Subgroups — C₆₇×#22105 #30701356

Order 17152.

708 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2396	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1940	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1924	C₂	1	2	yes	maximal
8	C₆₇×#1924	C₂	1	2	yes	maximal
9	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	#446×C₆₇	—	2	4	no	—
15	#433×C₆₇	—	2	4	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	C₂×#97×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	C₄	1	1	yes	characteristic
30	#399×C₆₇	C₄	1	1	yes	characteristic
31	#398×C₆₇	C₄	1	2	yes	—
32	#396×C₆₇	C₄	1	2	yes	—
33	#395×C₆₇	C₄	1	1	yes	characteristic
34	#393×C₆₇	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₆₇	—	2	2	no	—
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	#117×C₆₇	—	2	2	no	—
54	#117×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	C₄⋊C₈×C₆₇	—	4	2	no	—
76	C₂²⋊C₈×C₆₇	—	2	2	no	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	—	4	2	no	—
79	#97×C₆₇	—	2	4	no	—
80	#97×C₆₇	—	2	2	no	—
81	#97×C₆₇	—	4	2	no	—
82	C₄×C₈×C₆₇	D₄	1	4	yes	—
83	#95×C₆₇	C₂×C₄	1	2	yes	—
84	#95×C₆₇	C₂×C₄	1	2	yes	—
85	#95×C₆₇	C₂×C₄	1	2	yes	—
86	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
87	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
88	#95×C₆₇	C₂×C₄	1	2	yes	—
89	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	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	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	4	2	no	—
131	C₄⋊C₄×C₆₇	—	4	2	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₂²⋊C₄×C₆₇	—	2	2	no	—
134	C₂²⋊C₄×C₆₇	—	2	2	no	—
135	C₂²⋊C₄×C₆₇	—	4	2	no	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₄²×C₆₇	—	2	1	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	1	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	C₂×D₄	1	2	yes	—
144	C₄²×C₆₇	#41	1	2	yes	—
145	C₂³×C₆₇	—	2	1	no	—
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	1	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	1	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	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
183	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
184	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
185	C₈×C₆₇	—	2	2	no	—
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	—	4	1	no	—
188	C₈×C₆₇	—	4	2	no	—
189	C₈×C₆₇	—	4	1	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	1	no	—
192	C₂²×C₆₇	—	4	1	no	—
193	C₂²×C₆₇	—	4	1	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₆₇	#519	1	1	yes	characteristic
198	C₂²×C₆₇	#520	1	1	yes	characteristic
199	C₂²×C₆₇	#568	1	2	yes	—
200	C₂²×C₆₇	#568	1	2	yes	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	4	1	no	—
204	C₄×C₆₇	—	4	1	no	—
205	C₄×C₆₇	—	4	2	no	—
206	C₄×C₆₇	—	4	1	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	#536	1	1	yes	characteristic
211	C₄×C₆₇	#537	1	1	yes	characteristic
212	C₄×C₆₇	#541	1	1	yes	characteristic
213	C₄×C₆₇	#542	1	1	yes	characteristic
214	#22105	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₆₇	—	8	1	no	—
216	C₂×C₆₇	#2323	1	1	yes	characteristic
217	C₂×C₆₇	#2930	1	1	yes	characteristic
218	C₂×C₆₇	#2932	1	1	yes	characteristic
219	C₂×C₆₇	#2987	1	2	yes	—
220	C₂×C₆₇	#2995	1	2	yes	—
221	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₂×#433	C₂×C₆₇	1	1	yes	characteristic
223	#2396	C₂×C₆₇	1	1	yes	characteristic
224	#1985	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1940	C₂×C₆₇	1	2	yes	complemented
226	#1940	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#1924	C₂×C₆₇	1	2	yes	—
228	#1924	C₂×C₆₇	1	2	yes	—
229	#1917	C₂×C₆₇	1	2	yes	complemented
230	#1798	C₂×C₆₇	1	1	yes	characteristic
231	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₆₇	#22105	1	1	yes	characteristic, direct product factor, Sylow
233	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
235	#446	—	2	4	no	—
236	#433	—	2	4	no	—
237	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
242	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
243	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
246	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#97	C₂²×C₆₇	1	2	yes	—
248	C₂×#97	C₂²×C₆₇	1	2	yes	—
