Subgroups — C₆₇×#22688 #30701939

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₆₇	C₂	1	2	yes	maximal
2	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2906	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1945	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1866	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1806	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₇	C₂²	1	4	yes	—
11	C₄×#41×C₆₇	C₂²	1	2	yes	—
12	#443×C₆₇	—	2	8	no	—
13	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
14	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
15	C₄×SD₁₆×C₆₇	C₄	1	2	yes	complemented
16	C₄×SD₁₆×C₆₇	C₄	1	2	yes	complemented
17	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
18	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
19	C₂×#108×C₆₇	C₂²	1	2	yes	—
20	C₂×#108×C₆₇	C₂²	1	2	yes	—
21	#427×C₆₇	C₂²	1	1	yes	characteristic
22	#424×C₆₇	—	2	4	no	—
23	#424×C₆₇	C₂²	1	4	yes	—
24	#423×C₆₇	—	2	2	no	—
25	#423×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#104×C₆₇	C₂²	1	4	yes	—
29	#416×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	#339×C₆₇	C₂²	1	4	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	C₂×#41×C₆₇	C₂³	1	2	yes	—
36	#135×C₆₇	D₄	1	4	yes	—
37	#135×C₆₇	Q₈	1	4	yes	—
38	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	#131×C₆₇	—	2	8	no	—
42	#131×C₆₇	—	4	4	no	—
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₆₇	—	2	2	no	—
52	C₄×D₄×C₆₇	—	2	2	no	—
53	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
55	#117×C₆₇	—	2	2	no	—
56	#117×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#108×C₆₇	—	2	4	no	—
62	#108×C₆₇	—	4	2	no	—
63	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#106×C₆₇	C₂×C₄	1	2	yes	—
66	C₄⋊C₈×C₆₇	—	2	4	no	—
67	C₄⋊C₈×C₆₇	—	2	4	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
73	#104×C₆₇	—	2	8	no	—
74	#103×C₆₇	C₂×C₄	1	2	yes	—
75	#103×C₆₇	C₂×C₄	1	2	yes	—
76	#102×C₆₇	C₂×C₄	1	2	yes	—
77	#102×C₆₇	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	#95×C₆₇	—	2	1	no	—
84	#41×C₆₇	—	2	2	no	—
85	#41×C₆₇	—	2	2	no	—
86	#41×C₆₇	C₂×D₄	1	4	yes	—
87	#41×C₆₇	C₂×Q₈	1	4	yes	—
88	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
99	SD₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
100	D₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
101	M₁₆×C₆₇	—	2	4	no	—
102	M₁₆×C₆₇	—	2	4	no	—
103	M₁₆×C₆₇	—	4	8	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	4	8	no	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²⋊C₄×C₆₇	—	2	2	no	—
124	C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
137	Q₈×C₆₇	—	2	2	no	—
138	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
139	D₄×C₆₇	—	2	4	no	—
140	D₄×C₆₇	—	2	4	no	—
141	D₄×C₆₇	—	2	2	no	—
142	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	4	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	8	2	no	—
155	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	—	2	2	no	—
168	C₈×C₆₇	—	2	2	no	—
169	C₈×C₆₇	—	4	4	no	—
170	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
171	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
172	C₂²×C₆₇	—	2	1	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	4	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂²×C₆₇	#547	1	1	yes	characteristic
179	C₂²×C₆₇	#554	1	1	yes	characteristic
180	C₄×C₆₇	—	2	4	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	4	4	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	#443	1	2	yes	complemented
188	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₆₇	#520	1	1	yes	characteristic
190	C₄×C₆₇	D₄²	1	1	yes	characteristic
191	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
192	#22688	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	2	1	no	—
194	C₂×C₆₇	—	2	1	no	—
195	C₂×C₆₇	—	4	4	no	—
196	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
198	C₂×C₆₇	#2957	1	1	yes	characteristic
199	C₂×#443	C₂×C₆₇	1	2	yes	—
200	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
201	#2906	C₂×C₆₇	1	2	yes	complemented
202	#1945	C₂×C₆₇	1	2	yes	complemented
203	#1866	C₂×C₆₇	1	4	yes	complemented
204	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#1806	C₂×C₆₇	1	2	yes	complemented
206	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
207	C₆₇	#22688	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
209	C₂×#131	C₂²×C₆₇	1	4	yes	—
210	C₄×#41	C₂²×C₆₇	1	2	yes	—
211	#443	—	2	8	no	—
212	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
213	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
214	C₄×SD₁₆	C₄×C₆₇	1	2	yes	complemented
215	C₄×SD₁₆	C₄×C₆₇	1	2	yes	complemented
216	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
217	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
218	C₂×#108	C₂²×C₆₇	1	2	yes	—
219	C₂×#108	C₂²×C₆₇	1	2	yes	—
220	#427	C₂²×C₆₇	1	1	yes	characteristic
221	#424	—	2	4	no	—
222	#424	C₂²×C₆₇	1	4	yes	—
223	#423	—	2	2	no	—
224	#423	C₂²×C₆₇	1	2	yes	—
225	C₂×C₄⋊C₈	—	2	2	no	—
226	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#104	C₂²×C₆₇	1	4	yes	—
228	#416	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
230	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
231	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
232	#339	C₂²×C₆₇	1	4	yes	—
