Subgroups — C₆₇×#25058 #30704309

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2538	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2094	C₂	1	2	yes	maximal
4	C₆₇×#2084	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1926	C₂	1	2	yes	maximal
8	C₆₇×#1919	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#396×C₆₇	—	2	2	no	—
25	#390×C₆₇	C₂²	1	2	yes	—
26	#389×C₆₇	—	2	1	no	—
27	#389×C₆₇	—	2	2	no	—
28	#389×C₆₇	C₂²	1	2	yes	—
29	#384×C₆₇	C₂²	1	1	yes	characteristic
30	#384×C₆₇	C₂²	1	1	yes	characteristic
31	#384×C₆₇	C₂²	1	2	yes	—
32	#380×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	1	yes	characteristic
36	#340×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²⋊Q₈×C₆₇	—	4	4	no	—
48	C₄×Q₈×C₆₇	—	2	2	no	—
49	C₄×Q₈×C₆₇	—	2	2	no	—
50	C₄×Q₈×C₆₇	—	4	4	no	—
51	C₄×D₄×C₆₇	—	2	2	no	—
52	C₄×D₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#106×C₆₇	—	2	4	no	—
75	#106×C₆₇	—	4	2	no	—
76	#106×C₆₇	—	4	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	#102×C₆₇	—	2	2	no	—
79	#102×C₆₇	—	4	2	no	—
80	#102×C₆₇	Q₈	1	4	yes	complemented
81	C₄×C₈×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	1	no	—
84	#95×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	—	2	1	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	4	2	no	—
101	C₂²×C₄×C₆₇	—	4	2	no	—
102	C₂²×C₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₆₇	#41	1	2	yes	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	4	4	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	—	4	4	no	—
131	C₄⋊C₄×C₆₇	—	4	2	no	—
132	C₄⋊C₄×C₆₇	—	8	1	no	—
133	C₄⋊C₄×C₆₇	—	8	1	no	—
134	C₄⋊C₄×C₆₇	—	8	2	no	—
135	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₂²⋊C₄×C₆₇	—	4	4	no	—
138	C₂²⋊C₄×C₆₇	—	8	2	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	4	4	no	—
142	C₄²×C₆₇	—	4	2	no	—
143	C₄²×C₆₇	#41	1	2	yes	—
144	C₄²×C₆₇	#41	1	2	yes	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	4	1	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	4	1	no	—
150	C₂³×C₆₇	—	4	1	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₆₇	—	2	8	no	—
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	4	1	no	—
157	D₄×C₆₇	—	4	1	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	8	4	no	—
181	C₂×C₄×C₆₇	#123	1	2	yes	—
182	C₂×C₄×C₆₇	#123	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
184	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
185	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	2	2	no	—
194	C₈×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	8	1	no	—
199	C₂²×C₆₇	—	8	1	no	—
200	C₂²×C₆₇	—	8	1	no	—
201	C₂²×C₆₇	—	8	1	no	—
202	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
203	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
204	C₂²×C₆₇	#550	1	1	yes	characteristic
205	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
206	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂²×C₆₇	#577	1	1	yes	characteristic
208	C₂²×C₆₇	#577	1	1	yes	characteristic
209	C₄×C₆₇	—	2	4	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	8	1	no	—
214	C₄×C₆₇	—	8	2	no	—
215	C₄×C₆₇	—	8	1	no	—
216	C₄×C₆₇	—	8	2	no	—
217	C₄×C₆₇	#481	1	4	yes	—
218	C₄×C₆₇	#533	1	1	yes	characteristic
219	C₄×C₆₇	#542	1	1	yes	characteristic
220	C₄×C₆₇	#552	1	1	yes	characteristic
221	C₄×C₆₇	#554	1	1	yes	characteristic
222	#25058	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂×C₆₇	—	8	2	no	—
224	C₂×C₆₇	#2423	1	1	yes	characteristic
225	C₂×C₆₇	C₂×#481	1	1	yes	characteristic
226	C₂×C₆₇	#3099	1	1	yes	characteristic
227	C₂×C₆₇	Q₈×SD₁₆	1	1	yes	characteristic
228	C₂×C₆₇	#3384	1	1	yes	characteristic
229	C₂×C₆₇	#3392	1	1	yes	characteristic
230	C₂×C₆₇	#3397	1	1	yes	characteristic
231	#2538	C₂×C₆₇	1	1	yes	characteristic
232	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
233	#2094	C₂×C₆₇	1	2	yes	—
234	#2084	C₂×C₆₇	1	2	yes	complemented
235	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
237	#1926	C₂×C₆₇	1	2	yes	—
238	#1919	C₂×C₆₇	1	2	yes	complemented
239	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
240	#1802	C₂×C₆₇	1	1	yes	characteristic
241	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
242	C₆₇	#25058	1	1	yes	characteristic, direct product factor, Sylow
243	C₂×C₂²⋊Q₈	—	2	1	no	—
244	C₂×C₄×Q₈	—	2	1	no	—
245	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#106	C₂²×C₆₇	1	2	yes	—
248	C₂×#106	C₂²×C₆₇	1	2	yes	—
249	C₂×#106	C₂²×C₆₇	1	2	yes	—
250	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×#102	C₂²×C₆₇	1	2	yes	—
253	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
255	#396	—	2	2	no	—
256	#390	C₂²×C₆₇	1	2	yes	—
257	#389	—	2	1	no	—
258	#389	—	2	2	no	—
259	#389	C₂²×C₆₇	1	2	yes	—
260	#384	C₂²×C₆₇	1	1	yes	characteristic
261	#384	C₂²×C₆₇	1	2	yes	—
