#384×C₆₇	—	2	1	no	—
28	#384×C₆₇	—	2	1	no	—
29	#384×C₆₇	C₂²	1	2	yes	—
30	#384×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂×#41×C₆₇	—	2	8	no	—
33	C₂×#41×C₆₇	—	4	4	no	—
34	C₂×#41×C₆₇	—	4	4	no	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₆₇	—	2	2	no	—
41	C₂³×C₄×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
48	#127×C₆₇	—	2	2	no	—
49	#127×C₆₇	C₂×C₄	1	4	yes	complemented
50	#124×C₆₇	—	2	8	no	—
51	#121×C₆₇	—	2	8	no	—
52	#121×C₆₇	—	2	4	no	—
53	#121×C₆₇	C₂×C₄	1	8	yes	—
54	C₄×D₄×C₆₇	—	2	8	no	—
55	C₄×D₄×C₆₇	—	2	4	no	—
56	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	D₄	1	4	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	4	8	no	—
78	C₂²⋊C₈×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	—	2	2	no	—
81	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
82	#41×C₆₇	—	4	32	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×Q₈×C₆₇	—	4	2	no	—
86	C₂×D₄×C₆₇	—	2	8	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	4	4	no	—
93	C₂×D₄×C₆₇	—	4	4	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂×D₄×C₆₇	—	4	2	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	4	4	no	—
107	C₂²×C₄×C₆₇	—	4	4	no	—
108	C₂²×C₄×C₆₇	—	4	4	no	—
109	C₂²×C₄×C₆₇	—	4	4	no	—
110	C₂²×C₄×C₆₇	—	4	4	no	—
111	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
113	C₂²×C₄×C₆₇	D₈	1	4	yes	—
114	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	4	no	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₄⋊C₄×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₄⋊C₄×C₆₇	—	4	4	no	—
130	C₄⋊C₄×C₆₇	—	4	4	no	—
131	C₄⋊C₄×C₆₇	—	4	4	no	—
132	C₄⋊C₄×C₆₇	—	4	4	no	—
133	C₄⋊C₄×C₆₇	—	4	4	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂²⋊C₄×C₆₇	—	2	8	no	—
136	C₂²⋊C₄×C₆₇	—	2	8	no	—
137	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
138	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
139	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
140	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂×D₄	1	2	yes	—
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	2	1	no	—
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	4	2	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂³×C₆₇	C₂×D₈	1	2	yes	—
156	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
157	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₆₇	—	4	8	no	—
159	D₄×C₆₇	—	2	8	no	—
160	D₄×C₆₇	—	4	16	no	—
161	D₄×C₆₇	—	4	4	no	—
162	D₄×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	4	8	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	16	no	—
175	C₂×C₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	4	no	—
178	C₂×C₄×C₆₇	—	8	8	no	—
179	C₂×C₄×C₆₇	#102	1	16	yes	complemented
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
186	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
187	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
188	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
189	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₆₇	#137	1	2	yes	—
191	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
194	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
195	C₈×C₆₇	—	4	4	no	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	8	2	no	—
206	C₂²×C₆₇	—	8	2	no	—
207	C₂²×C₆₇	—	8	2	no	—
208	C₂²×C₆₇	—	8	2	no	—
209	C₂²×C₆₇	C₂×#102	1	4	yes	—
210	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₆₇	#535	1	1	yes	characteristic
212	C₂²×C₆₇	#536	1	1	yes	characteristic
213	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
214	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
215	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
217	C₄×C₆₇	—	2	2	no	—
218	C₄×C₆₇	—	2	2	no	—
219	C₄×C₆₇	—	4	4	no	—
220	C₄×C₆₇	—	8	4	no	—
221	C₄×C₆₇	C₂×#102	1	4	yes	—
222	C₄×C₆₇	C₂×#102	1	4	yes	—
223	C₄×C₆₇	#417	1	4	yes	—
224	C₄×C₆₇	#520	1	2	yes	—
225	C₄×C₆₇	#537	1	2	yes	—
226	#21859	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂×C₆₇	—	2	4	no	—
228	C₂×C₆₇	—	8	4	no	—
229	C₂×C₆₇	#2311	1	1	yes	characteristic
230	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
231	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
232	C₂×C₆₇	#3205	1	1	yes	characteristic
233	C₂×C₆₇	#3206	1	1	yes	characteristic
234	C₂×C₆₇	#3226	1	1	yes	characteristic
235	C₂×C₆₇	#3227	1	1	yes	characteristic
236	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
237	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#1990	C₂×C₆₇	1	2	yes	complemented
240	#1957	C₂×C₆₇	1	2	yes	complemented
241	#1874	C₂×C₆₇	1	4	yes	complemented
242	#1802	C₂×C₆₇	1	4	yes	complemented
243	C₆₇	#21859	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²×#41	—	2	1	no	—
245	#532	—	2	4	no	—
246	#532	C₄×C₆₇	1	8	yes	complemented
247	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#124	—	2	1	no	—
250	C₂×#121	—	2	1	no	—
251	C₂×#121	C₂²×C₆₇	1	2	yes	—
252	C₂×C₄×D₄	—	2	1	no	—
253	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
254	C₂²×C₄⋊C₄	—	2	1	no	—
255	C₂²×C₄⋊C₄	—	2	1	no	—
256	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
257	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
258	#431	—	2	4	no	—
259	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
260	C₂×#102	C₂²×C₆₇	1	8	yes	—
261	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
262	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
263	#384	—	2	1	no	—
264	#384	—	2	1	no	—
265	#384	C₂²×C₆₇	1	2	yes	—
266	#384	C₂²×C₆₇	1	2	yes	—
267	#340	C₂²×C₆₇	1	8	yes	—
268	C₂×#41	—	2	8	no	—
269	C₂×#41	—	4	4	no	—
270	C₂×#41	—	4	4	no	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
276	C₂³×C₄	—	2	2	no	—
277	C₂³×C₄	—	2	2	no	—
278	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