233	#336	C₂²×C₆₇	1	2	yes	—
234	C₂×#41	C₂³×C₆₇	1	2	yes	—
235	#135	D₄×C₆₇	1	4	yes	—
236	#135	Q₈×C₆₇	1	4	yes	—
237	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
238	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
239	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
240	#131	—	2	8	no	—
241	#131	—	4	4	no	—
242	C₂×M₁₆	—	2	4	no	—
243	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
244	C₂²×C₈	—	2	4	no	—
245	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
247	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
248	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
249	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
253	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
254	#117	—	2	2	no	—
255	#117	C₂³×C₆₇	1	2	yes	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄²	—	2	2	no	—
258	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
259	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
260	#108	—	2	4	no	—
261	#108	—	4	2	no	—
262	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
263	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
264	#106	C₂×C₄×C₆₇	1	2	yes	—
265	C₄⋊C₈	—	2	4	no	—
266	C₄⋊C₈	—	2	4	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
270	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
271	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
272	#104	—	2	8	no	—
273	#103	C₂×C₄×C₆₇	1	2	yes	—
274	#103	C₂×C₄×C₆₇	1	2	yes	—
275	#102	C₂×C₄×C₆₇	1	2	yes	—
276	#102	C₂×C₄×C₆₇	1	2	yes	—
277	C₄×C₈	—	2	4	no	—
278	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
279	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
280	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
281	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
282	#95	—	2	1	no	—
283	#41	—	2	2	no	—
284	#41	—	2	2	no	—
285	#41	C₂×D₄×C₆₇	1	4	yes	—
286	#41	C₂×Q₈×C₆₇	1	4	yes	—
287	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
297	Q₁₆	C₄⋊C₄×C₆₇	1	4	yes	complemented
298	SD₁₆	C₄⋊C₄×C₆₇	1	8	yes	complemented
299	D₈	C₄⋊C₄×C₆₇	1	4	yes	complemented
300	M₁₆	—	2	4	no	—
301	M₁₆	—	2	4	no	—
302	M₁₆	—	4	8	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	4	8	no	—
309	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
311	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
314	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
315	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
316	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
321	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
331	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
336	Q₈	—	2	2	no	—
337	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
338	D₄	—	2	4	no	—
339	D₄	—	2	4	no	—
340	D₄	—	2	2	no	—
341	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	8	2	no	—
354	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
361	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
362	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
363	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	2	no	—
367	C₈	—	2	2	no	—
368	C₈	—	4	4	no	—
369	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
370	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
377	C₂²	#547×C₆₇	1	1	yes	characteristic
378	C₂²	#554×C₆₇	1	1	yes	characteristic
379	C₄	—	2	4	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	4	4	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	#443×C₆₇	1	2	yes	complemented
387	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
388	C₄	#520×C₆₇	1	1	yes	characteristic
389	C₄	D₄²×C₆₇	1	1	yes	characteristic
390	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	4	4	no	—
394	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
395	C₂	C₂×#443×C₆₇	1	1	yes	characteristic
396	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄⋊C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2906	C₂	semidirect
4. C₆₇×#1945	C₂	semidirect
5. C₆₇×#1866	C₂	semidirect
6. C₆₇×#1848	C₂	semidirect
7. C₆₇×#1806	C₂	semidirect
8. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
13. C₄×Q₁₆×C₆₇	C₄	semidirect
14. C₄×Q₁₆×C₆₇	C₄	semidirect
15. C₄×SD₁₆×C₆₇	C₄	semidirect
16. C₄×SD₁₆×C₆₇	C₄	semidirect
17. C₄×D₈×C₆₇	C₄	semidirect
18. C₄×D₈×C₆₇	C₄	semidirect
98. Q₁₆×C₆₇	C₄⋊C₄	semidirect
99. SD₁₆×C₆₇	C₄⋊C₄	semidirect
100. D₈×C₆₇	C₄⋊C₄	semidirect
187. C₄×C₆₇	#443	semidirect
192. #22688	C₆₇	direct (N × H)
201. #2906	C₂×C₆₇	semidirect
202. #1945	C₂×C₆₇	semidirect
203. #1866	C₂×C₆₇	semidirect
204. #1848	C₂×C₆₇	semidirect
205. #1806	C₂×C₆₇	semidirect
206. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
207. C₆₇	#22688	direct (N × H)
212. C₄×Q₁₆	C₄×C₆₇	semidirect
213. C₄×Q₁₆	C₄×C₆₇	semidirect
214. C₄×SD₁₆	C₄×C₆₇	semidirect
215. C₄×SD₁₆	C₄×C₆₇	semidirect
216. C₄×D₈	C₄×C₆₇	semidirect
217. C₄×D₈	C₄×C₆₇	semidirect
297. Q₁₆	C₄⋊C₄×C₆₇	semidirect
298. SD₁₆	C₄⋊C₄×C₆₇	semidirect
299. D₈	C₄⋊C₄×C₆₇	semidirect
386. C₄	#443×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443×C₆₇	2
C₄×#135×C₆₇	2
C₆₇×#2906	2
C₆₇×#1945	2
C₆₇×#1866	2
C₆₇×#1848	2
C₆₇×#1806	2
C₈×C₄⋊C₄×C₆₇	2
#22688	67

Subgroups — C67×#22688 #30701939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22688 #30701939