262	#384	C₂²×C₆₇	1	1	yes	characteristic
263	#380	—	2	1	no	—
264	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
265	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
266	#340	C₂²×C₆₇	1	2	yes	—
267	#340	C₂²×C₆₇	1	1	yes	characteristic
268	#340	C₂²×C₆₇	1	1	yes	characteristic
269	#340	C₂²×C₆₇	1	2	yes	—
270	#336	C₂²×C₆₇	1	2	yes	—
271	#336	C₂²×C₆₇	1	2	yes	—
272	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
273	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
274	C₂³×C₄	—	2	1	no	—
275	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
276	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²⋊Q₈	—	4	4	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×Q₈	—	4	4	no	—
282	C₄×D₄	—	2	2	no	—
283	C₄×D₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
292	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₄²	—	2	2	no	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
305	#106	—	2	4	no	—
306	#106	—	4	2	no	—
307	#106	—	4	2	no	—
308	C₄⋊C₈	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	4	2	no	—
311	#102	Q₈×C₆₇	1	4	yes	complemented
312	C₄×C₈	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	—	2	1	no	—
315	#95	—	2	2	no	—
316	C₂⁴	—	2	1	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	SD₁₆×C₆₇	1	4	yes	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	—	4	2	no	—
333	C₂²×C₄	—	4	2	no	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
347	C₂×C₈	#41×C₆₇	1	2	yes	—
348	C₄⋊C₄	—	2	4	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	4	4	no	—
360	C₄⋊C₄	—	4	2	no	—
361	C₄⋊C₄	—	4	4	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	—	8	1	no	—
364	C₄⋊C₄	—	8	1	no	—
365	C₄⋊C₄	—	8	2	no	—
366	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	4	4	no	—
369	C₂²⋊C₄	—	8	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	4	4	no	—
373	C₄²	—	4	2	no	—
374	C₄²	#41×C₆₇	1	2	yes	—
375	C₄²	#41×C₆₇	1	2	yes	—
376	C₂³	—	2	2	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	1	no	—
379	C₂³	—	4	1	no	—
380	C₂³	—	4	1	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	1	no	—
383	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
384	Q₈	—	2	8	no	—
385	D₄	—	2	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	4	1	no	—
388	D₄	—	4	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	8	4	no	—
412	C₂×C₄	#123×C₆₇	1	2	yes	—
413	C₂×C₄	#123×C₆₇	1	2	yes	—
414	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
415	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
416	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
417	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
418	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
421	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
422	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
423	C₈	—	2	2	no	—
424	C₈	—	2	2	no	—
425	C₈	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	8	1	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	8	1	no	—
433	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
434	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
435	C₂²	#550×C₆₇	1	1	yes	characteristic
436	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
437	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
438	C₂²	#577×C₆₇	1	1	yes	characteristic
439	C₂²	#577×C₆₇	1	1	yes	characteristic
440	C₄	—	2	4	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	8	1	no	—
445	C₄	—	8	2	no	—
446	C₄	—	8	1	no	—
447	C₄	—	8	2	no	—
448	C₄	#481×C₆₇	1	4	yes	—
449	C₄	#533×C₆₇	1	1	yes	characteristic
450	C₄	#542×C₆₇	1	1	yes	characteristic
451	C₄	#552×C₆₇	1	1	yes	characteristic
452	C₄	#554×C₆₇	1	1	yes	characteristic
453	C₂	—	8	2	no	—
454	C₂	C₆₇×#2423	1	1	yes	characteristic
455	C₂	C₂×#481×C₆₇	1	1	yes	characteristic
456	C₂	C₆₇×#3099	1	1	yes	characteristic
457	C₂	Q₈×SD₁₆×C₆₇	1	1	yes	characteristic
458	C₂	C₆₇×#3384	1	1	yes	characteristic
459	C₂	C₆₇×#3392	1	1	yes	characteristic
460	C₂	C₆₇×#3397	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#2084	C₂	semidirect
5. C₆₇×#1984	C₂	semidirect
6. C₆₇×#1948	C₂	semidirect
8. C₆₇×#1919	C₂	semidirect
9. C₆₇×#1844	C₂	semidirect
80. #102×C₆₇	Q₈	semidirect
222. #25058	C₆₇	direct (N × H)
232. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
234. #2084	C₂×C₆₇	semidirect
235. #1984	C₂×C₆₇	semidirect
236. #1948	C₂×C₆₇	semidirect
238. #1919	C₂×C₆₇	semidirect
239. #1844	C₂×C₆₇	semidirect
242. C₆₇	#25058	direct (N × H)
311. #102	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2538	2
Q₈×C₄⋊C₄×C₆₇	2
C₆₇×#2094	2
C₆₇×#2084	2
C₆₇×#1984	2
C₆₇×#1948	2
C₆₇×#1926	2
C₆₇×#1919	2
C₆₇×#1844	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
#25058	67

Subgroups — C67×#25058 #30704309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25058 #30704309