279	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
282	C₄⋊Q₈	—	2	2	no	—
283	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
284	#127	—	2	2	no	—
285	#127	C₂×C₄×C₆₇	1	4	yes	complemented
286	#124	—	2	8	no	—
287	#121	—	2	8	no	—
288	#121	—	2	4	no	—
289	#121	C₂×C₄×C₆₇	1	8	yes	—
290	C₄×D₄	—	2	8	no	—
291	C₄×D₄	—	2	4	no	—
292	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	4	4	no	—
300	C₂×C₄⋊C₄	—	4	4	no	—
301	C₂×C₄⋊C₄	—	4	4	no	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
307	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₄²	D₄×C₆₇	1	4	yes	—
310	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	C₄⋊C₈	—	2	4	no	—
313	#102	—	4	8	no	—
314	C₂²⋊C₈	—	2	4	no	—
315	C₄×C₈	—	2	4	no	—
316	C₂⁴	—	2	2	no	—
317	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
318	#41	—	4	32	no	—
319	C₂×Q₈	—	2	4	no	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×D₄	—	2	8	no	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	4	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	4	4	no	—
329	C₂×D₄	—	4	4	no	—
330	C₂×D₄	—	4	2	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	4	2	no	—
334	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	8	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	—	4	4	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	—	4	4	no	—
347	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
348	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
349	C₂²×C₄	D₈×C₆₇	1	4	yes	—
350	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
351	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂×C₈	—	2	4	no	—
360	C₂×C₈	—	2	4	no	—
361	C₂×C₈	—	2	4	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	8	no	—
365	C₄⋊C₄	—	4	4	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	—	4	4	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
371	C₂²⋊C₄	—	2	8	no	—
372	C₂²⋊C₄	—	2	8	no	—
373	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
374	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
375	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
376	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
377	C₄²	C₂×D₄×C₆₇	1	2	yes	—
378	C₄²	C₂×D₄×C₆₇	1	2	yes	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	1	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
391	C₂³	C₂×D₈×C₆₇	1	2	yes	—
392	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
393	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	Q₈	—	4	8	no	—
395	D₄	—	2	8	no	—
396	D₄	—	4	16	no	—
397	D₄	—	4	4	no	—
398	D₄	—	4	8	no	—
399	D₄	—	4	8	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	16	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	—	8	8	no	—
415	C₂×C₄	#102×C₆₇	1	16	yes	complemented
416	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
417	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
418	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
419	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
420	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
421	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
422	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
423	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
424	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
425	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
426	C₂×C₄	#137×C₆₇	1	2	yes	—
427	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
430	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
431	C₈	—	4	4	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	C₂×#102×C₆₇	1	4	yes	—
446	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
447	C₂²	#535×C₆₇	1	1	yes	characteristic
448	C₂²	#536×C₆₇	1	1	yes	characteristic
449	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
450	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
451	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
452	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	4	4	no	—
456	C₄	—	8	4	no	—
457	C₄	C₂×#102×C₆₇	1	4	yes	—
458	C₄	C₂×#102×C₆₇	1	4	yes	—
459	C₄	#417×C₆₇	1	4	yes	—
460	C₄	#520×C₆₇	1	2	yes	—
461	C₄	#537×C₆₇	1	2	yes	—
462	C₂	—	2	4	no	—
463	C₂	—	8	4	no	—
464	C₂	C₆₇×#2311	1	1	yes	characteristic
465	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic
466	C₂	C₂×#417×C₆₇	1	1	yes	characteristic
467	C₂	C₆₇×#3205	1	1	yes	characteristic
468	C₂	C₆₇×#3206	1	1	yes	characteristic
469	C₂	C₆₇×#3226	1	1	yes	characteristic
470	C₂	C₆₇×#3227	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×#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. C₂×#431×C₆₇	C₂	semidirect
3. C₂×#384×C₆₇	C₂	semidirect
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1957	C₂	semidirect
6. C₆₇×#1874	C₂	semidirect
7. C₆₇×#1802	C₂	semidirect
10. #532×C₆₇	C₄	semidirect
47. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
49. #127×C₆₇	C₂×C₄	semidirect
56. C₄×D₄×C₆₇	C₂×C₄	semidirect
179. C₂×C₄×C₆₇	#102	semidirect
226. #21859	C₆₇	direct (N × H)
237. C₂×#431	C₂×C₆₇	semidirect
238. C₂×#384	C₂×C₆₇	semidirect
239. #1990	C₂×C₆₇	semidirect
240. #1957	C₂×C₆₇	semidirect
241. #1874	C₂×C₆₇	semidirect
242. #1802	C₂×C₆₇	semidirect
243. C₆₇	#21859	direct (N × H)
246. #532	C₄×C₆₇	semidirect
283. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
285. #127	C₂×C₄×C₆₇	semidirect
292. C₄×D₄	C₂×C₄×C₆₇	semidirect
415. C₂×C₄	#102×C₆₇	semidirect

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
C₂×#431×C₆₇	2
C₂×#384×C₆₇	2
C₆₇×#1990	2
C₆₇×#1957	2
C₆₇×#1874	2
C₆₇×#1802	2
#21859	67

Subgroups — C67×#21859 #30701110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21859 #30701110