249	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
250	#400	C₄×C₆₇	1	1	yes	characteristic
251	#399	C₄×C₆₇	1	1	yes	characteristic
252	#398	C₄×C₆₇	1	2	yes	—
253	#396	C₄×C₆₇	1	2	yes	—
254	#395	C₄×C₆₇	1	1	yes	characteristic
255	#393	C₄×C₆₇	1	1	yes	characteristic
256	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
257	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
258	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
259	#336	C₂²×C₆₇	1	2	yes	—
260	#336	C₂²×C₆₇	1	2	yes	—
261	#336	C₂²×C₆₇	1	2	yes	—
262	#336	C₂²×C₆₇	1	2	yes	—
263	#336	C₂²×C₆₇	1	2	yes	—
264	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
268	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	2	no	—
274	#117	—	2	2	no	—
275	#117	—	2	2	no	—
276	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
278	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
282	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
283	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
286	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
287	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
289	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
290	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	4	no	—
294	C₄⋊C₈	—	2	2	no	—
295	C₄⋊C₈	—	2	2	no	—
296	C₄⋊C₈	—	4	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	—	4	2	no	—
300	#97	—	2	4	no	—
301	#97	—	2	2	no	—
302	#97	—	4	2	no	—
303	C₄×C₈	D₄×C₆₇	1	4	yes	—
304	#95	C₂×C₄×C₆₇	1	2	yes	—
305	#95	C₂×C₄×C₆₇	1	2	yes	—
306	#95	C₂×C₄×C₆₇	1	2	yes	—
307	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
308	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
309	#95	C₂×C₄×C₆₇	1	2	yes	—
310	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	4	2	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
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	C₂×D₄×C₆₇	1	2	yes	—
346	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	4	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	2	2	no	—
355	C₂²⋊C₄	—	2	2	no	—
356	C₂²⋊C₄	—	4	2	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	2	no	—
364	C₄²	C₂×D₄×C₆₇	1	2	yes	—
365	C₄²	#41×C₆₇	1	2	yes	—
366	C₂³	—	2	1	no	—
367	C₂³	—	2	1	no	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	2	2	no	—
371	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
372	Q₈	—	4	4	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
399	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
400	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
402	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
403	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
404	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
405	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	1	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	2	no	—
411	C₈	—	4	2	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
418	C₂²	#519×C₆₇	1	1	yes	characteristic
419	C₂²	#520×C₆₇	1	1	yes	characteristic
420	C₂²	#568×C₆₇	1	2	yes	—
421	C₂²	#568×C₆₇	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	2	no	—
426	C₄	—	4	1	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	#536×C₆₇	1	1	yes	characteristic
432	C₄	#537×C₆₇	1	1	yes	characteristic
433	C₄	#541×C₆₇	1	1	yes	characteristic
434	C₄	#542×C₆₇	1	1	yes	characteristic
435	C₂	—	8	1	no	—
436	C₂	C₆₇×#2323	1	1	yes	characteristic
437	C₂	C₆₇×#2930	1	1	yes	characteristic
438	C₂	C₆₇×#2932	1	1	yes	characteristic
439	C₂	C₆₇×#2987	1	2	yes	—
440	C₂	C₆₇×#2995	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#446×C₆₇	C₂	semidirect
4. C₆₇×#1985	C₂	semidirect
5. C₆₇×#1940	C₂	semidirect
6. C₆₇×#1940	C₂	semidirect
9. C₆₇×#1917	C₂	semidirect
11. C₆₇×#1770	C₂	semidirect
214. #22105	C₆₇	direct (N × H)
221. C₂×#446	C₂×C₆₇	semidirect
224. #1985	C₂×C₆₇	semidirect
225. #1940	C₂×C₆₇	semidirect
226. #1940	C₂×C₆₇	semidirect
229. #1917	C₂×C₆₇	semidirect
231. #1770	C₂×C₆₇	semidirect
232. C₆₇	#22105	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×#446×C₆₇	2
C₂×#433×C₆₇	2
C₆₇×#2396	2
C₆₇×#1985	2
C₆₇×#1940	2
C₆₇×#1924	2
C₆₇×#1917	2
C₆₇×#1798	2
C₆₇×#1770	2
#22105	67

Subgroups — C67×#22105 #30701356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22105 #30701356